Research Article Output Feedback and Single-Phase Sliding...
17
Research Article Output Feedback and Single-Phase Sliding Mode Control for Complex Interconnected Systems Yao-Wen Tsai 1 and Van Van Huynh 2 1 Department of Mechanical and Automation Engineering, Da-Yeh University, No. 168, University Road, Changhua 51591, Taiwan 2 Faculty of Electrical and Electronics Engineering, Ton Duc ang University, 19 Nguyen Huu o Street, 7th District, Ho Chi Minh City, Vietnam Correspondence should be addressed to Yao-Wen Tsai; [email protected] Received 29 August 2014; Accepted 25 December 2014 Academic Editor: Xudong Zhao Copyright © 2015 Y.-W. Tsai and V. V. Huynh. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper generalized a new sliding mode control (SMC) without reaching phase to solve two important problems in the stability of complex interconnected systems: (1) a decentralized controller that uses only output variables directly and (2) the stability of complex interconnected systems ensured for all time. A new sliding surface is firstly designed to construct a single-phase SMC in which the desired motion is determined from the initial time instant. A new lemma is secondly established for the controller design using only output variables. e proposed single-phase SMC and the decentralized output feedback controller ensure the robust stability of complex interconnected systems from the beginning to the end. One of the key features of the single phase SMC scheme is that reaching time, which is required in most of the existing two phases of SMC approaches to stabilize the interconnected systems, is removed. Finally, a numerical example is used to demonstrate the efficacy of the method. 1. Introduction e theory of sliding mode control (SMC) is known to be an effective robust control technique and has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircraſts, underwater vehicles, spacecraſts, flexible space structures, electrical motors, power systems, and automotive engines [1]. e main advantages of SMC are fast response and strong robustness with respect to uncertainties and external disturbances [2–4]. Generally speaking, the traditional SMC design can be divided into two phases: the reaching phase and the sliding phase. Firstly, in the reaching phase, the feature of SMC is to use a switching control law to drive system state trajectories onto a switching surface and remain on it thereaſter. Secondly, in the sliding phase, the essence of SMC is to keep the state trajectories moving along the surface towards the origin with desired performance [5, 6]. Unfortunately, the applications of two phases SMC for the stability of complex interconnected system have some drawbacks. Firstly, the system stability is not ensured for all time because the motion equation in sliding mode is determined aſter the system state hits the sliding surface [6, 7]. Secondly, the performance of system in the reaching phase is unknown and, subsequently, global performance may be seriously degraded [6, 7]. In addition, the state variables of complex interconnected system are not always accessible in many practical systems. erefore, for complex intercon- nected systems, there are some important tasks should be solved: (1) the creation of a decentralized controller that uses only output variables directly; (2) guaranteed stability of complex interconnected systems for all time. In order to solve the above problems, first we develop a new SMC such that the reaching time is equal to zero and the desired motion is determined from the beginning time. Second, appropriate LMI stability conditions by the Lyapunov method are derived to guarantee the stability of the system. ird, a new lemma is established for controller design using only output variables directly. Consequently, the stability of complex interconnected systems driven by single- phase SMC law can be ensured throughout an entire response of the system starting from the initial time instance. Before Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 946385, 16 pages http://dx.doi.org/10.1155/2015/946385
Transcript of Research Article Output Feedback and Single-Phase Sliding...
Research ArticleOutput Feedback and Single-Phase Sliding Mode Control forComplex Interconnected Systems
Yao-Wen Tsai1 and Van Van Huynh2
1Department of Mechanical and Automation Engineering Da-Yeh University No 168 University Road Changhua 51591 Taiwan2Faculty of Electrical and Electronics Engineering Ton Duc Thang University 19 Nguyen HuuTho Street 7th DistrictHo Chi Minh City Vietnam
Correspondence should be addressed to Yao-Wen Tsai ywtsaitwgmailcom
Received 29 August 2014 Accepted 25 December 2014
Academic Editor Xudong Zhao
Copyright copy 2015 Y-W Tsai and V V Huynh This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper generalized a new sliding mode control (SMC) without reaching phase to solve two important problems in the stabilityof complex interconnected systems (1) a decentralized controller that uses only output variables directly and (2) the stability ofcomplex interconnected systems ensured for all time A new sliding surface is firstly designed to construct a single-phase SMCin which the desired motion is determined from the initial time instant A new lemma is secondly established for the controllerdesign using only output variables The proposed single-phase SMC and the decentralized output feedback controller ensure therobust stability of complex interconnected systems from the beginning to the end One of the key features of the single phase SMCscheme is that reaching time which is required inmost of the existing two phases of SMC approaches to stabilize the interconnectedsystems is removed Finally a numerical example is used to demonstrate the efficacy of the method
1 Introduction
The theory of sliding mode control (SMC) is known to be aneffective robust control technique and has been successfullyapplied to a wide variety of practical engineering systemssuch as robot manipulators aircrafts underwater vehiclesspacecrafts flexible space structures electrical motors powersystems and automotive engines [1] The main advantagesof SMC are fast response and strong robustness with respectto uncertainties and external disturbances [2ndash4] Generallyspeaking the traditional SMC design can be divided into twophases the reaching phase and the sliding phase Firstly inthe reaching phase the feature of SMC is to use a switchingcontrol law to drive system state trajectories onto a switchingsurface and remain on it thereafter Secondly in the slidingphase the essence of SMC is to keep the state trajectoriesmoving along the surface towards the origin with desiredperformance [5 6]
Unfortunately the applications of two phases SMC forthe stability of complex interconnected system have somedrawbacks Firstly the system stability is not ensured for
all time because the motion equation in sliding mode isdetermined after the system state hits the sliding surface [67] Secondly the performance of system in the reaching phaseis unknown and subsequently global performance may beseriously degraded [6 7] In addition the state variablesof complex interconnected system are not always accessiblein many practical systems Therefore for complex intercon-nected systems there are some important tasks should besolved (1) the creation of a decentralized controller thatuses only output variables directly (2) guaranteed stability ofcomplex interconnected systems for all time
In order to solve the above problems first we developa new SMC such that the reaching time is equal to zeroand the desired motion is determined from the beginningtime Second appropriate LMI stability conditions by theLyapunov method are derived to guarantee the stability ofthe system Third a new lemma is established for controllerdesign using only output variables directly Consequently thestability of complex interconnected systems driven by single-phase SMC law can be ensured throughout an entire responseof the system starting from the initial time instance Before
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 946385 16 pageshttpdxdoiorg1011552015946385
2 Mathematical Problems in Engineering
demonstrating the advantages of the application of single-phase SMC to complex interconnected systems one wants topoint out some previous results about the stability analysis ofuncertain systems
The design of SMC without reaching phase can be foundin [1 7ndash12]The authors of [1 8] have presented a newmethodto design an integral sliding mode control law This nicefeature of the integral SMC law compensates the generallyslower and more oscillatory transient [1] In order to reducedisturbance Rubagotti et al [9] developed an integral slidingmode controller with state-dependent drift and input matrixMore recently the researchers in [10] proposed a universalfuzzy integral SMC formismatched uncertain systems whichdoes not require that all local linear systems share a commoninput matrix The authors of [11] developed an integral SMCfor handling a larger class of mismatched uncertainties In[12] a new approach was proposed for approximating thesystem states and disturbance vectors using observer-basedintegral SMC In addition the stability of the sliding mode interms of linear matrix inequalities (LMI) has some benefitsover conventional approach methods where LMI problemscan be easily determined and efficiently solved using the LMIToolbox in MATLAB software As a result the robustnessof the integral SMC via the LMI technique is guaranteedthroughout its entire trajectories starting from the initialtime
Thus the approaches in [1 7ndash12] cannot be directlyapplied to complex interconnected systems in which onlyoutput information is available In the limited available lit-erature the associated decentralized output feedback resultsare few In particular when the mismatched uncertaintiesare included only a few results are available [13ndash19] Earlierworks on decentralized SMC were mainly focused on inter-connected systems or nonlinear systems with the matchingcondition [20ndash23] A decentralized model reference adaptivecontrol scheme is proposed in [24] in which the inter-connections considered are linear and matched In [25]sufficient stability conditions were derived for the switchedinterconnected time-delayed systems The authors in [13]proposed a decentralized slidingmode controller for a class ofmismatched uncertain interconnected systems by using twosets of switching surfaces where the exogenous disturbancewas not mentioned In [14 15] a decentralized SMC schemewas proposed for a class of interconnected time-delayedsystems with dead-zone input In [16] a multiple-slidingsurface control scheme is presented for a class of multi-input perturbed systems In [17] a decentralized dynamicoutput feedback sliding mode controller is designed for mis-matched uncertain interconnected systems In [18] a globaldecentralised static output feedback SMC control schemeis proposed for interconnected time-delayed systems wherethe interconnection terms are functions of the output In[19] a state observer-based sliding mode control is designedfor a class of switched systems in which the system statesare unmeasurable The above works obtained importantresults related to handling the effects of interconnections anddisturbances of interconnected systems using SMC theory Asa result the stability of interconnected systems was assuredunder certain conditions
However it is worth to point out that there are somelimitations in the existing design methods of SMC in appli-cation for the stability of interconnected systems First theapproaches proposed in [20ndash25] could not be applied formismatched uncertain interconnected systems Second thecontrol schemes given in [13ndash24] are based on the traditionalSMCmethod which only yields the desired motion after slid-ing motion has occurred Therefore the global performancemay be seriously degraded Hence it is necessary to developa new SMC without reaching phase to stabilize complexinterconnected systems for all time
This study therefore developed a new single-phase SMCfor robust stability of a class of complex interconnectedsystems from beginning to end First a new sliding surfaceis designed to construct the single-phase SMC which thedesired motion is determined from the initial time instantSecond appropriate LMI stability conditions by the Lyapunovmethod are derived to guarantee the stability of the systemThird a new lemma is established for controller designusing only output variables Fourth a decentralized outputfeedback controller is designed to force the system states tostay on the sliding surface for all time Unlike the existingrelated works such as [1 7ndash12] this method can be directlyapplied for complex interconnected systems in which onlyoutput information is available In contrast to the other SMCapproaches given in [13ndash24] this approach guarantees thestability of complex interconnected systems for all time Inaddition the complex interconnected systems investigatedin this study include exogenous disturbance mismatchedparameter uncertainties in the state matrix and mismatchedinterconnections Therefore we consider a more generalstructure than [13ndash25] To summarize themain contributionsof this paper are as follows
(i) Design of a new sliding surface to construct a single-phase SMC such that the desired motion is deter-mined from the initial time instant
(ii) Derivation of appropriate LMI stability conditions bythe Lyapunov method to guarantee the stability of thesystem
(iii) Establishment of a lemma for controller design usingonly output variables
(iv) Development of a new approach (single-phase SMCand decentralized output feedback controller) guar-antees that sliding mode exists from the initial timeinstant and the closed loop of the complex inter-connected systems in sliding mode is asymptoticallystable
Notation The notation used throughout this paper is fairlystandard 119883119879 denotes the transpose of matrix 119883 119868
119899times119898and
0119899times119898
are used to denote the 119899 times 119898 identity matrix andthe 119899 times 119898 zero matrix respectively The subscripts 119899 and119899 times 119898 are omitted where the dimension is irrelevant orcan be determined from the context 119909 stands for theEuclidean norm of vector 119909 and 119860 stands for the matrixinduced norm of the matrix 119860 The expression 119860 gt 0
means that 119860 is symmetric positive definite 119877119899 denotes the
Mathematical Problems in Engineering 3
119899-dimensional Euclidean space For the sake of simplicitysometimes function 119909
119894(119905) is denoted by 119909
119894
2 Problem Formulation and Preliminaries
In this paper we consider a class of complex interconnectedsystems with exogenous disturbance and mismatched uncer-tainties of each isolated subsystem and interconnection Thesystem is decomposed into 119871 subsystems and the state spacerepresentation of each subsystem is described as follows
119894= (119860119894+ Δ119860119894) 119909119894+ 119861119894(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
(119867119894119895+ Δ119867119894119895) 119909119895
119910119894= 119862119894119909119894
(1)
where 119909119894isin 119877119899119894 119906119894isin 119877119898119894 and 119910
119894isin 119877119901119894 with 119898
119894lt 119901119894lt 119899119894are
the state variables inputs and outputs of the 119894th subsystemrespectively The triples (119860
119894 119861119894 119862119894) and119867
119894119895represent known
constant matrices of appropriate dimensions The matricesΔ119860119894119909119894and Δ119867
119894119895119909119895represent the mismatched parameter
uncertainty in the state matrix in each isolated subsystemsand mismatched interconnections respectively The matrix119861119894120585119894(119909119894 119905) is disturbance input In this paper only the output
variables 119910119894are assumed to be known
In order to modify the existing two phases SMC wedenote the sliding surface by 120590
119894(119909119894(119905) 119905) = 0 119894 = 1 2 119871
where the single-phase sliding function is given as
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (0) 0) exp (minus120573119894119905) (2)
with constant 120573119894gt 0 The function 120590
119894(119910119894(119905) 119905) is defined
later The sliding mode is defined by 120590119894(119909119894(119905) 119905) = 0 and
119894(119909119894(119905) 119905) = 0 From (2) one can see that there are only
output variables used and the system states are in the slidingmode from the initial time 120590
119894(119909119894(0) 0) = 0 Therefore this is
to say that the SMC is single-phase (without reaching phase)This can be formally defined as follows
Definition 1 A sliding mode control is said to be a single-phase SMC if and only if the following two conditions aresatisfied
(1) the reaching time is equal to zero 120590119894(119909119894(0) 0) = 0
(2) the order of the motion equation in sliding mode isequal to the order of the original system
Remark 2 The concept of single-phase sliding mode controlfocusses on the robustness of the motion in the entire statespace The order of the motion equation in sliding modeis equal to the dimension of the state space Therefore therobustness of complex interconnected systems can be assuredthroughout an entire response of the system starting from theinitial time instance
In order to apply the concept of single-phase SMC for thesystem (1) we assume the following to be valid
Assumption 3 The mismatched parameter uncertainties inthe state matrix of each isolated subsystem are satisfied asΔ119860119894= 119863119894119865119894(119909119894 119905)119864119894where 119865
119894(119909119894 119905) is unknown but bounded
as 119865119894(119909119894 119905) le 1 and119863
119894119864119894are knownmatrices of appropriate
dimensions
Assumption 4 The matrices 119861119894and 119862
119894are full rank and
rank(119862119894119861119894) = 119898
119894
From [18] Assumption 4 implies that there exists anonsingular linear coordinate transformation
119894= 119894119909119894such
that the triple (119860119894 119861119894 119862119894)with respect to the new coordinates
has the structure
119860119894= [
11986011989411198601198942
11986011989431198601198944
] 119861119894= [
0
1198611198942
] 119862119894= [0 119862
1198942] (3)
where1198601198941isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) 119861
1198942isin 119877119898119894times119898119894 are nonsingular and
1198621198942isin 119877119901119894times119901119894 is orthogonal
Assumption 5 The triple (1198601198941 1198601198942 Ξ119894) is output feedback
stabilisable where Ξ119894
= [0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)
] 119894 =
1 2 3 119871From [18] Assumption 5 implies that there exist matrices
119894such that the matrices 119860
1198941= 1198601198941minus 1198601198942119894Ξ119894are stable
Assumption 6 There exist known nonnegative constants 119888119894
and 119887119894such that 120585
119894(119909119894 119905) le 119888
119894+ 119887119894119909119894(119905)
Assumption 7 The mismatched interconnections are givenas Δ119867
119894119895= 119872119894119895119865119894119895(119909119895 119905)119873119894119895 where 119865
119894119895(119909119895 119905) is unknown but
bounded 119865119894119895(119909119895 119905) le 1 and119872
119894119895 119873119894119895are known matrices of
appropriate dimensions
Remark 8 Assumptions 4 and 5 have been utilized in [18]The assumption of the norm boundedness of 119865
119894(119909119894 119905) and
120585119894(119909119894 119905) can be found in [19 26 27]
3 Single-Phase Sliding Mode Control forComplex Interconnected Systems
In this section we develop a single-phase SMC to stabilizethe complex interconnected system (1) for all time There arefour steps involved in the design of our single-phase SMCusing only output variables In the first step a proper slidingsurface is designed to construct the single-phase SMC suchthat the desired motion is determined from the initial timeinstant In the second step sufficient conditions in termsof LMI are derived for the existence of a sliding surfaceguaranteeing asymptotic stability In the third step a newlemma is established for controller design using only outputvariables In the fourth step a decentralized output feedbackcontroller is designed to force the system states to stay on thesliding surface for all time
31 Single-Phase Sliding Surface Design Let us first designa new sliding surface without reaching phase that uses onlyoutput variables and the desired motion is determined fromthe initial time instant Under Assumptions 4 and 5 it followsfrom (11) (12) and (13) of paper [18] that there exists a
4 Mathematical Problems in Engineering
coordinate transformation 119911119894= 119879119894119909119894such that the system (1)
has the following regular form
119894= ([
11986011989411198601198942
11986011989431198601198944
] + [1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]) 119911119894
+ [0
1198611198942
] (119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
([1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] + [1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]) 119911119895
(4)
119910119894= [0 119862
1198942] 119911119894 (5)
where
119879119894119860119894119879minus1
119894= [
11986011989411198601198942
11986011989431198601198944
]
119879119894119863119894119865119894119864119894119879minus1
119894= [
1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]
119879119894119867119894119895119879minus1
119895= [
1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] 119879119894119861119894= [
0
1198611198942
]
119879119894119872119894119895119865119894119895119873119894119895119879minus1
119895= [
1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]
119862119894119879119894
minus1= [0 119862
1198942]
(6)
The matrices 1198611198942isin 119877119898119894times119898119894 and 119862
1198942isin 119877119901119894times119901119894 are nonsingular
and 1198601198941= 1198601198941minus 1198601198942119894Ξ119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is stable Then
by using the sliding function (2) the sliding surface can bedefined as follows
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (119905) 0) exp (minus120573119894119905) = 0
(7)
where the solution of 120590119894(119910119894 119905) is given by
120590119894(119910119894 119905) = 119870
119894119862minus1
1198942119910119894
= 119870119894[
119873119894
0(119901119894minus119898119894)times119898119894
0119898119894times(119899119894minus119898119894)
119868119898119894times119898119894
] [1199111198941
1199111198942
] = 11987011989421199111198942
(8)
in which 119911119894= [11991111989411199111198942] 1199111198941
isin 119877119899119894minus119898119894 119911
1198942isin 119877119898119894 119873
119894=
[0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)times(119901119894minus119898119894)
] and 119870119894= [0119898119894times(119901119894minus119898119894)
1198701198942]
The matrix1198701198942isin 119877119898119894times119898119894 is given as
1198701198942= Π119894119875119894Π119879
119894 (9)
where the matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined later and
the matrix Π119894isin 119877119898119894times(119899119894minus119898119894) is selected such that 119870
1198942is
nonsingular Using (8) sliding surface (7) can be rewrittenas
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 119905) minus 120590119894 (119910119894 0) exp (minus120573119894119905)
= 11987011989421199111198942 (119905) minus 11987011989421199111198942 (0) exp (minus120573119894119905) = 0
(10)
Since 1198701198942isin 119877119898119894times119898119894 is nonsingular sliding surface (7) can be
described by
col (1199111 1199112 119911
119871) | 1199111198942= 1199111198942 (0) exp (minus120573119894119905)
119894 = 1 2 119871
(11)
From (11) it is clear that in sliding mode
1198942= minus1205731198941199111198942 (12)
Then from the structure of system (4)-(5) and (12) the slidingmode dynamics of system (1) associated with sliding surface(7) are described by
119894= [
1198601198941
1198601198942
0 minus120573119894119868119898119894times119898119894
] 119911119894+
119871
sum
119895=1
119895 =119894
[1198671198941198951
1198671198941198952
0 0] 119911119895 (13)
where 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+ 11986311989411198651198941198641198942 1198671198941198951
=
1198671198941198951+11987211989411989511198651198941198951198731198941198951 and119867
1198941198952= 1198671198941198952+11987211989411989511198651198941198951198731198941198952
Remark 9 It is obvious that 120590119894(119909119894(0) 0) = 0 which means
that the reaching time is equal to zero and the sliding modeexists from the initial time instant In other words the desiredmotion is determined from the beginning of the time
Remark 10 This approach concentrates on the robustness ofthe motion in the entire state space The order of the motionequation in sliding mode is equal to the order of the originalsystemTherefore the robustness of the system can be assuredthroughout an entire response of the system starting from theinitial time instance
32 Single-Phase Sliding Mode Stability Analysis Followingdesign of the sliding surface two tasks remain First forstability analysis appropriate LMI stability conditions by theLyapunov method must be derived to ensure the stability ofslidingmotion (13) Second we design a decentralized outputfeedback sliding mode controller to keep the system states tostay on the sliding surface for all time
This section focuses on the former task We begin byconsidering the following LMI
[[[[[
[
Ψ119894
1198751198941198631198941
119864119879
1198941
119863119879
1198941119875119894minus120592minus1
119894119868 0
1198641198941
0 minus120593minus1
119894119868
]]]]]
]
lt 0 (14)
whereΨ119894= 119860119879
1198941119875119894+ 1198751198941198601198941+ 120576119894119875119894119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
(15)
119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is any positive matrix and 120593
119894 120576119894 120592119894
120593119895 120593119895 120588119894are positive constants
We also recall the following lemmas which will be usedin proving the stability of sliding motion (13)
Mathematical Problems in Engineering 5
Lemma 11 (see [26]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 and then for any scalar120593 gt 0 the following matrix inequality holds
119883119865119884 + 119884119879119865119879119883119879le 120593minus1119883119883119879+ 120593119884119879119884 (16)
Lemma 12 (see [27]) Let119883 and 119884 be real matrices of suitabledimension and then for any scalar 120583 gt 0 the following matrixinequality holds
119883119879119884 + 119884
119879119883 le 120583119883
119879119883 + 120583
minus1119884119879119884 (17)
Lemma 13 (see [28]) The linear matrix inequality
[Θ (119909) Γ (119909)
Γ (119909)119879119877 (119909)
] gt 0 (18)
where Θ(119909) = Θ(119909)119879 119877(119909) = 119877(119909)119879 and Γ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0 Θ(119909) minus Γ(119909)119877(119909)minus1Γ(119909)119879 gt 0
Then we can establish the following theorem
Theorem 14 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 And the
sliding surface is given by (7) Then the sliding motion (13) isasymptotically stable
Proof of Theorem 14 Let us consider the following positivedefinition function
119881 =
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894 (19)
where the positive constant 120574119894will be selected later the
positive matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in LMI (14)
and 119876119894isin 119877119898119894times119898119894 is any positive matrix Then taking the time
derivative of 119881 along the state trajectory of system (13) wecan obtain that
=
119871
sum
119894=1
119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894+
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119894 (20)
Hence substituting (13) into (20) we derive
=
119871
sum
119894=1
119911119879
119894
[[
[
119860119879
1198941119875119894+ 1198751198941198601198941
1198751198941198601198942
119860119879
1198942119875119894
minus2120573119894120574119894119876119894
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119895
[[
[
119867119879
11989411989511198751198940
119867119879
11989411989521198751198940
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119894[
[
1198751198941198671198941198951
1198751198941198671198941198952
0 0
]
]
119911119895
(21)
Since 119911119894= [1199111198941 1199111198942]
119879 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+
11986311989411198651198941198641198942 1198671198941198951
= 1198671198941198951
+ 11987211989411989511198651198941198951198731198941198951 and 119867
1198941198952= 1198671198941198952
+
11987211989411989511198651198941198951198731198941198952 the above equation can be rewritten as
=
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 119864119879
1198941119865119879
119894119863119879
1198941119875119894
+ 1198751198941198601198941+ 11987511989411986311989411198651198941198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 119911119879
1198942119864119879
1198942119865119879
119894
sdot 119863119879
11989411198751198941199111198941+ 119911119879
1198941119875119894119863119894111986511989411986411989421199111198942minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
1198951119873119879
1198941198951119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989511199111198951+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989511199111198951)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
1198952119873119879
1198941198952119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989521199111198952+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989521199111198952)
(22)
Using Lemma 11 and (22) we achieve
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941
+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
119894111987511989411986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
119894111987511989411986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(23)
where 120593119894gt 0 120593
119894gt 0 120593
119894gt 0 and 120575
119894gt 0 are scalars Applying
Lemma 12 to (23) we can obtain
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
demonstrating the advantages of the application of single-phase SMC to complex interconnected systems one wants topoint out some previous results about the stability analysis ofuncertain systems
The design of SMC without reaching phase can be foundin [1 7ndash12]The authors of [1 8] have presented a newmethodto design an integral sliding mode control law This nicefeature of the integral SMC law compensates the generallyslower and more oscillatory transient [1] In order to reducedisturbance Rubagotti et al [9] developed an integral slidingmode controller with state-dependent drift and input matrixMore recently the researchers in [10] proposed a universalfuzzy integral SMC formismatched uncertain systems whichdoes not require that all local linear systems share a commoninput matrix The authors of [11] developed an integral SMCfor handling a larger class of mismatched uncertainties In[12] a new approach was proposed for approximating thesystem states and disturbance vectors using observer-basedintegral SMC In addition the stability of the sliding mode interms of linear matrix inequalities (LMI) has some benefitsover conventional approach methods where LMI problemscan be easily determined and efficiently solved using the LMIToolbox in MATLAB software As a result the robustnessof the integral SMC via the LMI technique is guaranteedthroughout its entire trajectories starting from the initialtime
Thus the approaches in [1 7ndash12] cannot be directlyapplied to complex interconnected systems in which onlyoutput information is available In the limited available lit-erature the associated decentralized output feedback resultsare few In particular when the mismatched uncertaintiesare included only a few results are available [13ndash19] Earlierworks on decentralized SMC were mainly focused on inter-connected systems or nonlinear systems with the matchingcondition [20ndash23] A decentralized model reference adaptivecontrol scheme is proposed in [24] in which the inter-connections considered are linear and matched In [25]sufficient stability conditions were derived for the switchedinterconnected time-delayed systems The authors in [13]proposed a decentralized slidingmode controller for a class ofmismatched uncertain interconnected systems by using twosets of switching surfaces where the exogenous disturbancewas not mentioned In [14 15] a decentralized SMC schemewas proposed for a class of interconnected time-delayedsystems with dead-zone input In [16] a multiple-slidingsurface control scheme is presented for a class of multi-input perturbed systems In [17] a decentralized dynamicoutput feedback sliding mode controller is designed for mis-matched uncertain interconnected systems In [18] a globaldecentralised static output feedback SMC control schemeis proposed for interconnected time-delayed systems wherethe interconnection terms are functions of the output In[19] a state observer-based sliding mode control is designedfor a class of switched systems in which the system statesare unmeasurable The above works obtained importantresults related to handling the effects of interconnections anddisturbances of interconnected systems using SMC theory Asa result the stability of interconnected systems was assuredunder certain conditions
However it is worth to point out that there are somelimitations in the existing design methods of SMC in appli-cation for the stability of interconnected systems First theapproaches proposed in [20ndash25] could not be applied formismatched uncertain interconnected systems Second thecontrol schemes given in [13ndash24] are based on the traditionalSMCmethod which only yields the desired motion after slid-ing motion has occurred Therefore the global performancemay be seriously degraded Hence it is necessary to developa new SMC without reaching phase to stabilize complexinterconnected systems for all time
This study therefore developed a new single-phase SMCfor robust stability of a class of complex interconnectedsystems from beginning to end First a new sliding surfaceis designed to construct the single-phase SMC which thedesired motion is determined from the initial time instantSecond appropriate LMI stability conditions by the Lyapunovmethod are derived to guarantee the stability of the systemThird a new lemma is established for controller designusing only output variables Fourth a decentralized outputfeedback controller is designed to force the system states tostay on the sliding surface for all time Unlike the existingrelated works such as [1 7ndash12] this method can be directlyapplied for complex interconnected systems in which onlyoutput information is available In contrast to the other SMCapproaches given in [13ndash24] this approach guarantees thestability of complex interconnected systems for all time Inaddition the complex interconnected systems investigatedin this study include exogenous disturbance mismatchedparameter uncertainties in the state matrix and mismatchedinterconnections Therefore we consider a more generalstructure than [13ndash25] To summarize themain contributionsof this paper are as follows
(i) Design of a new sliding surface to construct a single-phase SMC such that the desired motion is deter-mined from the initial time instant
(ii) Derivation of appropriate LMI stability conditions bythe Lyapunov method to guarantee the stability of thesystem
(iii) Establishment of a lemma for controller design usingonly output variables
(iv) Development of a new approach (single-phase SMCand decentralized output feedback controller) guar-antees that sliding mode exists from the initial timeinstant and the closed loop of the complex inter-connected systems in sliding mode is asymptoticallystable
Notation The notation used throughout this paper is fairlystandard 119883119879 denotes the transpose of matrix 119883 119868
119899times119898and
0119899times119898
are used to denote the 119899 times 119898 identity matrix andthe 119899 times 119898 zero matrix respectively The subscripts 119899 and119899 times 119898 are omitted where the dimension is irrelevant orcan be determined from the context 119909 stands for theEuclidean norm of vector 119909 and 119860 stands for the matrixinduced norm of the matrix 119860 The expression 119860 gt 0
means that 119860 is symmetric positive definite 119877119899 denotes the
Mathematical Problems in Engineering 3
119899-dimensional Euclidean space For the sake of simplicitysometimes function 119909
119894(119905) is denoted by 119909
119894
2 Problem Formulation and Preliminaries
In this paper we consider a class of complex interconnectedsystems with exogenous disturbance and mismatched uncer-tainties of each isolated subsystem and interconnection Thesystem is decomposed into 119871 subsystems and the state spacerepresentation of each subsystem is described as follows
119894= (119860119894+ Δ119860119894) 119909119894+ 119861119894(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
(119867119894119895+ Δ119867119894119895) 119909119895
119910119894= 119862119894119909119894
(1)
where 119909119894isin 119877119899119894 119906119894isin 119877119898119894 and 119910
119894isin 119877119901119894 with 119898
119894lt 119901119894lt 119899119894are
the state variables inputs and outputs of the 119894th subsystemrespectively The triples (119860
119894 119861119894 119862119894) and119867
119894119895represent known
constant matrices of appropriate dimensions The matricesΔ119860119894119909119894and Δ119867
119894119895119909119895represent the mismatched parameter
uncertainty in the state matrix in each isolated subsystemsand mismatched interconnections respectively The matrix119861119894120585119894(119909119894 119905) is disturbance input In this paper only the output
variables 119910119894are assumed to be known
In order to modify the existing two phases SMC wedenote the sliding surface by 120590
119894(119909119894(119905) 119905) = 0 119894 = 1 2 119871
where the single-phase sliding function is given as
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (0) 0) exp (minus120573119894119905) (2)
with constant 120573119894gt 0 The function 120590
119894(119910119894(119905) 119905) is defined
later The sliding mode is defined by 120590119894(119909119894(119905) 119905) = 0 and
119894(119909119894(119905) 119905) = 0 From (2) one can see that there are only
output variables used and the system states are in the slidingmode from the initial time 120590
119894(119909119894(0) 0) = 0 Therefore this is
to say that the SMC is single-phase (without reaching phase)This can be formally defined as follows
Definition 1 A sliding mode control is said to be a single-phase SMC if and only if the following two conditions aresatisfied
(1) the reaching time is equal to zero 120590119894(119909119894(0) 0) = 0
(2) the order of the motion equation in sliding mode isequal to the order of the original system
Remark 2 The concept of single-phase sliding mode controlfocusses on the robustness of the motion in the entire statespace The order of the motion equation in sliding modeis equal to the dimension of the state space Therefore therobustness of complex interconnected systems can be assuredthroughout an entire response of the system starting from theinitial time instance
In order to apply the concept of single-phase SMC for thesystem (1) we assume the following to be valid
Assumption 3 The mismatched parameter uncertainties inthe state matrix of each isolated subsystem are satisfied asΔ119860119894= 119863119894119865119894(119909119894 119905)119864119894where 119865
119894(119909119894 119905) is unknown but bounded
as 119865119894(119909119894 119905) le 1 and119863
119894119864119894are knownmatrices of appropriate
dimensions
Assumption 4 The matrices 119861119894and 119862
119894are full rank and
rank(119862119894119861119894) = 119898
119894
From [18] Assumption 4 implies that there exists anonsingular linear coordinate transformation
119894= 119894119909119894such
that the triple (119860119894 119861119894 119862119894)with respect to the new coordinates
has the structure
119860119894= [
11986011989411198601198942
11986011989431198601198944
] 119861119894= [
0
1198611198942
] 119862119894= [0 119862
1198942] (3)
where1198601198941isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) 119861
1198942isin 119877119898119894times119898119894 are nonsingular and
1198621198942isin 119877119901119894times119901119894 is orthogonal
Assumption 5 The triple (1198601198941 1198601198942 Ξ119894) is output feedback
stabilisable where Ξ119894
= [0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)
] 119894 =
1 2 3 119871From [18] Assumption 5 implies that there exist matrices
119894such that the matrices 119860
1198941= 1198601198941minus 1198601198942119894Ξ119894are stable
Assumption 6 There exist known nonnegative constants 119888119894
and 119887119894such that 120585
119894(119909119894 119905) le 119888
119894+ 119887119894119909119894(119905)
Assumption 7 The mismatched interconnections are givenas Δ119867
119894119895= 119872119894119895119865119894119895(119909119895 119905)119873119894119895 where 119865
119894119895(119909119895 119905) is unknown but
bounded 119865119894119895(119909119895 119905) le 1 and119872
119894119895 119873119894119895are known matrices of
appropriate dimensions
Remark 8 Assumptions 4 and 5 have been utilized in [18]The assumption of the norm boundedness of 119865
119894(119909119894 119905) and
120585119894(119909119894 119905) can be found in [19 26 27]
3 Single-Phase Sliding Mode Control forComplex Interconnected Systems
In this section we develop a single-phase SMC to stabilizethe complex interconnected system (1) for all time There arefour steps involved in the design of our single-phase SMCusing only output variables In the first step a proper slidingsurface is designed to construct the single-phase SMC suchthat the desired motion is determined from the initial timeinstant In the second step sufficient conditions in termsof LMI are derived for the existence of a sliding surfaceguaranteeing asymptotic stability In the third step a newlemma is established for controller design using only outputvariables In the fourth step a decentralized output feedbackcontroller is designed to force the system states to stay on thesliding surface for all time
31 Single-Phase Sliding Surface Design Let us first designa new sliding surface without reaching phase that uses onlyoutput variables and the desired motion is determined fromthe initial time instant Under Assumptions 4 and 5 it followsfrom (11) (12) and (13) of paper [18] that there exists a
4 Mathematical Problems in Engineering
coordinate transformation 119911119894= 119879119894119909119894such that the system (1)
has the following regular form
119894= ([
11986011989411198601198942
11986011989431198601198944
] + [1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]) 119911119894
+ [0
1198611198942
] (119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
([1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] + [1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]) 119911119895
(4)
119910119894= [0 119862
1198942] 119911119894 (5)
where
119879119894119860119894119879minus1
119894= [
11986011989411198601198942
11986011989431198601198944
]
119879119894119863119894119865119894119864119894119879minus1
119894= [
1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]
119879119894119867119894119895119879minus1
119895= [
1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] 119879119894119861119894= [
0
1198611198942
]
119879119894119872119894119895119865119894119895119873119894119895119879minus1
119895= [
1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]
119862119894119879119894
minus1= [0 119862
1198942]
(6)
The matrices 1198611198942isin 119877119898119894times119898119894 and 119862
1198942isin 119877119901119894times119901119894 are nonsingular
and 1198601198941= 1198601198941minus 1198601198942119894Ξ119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is stable Then
by using the sliding function (2) the sliding surface can bedefined as follows
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (119905) 0) exp (minus120573119894119905) = 0
(7)
where the solution of 120590119894(119910119894 119905) is given by
120590119894(119910119894 119905) = 119870
119894119862minus1
1198942119910119894
= 119870119894[
119873119894
0(119901119894minus119898119894)times119898119894
0119898119894times(119899119894minus119898119894)
119868119898119894times119898119894
] [1199111198941
1199111198942
] = 11987011989421199111198942
(8)
in which 119911119894= [11991111989411199111198942] 1199111198941
isin 119877119899119894minus119898119894 119911
1198942isin 119877119898119894 119873
119894=
[0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)times(119901119894minus119898119894)
] and 119870119894= [0119898119894times(119901119894minus119898119894)
1198701198942]
The matrix1198701198942isin 119877119898119894times119898119894 is given as
1198701198942= Π119894119875119894Π119879
119894 (9)
where the matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined later and
the matrix Π119894isin 119877119898119894times(119899119894minus119898119894) is selected such that 119870
1198942is
nonsingular Using (8) sliding surface (7) can be rewrittenas
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 119905) minus 120590119894 (119910119894 0) exp (minus120573119894119905)
= 11987011989421199111198942 (119905) minus 11987011989421199111198942 (0) exp (minus120573119894119905) = 0
(10)
Since 1198701198942isin 119877119898119894times119898119894 is nonsingular sliding surface (7) can be
described by
col (1199111 1199112 119911
119871) | 1199111198942= 1199111198942 (0) exp (minus120573119894119905)
119894 = 1 2 119871
(11)
From (11) it is clear that in sliding mode
1198942= minus1205731198941199111198942 (12)
Then from the structure of system (4)-(5) and (12) the slidingmode dynamics of system (1) associated with sliding surface(7) are described by
119894= [
1198601198941
1198601198942
0 minus120573119894119868119898119894times119898119894
] 119911119894+
119871
sum
119895=1
119895 =119894
[1198671198941198951
1198671198941198952
0 0] 119911119895 (13)
where 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+ 11986311989411198651198941198641198942 1198671198941198951
=
1198671198941198951+11987211989411989511198651198941198951198731198941198951 and119867
1198941198952= 1198671198941198952+11987211989411989511198651198941198951198731198941198952
Remark 9 It is obvious that 120590119894(119909119894(0) 0) = 0 which means
that the reaching time is equal to zero and the sliding modeexists from the initial time instant In other words the desiredmotion is determined from the beginning of the time
Remark 10 This approach concentrates on the robustness ofthe motion in the entire state space The order of the motionequation in sliding mode is equal to the order of the originalsystemTherefore the robustness of the system can be assuredthroughout an entire response of the system starting from theinitial time instance
32 Single-Phase Sliding Mode Stability Analysis Followingdesign of the sliding surface two tasks remain First forstability analysis appropriate LMI stability conditions by theLyapunov method must be derived to ensure the stability ofslidingmotion (13) Second we design a decentralized outputfeedback sliding mode controller to keep the system states tostay on the sliding surface for all time
This section focuses on the former task We begin byconsidering the following LMI
[[[[[
[
Ψ119894
1198751198941198631198941
119864119879
1198941
119863119879
1198941119875119894minus120592minus1
119894119868 0
1198641198941
0 minus120593minus1
119894119868
]]]]]
]
lt 0 (14)
whereΨ119894= 119860119879
1198941119875119894+ 1198751198941198601198941+ 120576119894119875119894119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
(15)
119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is any positive matrix and 120593
119894 120576119894 120592119894
120593119895 120593119895 120588119894are positive constants
We also recall the following lemmas which will be usedin proving the stability of sliding motion (13)
Mathematical Problems in Engineering 5
Lemma 11 (see [26]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 and then for any scalar120593 gt 0 the following matrix inequality holds
119883119865119884 + 119884119879119865119879119883119879le 120593minus1119883119883119879+ 120593119884119879119884 (16)
Lemma 12 (see [27]) Let119883 and 119884 be real matrices of suitabledimension and then for any scalar 120583 gt 0 the following matrixinequality holds
119883119879119884 + 119884
119879119883 le 120583119883
119879119883 + 120583
minus1119884119879119884 (17)
Lemma 13 (see [28]) The linear matrix inequality
[Θ (119909) Γ (119909)
Γ (119909)119879119877 (119909)
] gt 0 (18)
where Θ(119909) = Θ(119909)119879 119877(119909) = 119877(119909)119879 and Γ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0 Θ(119909) minus Γ(119909)119877(119909)minus1Γ(119909)119879 gt 0
Then we can establish the following theorem
Theorem 14 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 And the
sliding surface is given by (7) Then the sliding motion (13) isasymptotically stable
Proof of Theorem 14 Let us consider the following positivedefinition function
119881 =
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894 (19)
where the positive constant 120574119894will be selected later the
positive matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in LMI (14)
and 119876119894isin 119877119898119894times119898119894 is any positive matrix Then taking the time
derivative of 119881 along the state trajectory of system (13) wecan obtain that
=
119871
sum
119894=1
119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894+
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119894 (20)
Hence substituting (13) into (20) we derive
=
119871
sum
119894=1
119911119879
119894
[[
[
119860119879
1198941119875119894+ 1198751198941198601198941
1198751198941198601198942
119860119879
1198942119875119894
minus2120573119894120574119894119876119894
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119895
[[
[
119867119879
11989411989511198751198940
119867119879
11989411989521198751198940
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119894[
[
1198751198941198671198941198951
1198751198941198671198941198952
0 0
]
]
119911119895
(21)
Since 119911119894= [1199111198941 1199111198942]
119879 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+
11986311989411198651198941198641198942 1198671198941198951
= 1198671198941198951
+ 11987211989411989511198651198941198951198731198941198951 and 119867
1198941198952= 1198671198941198952
+
11987211989411989511198651198941198951198731198941198952 the above equation can be rewritten as
=
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 119864119879
1198941119865119879
119894119863119879
1198941119875119894
+ 1198751198941198601198941+ 11987511989411986311989411198651198941198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 119911119879
1198942119864119879
1198942119865119879
119894
sdot 119863119879
11989411198751198941199111198941+ 119911119879
1198941119875119894119863119894111986511989411986411989421199111198942minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
1198951119873119879
1198941198951119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989511199111198951+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989511199111198951)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
1198952119873119879
1198941198952119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989521199111198952+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989521199111198952)
(22)
Using Lemma 11 and (22) we achieve
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941
+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
119894111987511989411986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
119894111987511989411986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(23)
where 120593119894gt 0 120593
119894gt 0 120593
119894gt 0 and 120575
119894gt 0 are scalars Applying
Lemma 12 to (23) we can obtain
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
119899-dimensional Euclidean space For the sake of simplicitysometimes function 119909
119894(119905) is denoted by 119909
119894
2 Problem Formulation and Preliminaries
In this paper we consider a class of complex interconnectedsystems with exogenous disturbance and mismatched uncer-tainties of each isolated subsystem and interconnection Thesystem is decomposed into 119871 subsystems and the state spacerepresentation of each subsystem is described as follows
119894= (119860119894+ Δ119860119894) 119909119894+ 119861119894(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
(119867119894119895+ Δ119867119894119895) 119909119895
119910119894= 119862119894119909119894
(1)
where 119909119894isin 119877119899119894 119906119894isin 119877119898119894 and 119910
119894isin 119877119901119894 with 119898
119894lt 119901119894lt 119899119894are
the state variables inputs and outputs of the 119894th subsystemrespectively The triples (119860
119894 119861119894 119862119894) and119867
119894119895represent known
constant matrices of appropriate dimensions The matricesΔ119860119894119909119894and Δ119867
119894119895119909119895represent the mismatched parameter
uncertainty in the state matrix in each isolated subsystemsand mismatched interconnections respectively The matrix119861119894120585119894(119909119894 119905) is disturbance input In this paper only the output
variables 119910119894are assumed to be known
In order to modify the existing two phases SMC wedenote the sliding surface by 120590
119894(119909119894(119905) 119905) = 0 119894 = 1 2 119871
where the single-phase sliding function is given as
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (0) 0) exp (minus120573119894119905) (2)
with constant 120573119894gt 0 The function 120590
119894(119910119894(119905) 119905) is defined
later The sliding mode is defined by 120590119894(119909119894(119905) 119905) = 0 and
119894(119909119894(119905) 119905) = 0 From (2) one can see that there are only
output variables used and the system states are in the slidingmode from the initial time 120590
119894(119909119894(0) 0) = 0 Therefore this is
to say that the SMC is single-phase (without reaching phase)This can be formally defined as follows
Definition 1 A sliding mode control is said to be a single-phase SMC if and only if the following two conditions aresatisfied
(1) the reaching time is equal to zero 120590119894(119909119894(0) 0) = 0
(2) the order of the motion equation in sliding mode isequal to the order of the original system
Remark 2 The concept of single-phase sliding mode controlfocusses on the robustness of the motion in the entire statespace The order of the motion equation in sliding modeis equal to the dimension of the state space Therefore therobustness of complex interconnected systems can be assuredthroughout an entire response of the system starting from theinitial time instance
In order to apply the concept of single-phase SMC for thesystem (1) we assume the following to be valid
Assumption 3 The mismatched parameter uncertainties inthe state matrix of each isolated subsystem are satisfied asΔ119860119894= 119863119894119865119894(119909119894 119905)119864119894where 119865
119894(119909119894 119905) is unknown but bounded
as 119865119894(119909119894 119905) le 1 and119863
119894119864119894are knownmatrices of appropriate
dimensions
Assumption 4 The matrices 119861119894and 119862
119894are full rank and
rank(119862119894119861119894) = 119898
119894
From [18] Assumption 4 implies that there exists anonsingular linear coordinate transformation
119894= 119894119909119894such
that the triple (119860119894 119861119894 119862119894)with respect to the new coordinates
has the structure
119860119894= [
11986011989411198601198942
11986011989431198601198944
] 119861119894= [
0
1198611198942
] 119862119894= [0 119862
1198942] (3)
where1198601198941isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) 119861
1198942isin 119877119898119894times119898119894 are nonsingular and
1198621198942isin 119877119901119894times119901119894 is orthogonal
Assumption 5 The triple (1198601198941 1198601198942 Ξ119894) is output feedback
stabilisable where Ξ119894
= [0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)
] 119894 =
1 2 3 119871From [18] Assumption 5 implies that there exist matrices
119894such that the matrices 119860
1198941= 1198601198941minus 1198601198942119894Ξ119894are stable
Assumption 6 There exist known nonnegative constants 119888119894
and 119887119894such that 120585
119894(119909119894 119905) le 119888
119894+ 119887119894119909119894(119905)
Assumption 7 The mismatched interconnections are givenas Δ119867
119894119895= 119872119894119895119865119894119895(119909119895 119905)119873119894119895 where 119865
119894119895(119909119895 119905) is unknown but
bounded 119865119894119895(119909119895 119905) le 1 and119872
119894119895 119873119894119895are known matrices of
appropriate dimensions
Remark 8 Assumptions 4 and 5 have been utilized in [18]The assumption of the norm boundedness of 119865
119894(119909119894 119905) and
120585119894(119909119894 119905) can be found in [19 26 27]
3 Single-Phase Sliding Mode Control forComplex Interconnected Systems
In this section we develop a single-phase SMC to stabilizethe complex interconnected system (1) for all time There arefour steps involved in the design of our single-phase SMCusing only output variables In the first step a proper slidingsurface is designed to construct the single-phase SMC suchthat the desired motion is determined from the initial timeinstant In the second step sufficient conditions in termsof LMI are derived for the existence of a sliding surfaceguaranteeing asymptotic stability In the third step a newlemma is established for controller design using only outputvariables In the fourth step a decentralized output feedbackcontroller is designed to force the system states to stay on thesliding surface for all time
31 Single-Phase Sliding Surface Design Let us first designa new sliding surface without reaching phase that uses onlyoutput variables and the desired motion is determined fromthe initial time instant Under Assumptions 4 and 5 it followsfrom (11) (12) and (13) of paper [18] that there exists a
4 Mathematical Problems in Engineering
coordinate transformation 119911119894= 119879119894119909119894such that the system (1)
has the following regular form
119894= ([
11986011989411198601198942
11986011989431198601198944
] + [1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]) 119911119894
+ [0
1198611198942
] (119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
([1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] + [1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]) 119911119895
(4)
119910119894= [0 119862
1198942] 119911119894 (5)
where
119879119894119860119894119879minus1
119894= [
11986011989411198601198942
11986011989431198601198944
]
119879119894119863119894119865119894119864119894119879minus1
119894= [
1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]
119879119894119867119894119895119879minus1
119895= [
1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] 119879119894119861119894= [
0
1198611198942
]
119879119894119872119894119895119865119894119895119873119894119895119879minus1
119895= [
1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]
119862119894119879119894
minus1= [0 119862
1198942]
(6)
The matrices 1198611198942isin 119877119898119894times119898119894 and 119862
1198942isin 119877119901119894times119901119894 are nonsingular
and 1198601198941= 1198601198941minus 1198601198942119894Ξ119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is stable Then
by using the sliding function (2) the sliding surface can bedefined as follows
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (119905) 0) exp (minus120573119894119905) = 0
(7)
where the solution of 120590119894(119910119894 119905) is given by
120590119894(119910119894 119905) = 119870
119894119862minus1
1198942119910119894
= 119870119894[
119873119894
0(119901119894minus119898119894)times119898119894
0119898119894times(119899119894minus119898119894)
119868119898119894times119898119894
] [1199111198941
1199111198942
] = 11987011989421199111198942
(8)
in which 119911119894= [11991111989411199111198942] 1199111198941
isin 119877119899119894minus119898119894 119911
1198942isin 119877119898119894 119873
119894=
[0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)times(119901119894minus119898119894)
] and 119870119894= [0119898119894times(119901119894minus119898119894)
1198701198942]
The matrix1198701198942isin 119877119898119894times119898119894 is given as
1198701198942= Π119894119875119894Π119879
119894 (9)
where the matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined later and
the matrix Π119894isin 119877119898119894times(119899119894minus119898119894) is selected such that 119870
1198942is
nonsingular Using (8) sliding surface (7) can be rewrittenas
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 119905) minus 120590119894 (119910119894 0) exp (minus120573119894119905)
= 11987011989421199111198942 (119905) minus 11987011989421199111198942 (0) exp (minus120573119894119905) = 0
(10)
Since 1198701198942isin 119877119898119894times119898119894 is nonsingular sliding surface (7) can be
described by
col (1199111 1199112 119911
119871) | 1199111198942= 1199111198942 (0) exp (minus120573119894119905)
119894 = 1 2 119871
(11)
From (11) it is clear that in sliding mode
1198942= minus1205731198941199111198942 (12)
Then from the structure of system (4)-(5) and (12) the slidingmode dynamics of system (1) associated with sliding surface(7) are described by
119894= [
1198601198941
1198601198942
0 minus120573119894119868119898119894times119898119894
] 119911119894+
119871
sum
119895=1
119895 =119894
[1198671198941198951
1198671198941198952
0 0] 119911119895 (13)
where 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+ 11986311989411198651198941198641198942 1198671198941198951
=
1198671198941198951+11987211989411989511198651198941198951198731198941198951 and119867
1198941198952= 1198671198941198952+11987211989411989511198651198941198951198731198941198952
Remark 9 It is obvious that 120590119894(119909119894(0) 0) = 0 which means
that the reaching time is equal to zero and the sliding modeexists from the initial time instant In other words the desiredmotion is determined from the beginning of the time
Remark 10 This approach concentrates on the robustness ofthe motion in the entire state space The order of the motionequation in sliding mode is equal to the order of the originalsystemTherefore the robustness of the system can be assuredthroughout an entire response of the system starting from theinitial time instance
32 Single-Phase Sliding Mode Stability Analysis Followingdesign of the sliding surface two tasks remain First forstability analysis appropriate LMI stability conditions by theLyapunov method must be derived to ensure the stability ofslidingmotion (13) Second we design a decentralized outputfeedback sliding mode controller to keep the system states tostay on the sliding surface for all time
This section focuses on the former task We begin byconsidering the following LMI
[[[[[
[
Ψ119894
1198751198941198631198941
119864119879
1198941
119863119879
1198941119875119894minus120592minus1
119894119868 0
1198641198941
0 minus120593minus1
119894119868
]]]]]
]
lt 0 (14)
whereΨ119894= 119860119879
1198941119875119894+ 1198751198941198601198941+ 120576119894119875119894119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
(15)
119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is any positive matrix and 120593
119894 120576119894 120592119894
120593119895 120593119895 120588119894are positive constants
We also recall the following lemmas which will be usedin proving the stability of sliding motion (13)
Mathematical Problems in Engineering 5
Lemma 11 (see [26]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 and then for any scalar120593 gt 0 the following matrix inequality holds
119883119865119884 + 119884119879119865119879119883119879le 120593minus1119883119883119879+ 120593119884119879119884 (16)
Lemma 12 (see [27]) Let119883 and 119884 be real matrices of suitabledimension and then for any scalar 120583 gt 0 the following matrixinequality holds
119883119879119884 + 119884
119879119883 le 120583119883
119879119883 + 120583
minus1119884119879119884 (17)
Lemma 13 (see [28]) The linear matrix inequality
[Θ (119909) Γ (119909)
Γ (119909)119879119877 (119909)
] gt 0 (18)
where Θ(119909) = Θ(119909)119879 119877(119909) = 119877(119909)119879 and Γ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0 Θ(119909) minus Γ(119909)119877(119909)minus1Γ(119909)119879 gt 0
Then we can establish the following theorem
Theorem 14 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 And the
sliding surface is given by (7) Then the sliding motion (13) isasymptotically stable
Proof of Theorem 14 Let us consider the following positivedefinition function
119881 =
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894 (19)
where the positive constant 120574119894will be selected later the
positive matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in LMI (14)
and 119876119894isin 119877119898119894times119898119894 is any positive matrix Then taking the time
derivative of 119881 along the state trajectory of system (13) wecan obtain that
=
119871
sum
119894=1
119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894+
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119894 (20)
Hence substituting (13) into (20) we derive
=
119871
sum
119894=1
119911119879
119894
[[
[
119860119879
1198941119875119894+ 1198751198941198601198941
1198751198941198601198942
119860119879
1198942119875119894
minus2120573119894120574119894119876119894
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119895
[[
[
119867119879
11989411989511198751198940
119867119879
11989411989521198751198940
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119894[
[
1198751198941198671198941198951
1198751198941198671198941198952
0 0
]
]
119911119895
(21)
Since 119911119894= [1199111198941 1199111198942]
119879 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+
11986311989411198651198941198641198942 1198671198941198951
= 1198671198941198951
+ 11987211989411989511198651198941198951198731198941198951 and 119867
1198941198952= 1198671198941198952
+
11987211989411989511198651198941198951198731198941198952 the above equation can be rewritten as
=
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 119864119879
1198941119865119879
119894119863119879
1198941119875119894
+ 1198751198941198601198941+ 11987511989411986311989411198651198941198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 119911119879
1198942119864119879
1198942119865119879
119894
sdot 119863119879
11989411198751198941199111198941+ 119911119879
1198941119875119894119863119894111986511989411986411989421199111198942minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
1198951119873119879
1198941198951119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989511199111198951+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989511199111198951)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
1198952119873119879
1198941198952119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989521199111198952+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989521199111198952)
(22)
Using Lemma 11 and (22) we achieve
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941
+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
119894111987511989411986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
119894111987511989411986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(23)
where 120593119894gt 0 120593
119894gt 0 120593
119894gt 0 and 120575
119894gt 0 are scalars Applying
Lemma 12 to (23) we can obtain
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
coordinate transformation 119911119894= 119879119894119909119894such that the system (1)
has the following regular form
119894= ([
11986011989411198601198942
11986011989431198601198944
] + [1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]) 119911119894
+ [0
1198611198942
] (119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119895=1
119895 =119894
([1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] + [1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]) 119911119895
(4)
119910119894= [0 119862
1198942] 119911119894 (5)
where
119879119894119860119894119879minus1
119894= [
11986011989411198601198942
11986011989431198601198944
]
119879119894119863119894119865119894119864119894119879minus1
119894= [
1198631198941
1198631198942
]119865119894[1198641198941 1198641198942]
119879119894119867119894119895119879minus1
119895= [
1198671198941198951
1198671198941198952
1198671198941198953
1198671198941198954
] 119879119894119861119894= [
0
1198611198942
]
119879119894119872119894119895119865119894119895119873119894119895119879minus1
119895= [
1198721198941198951
1198721198941198952
]119865119894119895[1198731198941198951 1198731198941198952]
119862119894119879119894
minus1= [0 119862
1198942]
(6)
The matrices 1198611198942isin 119877119898119894times119898119894 and 119862
1198942isin 119877119901119894times119901119894 are nonsingular
and 1198601198941= 1198601198941minus 1198601198942119894Ξ119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is stable Then
by using the sliding function (2) the sliding surface can bedefined as follows
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 (119905) 119905) minus 120590119894 (119910119894 (119905) 0) exp (minus120573119894119905) = 0
(7)
where the solution of 120590119894(119910119894 119905) is given by
120590119894(119910119894 119905) = 119870
119894119862minus1
1198942119910119894
= 119870119894[
119873119894
0(119901119894minus119898119894)times119898119894
0119898119894times(119899119894minus119898119894)
119868119898119894times119898119894
] [1199111198941
1199111198942
] = 11987011989421199111198942
(8)
in which 119911119894= [11991111989411199111198942] 1199111198941
isin 119877119899119894minus119898119894 119911
1198942isin 119877119898119894 119873
119894=
[0(119901119894minus119898119894)times(119899119894minus119901119894)119868(119901119894minus119898119894)times(119901119894minus119898119894)
] and 119870119894= [0119898119894times(119901119894minus119898119894)
1198701198942]
The matrix1198701198942isin 119877119898119894times119898119894 is given as
1198701198942= Π119894119875119894Π119879
119894 (9)
where the matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined later and
the matrix Π119894isin 119877119898119894times(119899119894minus119898119894) is selected such that 119870
1198942is
nonsingular Using (8) sliding surface (7) can be rewrittenas
120590119894(119909119894 (119905) 119905) = 120590119894 (119910119894 119905) minus 120590119894 (119910119894 0) exp (minus120573119894119905)
= 11987011989421199111198942 (119905) minus 11987011989421199111198942 (0) exp (minus120573119894119905) = 0
(10)
Since 1198701198942isin 119877119898119894times119898119894 is nonsingular sliding surface (7) can be
described by
col (1199111 1199112 119911
119871) | 1199111198942= 1199111198942 (0) exp (minus120573119894119905)
119894 = 1 2 119871
(11)
From (11) it is clear that in sliding mode
1198942= minus1205731198941199111198942 (12)
Then from the structure of system (4)-(5) and (12) the slidingmode dynamics of system (1) associated with sliding surface(7) are described by
119894= [
1198601198941
1198601198942
0 minus120573119894119868119898119894times119898119894
] 119911119894+
119871
sum
119895=1
119895 =119894
[1198671198941198951
1198671198941198952
0 0] 119911119895 (13)
where 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+ 11986311989411198651198941198641198942 1198671198941198951
=
1198671198941198951+11987211989411989511198651198941198951198731198941198951 and119867
1198941198952= 1198671198941198952+11987211989411989511198651198941198951198731198941198952
Remark 9 It is obvious that 120590119894(119909119894(0) 0) = 0 which means
that the reaching time is equal to zero and the sliding modeexists from the initial time instant In other words the desiredmotion is determined from the beginning of the time
Remark 10 This approach concentrates on the robustness ofthe motion in the entire state space The order of the motionequation in sliding mode is equal to the order of the originalsystemTherefore the robustness of the system can be assuredthroughout an entire response of the system starting from theinitial time instance
32 Single-Phase Sliding Mode Stability Analysis Followingdesign of the sliding surface two tasks remain First forstability analysis appropriate LMI stability conditions by theLyapunov method must be derived to ensure the stability ofslidingmotion (13) Second we design a decentralized outputfeedback sliding mode controller to keep the system states tostay on the sliding surface for all time
This section focuses on the former task We begin byconsidering the following LMI
[[[[[
[
Ψ119894
1198751198941198631198941
119864119879
1198941
119863119879
1198941119875119894minus120592minus1
119894119868 0
1198641198941
0 minus120593minus1
119894119868
]]]]]
]
lt 0 (14)
whereΨ119894= 119860119879
1198941119875119894+ 1198751198941198601198941+ 120576119894119875119894119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
(15)
119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is any positive matrix and 120593
119894 120576119894 120592119894
120593119895 120593119895 120588119894are positive constants
We also recall the following lemmas which will be usedin proving the stability of sliding motion (13)
Mathematical Problems in Engineering 5
Lemma 11 (see [26]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 and then for any scalar120593 gt 0 the following matrix inequality holds
119883119865119884 + 119884119879119865119879119883119879le 120593minus1119883119883119879+ 120593119884119879119884 (16)
Lemma 12 (see [27]) Let119883 and 119884 be real matrices of suitabledimension and then for any scalar 120583 gt 0 the following matrixinequality holds
119883119879119884 + 119884
119879119883 le 120583119883
119879119883 + 120583
minus1119884119879119884 (17)
Lemma 13 (see [28]) The linear matrix inequality
[Θ (119909) Γ (119909)
Γ (119909)119879119877 (119909)
] gt 0 (18)
where Θ(119909) = Θ(119909)119879 119877(119909) = 119877(119909)119879 and Γ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0 Θ(119909) minus Γ(119909)119877(119909)minus1Γ(119909)119879 gt 0
Then we can establish the following theorem
Theorem 14 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 And the
sliding surface is given by (7) Then the sliding motion (13) isasymptotically stable
Proof of Theorem 14 Let us consider the following positivedefinition function
119881 =
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894 (19)
where the positive constant 120574119894will be selected later the
positive matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in LMI (14)
and 119876119894isin 119877119898119894times119898119894 is any positive matrix Then taking the time
derivative of 119881 along the state trajectory of system (13) wecan obtain that
=
119871
sum
119894=1
119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894+
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119894 (20)
Hence substituting (13) into (20) we derive
=
119871
sum
119894=1
119911119879
119894
[[
[
119860119879
1198941119875119894+ 1198751198941198601198941
1198751198941198601198942
119860119879
1198942119875119894
minus2120573119894120574119894119876119894
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119895
[[
[
119867119879
11989411989511198751198940
119867119879
11989411989521198751198940
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119894[
[
1198751198941198671198941198951
1198751198941198671198941198952
0 0
]
]
119911119895
(21)
Since 119911119894= [1199111198941 1199111198942]
119879 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+
11986311989411198651198941198641198942 1198671198941198951
= 1198671198941198951
+ 11987211989411989511198651198941198951198731198941198951 and 119867
1198941198952= 1198671198941198952
+
11987211989411989511198651198941198951198731198941198952 the above equation can be rewritten as
=
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 119864119879
1198941119865119879
119894119863119879
1198941119875119894
+ 1198751198941198601198941+ 11987511989411986311989411198651198941198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 119911119879
1198942119864119879
1198942119865119879
119894
sdot 119863119879
11989411198751198941199111198941+ 119911119879
1198941119875119894119863119894111986511989411986411989421199111198942minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
1198951119873119879
1198941198951119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989511199111198951+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989511199111198951)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
1198952119873119879
1198941198952119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989521199111198952+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989521199111198952)
(22)
Using Lemma 11 and (22) we achieve
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941
+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
119894111987511989411986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
119894111987511989411986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(23)
where 120593119894gt 0 120593
119894gt 0 120593
119894gt 0 and 120575
119894gt 0 are scalars Applying
Lemma 12 to (23) we can obtain
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Lemma 11 (see [26]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 and then for any scalar120593 gt 0 the following matrix inequality holds
119883119865119884 + 119884119879119865119879119883119879le 120593minus1119883119883119879+ 120593119884119879119884 (16)
Lemma 12 (see [27]) Let119883 and 119884 be real matrices of suitabledimension and then for any scalar 120583 gt 0 the following matrixinequality holds
119883119879119884 + 119884
119879119883 le 120583119883
119879119883 + 120583
minus1119884119879119884 (17)
Lemma 13 (see [28]) The linear matrix inequality
[Θ (119909) Γ (119909)
Γ (119909)119879119877 (119909)
] gt 0 (18)
where Θ(119909) = Θ(119909)119879 119877(119909) = 119877(119909)119879 and Γ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0 Θ(119909) minus Γ(119909)119877(119909)minus1Γ(119909)119879 gt 0
Then we can establish the following theorem
Theorem 14 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 And the
sliding surface is given by (7) Then the sliding motion (13) isasymptotically stable
Proof of Theorem 14 Let us consider the following positivedefinition function
119881 =
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894 (19)
where the positive constant 120574119894will be selected later the
positive matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in LMI (14)
and 119876119894isin 119877119898119894times119898119894 is any positive matrix Then taking the time
derivative of 119881 along the state trajectory of system (13) wecan obtain that
=
119871
sum
119894=1
119879
119894[119875119894
0
0 120574119894119876119894
] 119911119894+
119871
sum
119894=1
119911119879
119894[119875119894
0
0 120574119894119876119894
] 119894 (20)
Hence substituting (13) into (20) we derive
=
119871
sum
119894=1
119911119879
119894
[[
[
119860119879
1198941119875119894+ 1198751198941198601198941
1198751198941198601198942
119860119879
1198942119875119894
minus2120573119894120574119894119876119894
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119895
[[
[
119867119879
11989411989511198751198940
119867119879
11989411989521198751198940
]]
]
119911119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
119911119879
119894[
[
1198751198941198671198941198951
1198751198941198671198941198952
0 0
]
]
119911119895
(21)
Since 119911119894= [1199111198941 1199111198942]
119879 1198601198941= 1198601198941+ 11986311989411198651198941198641198941 1198601198942= 1198601198942+
11986311989411198651198941198641198942 1198671198941198951
= 1198671198941198951
+ 11987211989411989511198651198941198951198731198941198951 and 119867
1198941198952= 1198671198941198952
+
11987211989411989511198651198941198951198731198941198952 the above equation can be rewritten as
=
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 119864119879
1198941119865119879
119894119863119879
1198941119875119894
+ 1198751198941198601198941+ 11987511989411986311989411198651198941198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 119911119879
1198942119864119879
1198942119865119879
119894
sdot 119863119879
11989411198751198941199111198941+ 119911119879
1198941119875119894119863119894111986511989411986411989421199111198942minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
1198951119873119879
1198941198951119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989511199111198951+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989511199111198951)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
1198952119873119879
1198941198952119865119879
119894119895119872119879
11989411989511198751198941199111198941
+ 119911119879
119894111987511989411986711989411989521199111198952+ 119911119879
1198941119875119894119872119894119895111986511989411989511987311989411989521199111198952)
(22)
Using Lemma 11 and (22) we achieve
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941
+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[119911119879
1198942119860119879
11989421198751198941199111198941+ 119911119879
119894111987511989411986011989421199111198942+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198951119867119879
11989411989511198751198941199111198941+ 119911119879
119894111987511989411986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(119911119879
1198952119867119879
11989411989521198751198941199111198941+ 119911119879
119894111987511989411986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(23)
where 120593119894gt 0 120593
119894gt 0 120593
119894gt 0 and 120575
119894gt 0 are scalars Applying
Lemma 12 to (23) we can obtain
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
le
119871
sum
119894=1
119911119879
1198941(119860119879
1198941119875119894+ 1198751198941198601198941+ 120593minus1
1198941198751198941198631198941119863119879
1198941119875119894+ 120593119894119864119879
11989411198641198941) 1199111198941
+
119871
sum
119894=1
[120575119894119911119879
1198942119860119879
119894211986011989421199111198942+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198942119864119879
119894211986411989421199111198942
+ 120593minus1
119894119911119879
11989411198751198941198631198941119863119879
11989411198751198941199111198941minus 2120573119894120574119894119911119879
11989421198761198941199111198942]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120593119894119911119879
1198951119873119879
1198941198951
sdot 11987311989411989511199111198951+ 120593minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198751198941199111198941+ 120575119894119911119879
1198952119873119879
1198941198952
sdot 11987311989411989521199111198952+ 120575minus1
119894119911119879
11989411198751198941198721198941198951119872119879
11989411989511198751198941199111198941)
(24)
where 120593119894gt 0 120575
119894gt 0 and 120575
119894gt 0 are scalars From (24) and
property
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119894119911119879
1198951119867119879
119894119895111986711989411989511199111198951+ 120593119894119911119879
1198951119873119879
119894119895111987311989411989511199111198951)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120593119895119911119879
1198941119867119879
119895119894111986711989511989411199111198941+ 120593119895119911119879
1198941119873119879
119895119894111987311989511989411199111198941)
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119894119911119879
1198952119867119879
119894119895211986711989411989521199111198952+ 120575119894119911119879
1198952119873119879
119894119895211987311989411989521199111198952)
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(120575119895119911119879
1198942119867119879
119895119894211986711989511989421199111198942+ 120575119895119911119879
1198942119873119879
119895119894211987311989511989421199111198942)
(25)
it generates
le
119871
sum
119894=1
119911119879
1198941
[[[
[
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941
+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941
+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894)
]]]
]
1199111198941
+
119871
sum
119894=1
119911119879
1198942
[[[
[
minus2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989511989421198731198951198942)
]]]
]
1199111198942
(26)
where 120592119894= 120593minus1
119894+ 120593minus1
119894 120588119894= 120593minus1
119894+ 120575minus1
119894 and 120576
119894= (119871 minus 1)120593
minus1
119894+
120575minus1
119894+ (119871 minus 1)120575
minus1
119894and 119871 is the number of subsystems Let us
define Ω119894= 2120573119894119876119894gt 0 so that all the eigenvalues of Ω
119894are
greater than some positive ni119894 Then it is easy to select
120574119894gt1
ni119894
max119894=1119871119895=1119871
119894 =119895
[[[
[
120575119894
10038171003817100381710038171003817119860119879
11989421198601198942
10038171003817100381710038171003817+ 120593119894
10038171003817100381710038171003817119864119879
11989421198641198942
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(120575119895
10038171003817100381710038171003817119867119879
11989511989421198671198951198942
10038171003817100381710038171003817+ 120575119895
10038171003817100381710038171003817119873119879
11989511989421198731198951198942
10038171003817100381710038171003817)
]]]
]
(27)
According to (27) and the result of paper [29] we have
minus 2120573119894120574119894119876119894+ 120575119894119860119879
11989421198601198942+ 120593119894119864119879
11989421198641198942
+
119871
sum
119895=1
119895 =119894
(120575119895119867119879
11989511989421198671198951198942+ 120575119895119873119879
11989411989521198731198951198942) lt 0
(28)
By applying Lemma 13 LMI (14) is equivalent to the followinginequality
119860119879
1198941119875119894+ 1198751198941198601198941+ 120593119894119864119879
11989411198641198941+ 120576119894119875119894119875119894+ 1205921198941198751198941198631198941119863119879
1198941119875119894
+
119871
sum
119895=1
119895 =119894
(120593119895119867119879
11989511989411198671198951198941+ 120593119895119873119879
11989511989411198731198951198941+ 1205881198941198751198941198721198941198951119872119879
1198941198951119875119894) lt 0
(29)
From (26) (28) and (29) we have
lt 0 (30)
Inequality (30) implies that if LMI (14) holds then slidingmotion (13) is asymptotically stable
Remark 15 Theorem 14 provides an existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI Toolbox in MATLAB
Remark 16 It is seen that compared to the the recentLMI methods [17] the present LMI method shows lessconservative results and easily finds a feasible solution of theLMI
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
In order to design a new output feedback sliding modecontrol scheme for complex interconnected system (1) weestablish the following lemma
Lemma 17 Consider a class of interconnected systems that isdecomposed into 119871 subsystems
V119894= [119878119894+ 119866119894Δ119894(V119894 119905) 119883119894] V119894+ 119861119894(119906119894+ 120585119894(V119894 119905))
+
119871
sum
119895=1
119895 =119894
(119860119894119895+ 119863119894119895Δ119894119895(V119895 119905) 119864119894119895) V119895
(31)
where V119894= [
V1198941V1198942 ] are the state variables of the 119894th subsystem
with V1198941isin 119877119899119894minus119898119894 and V
1198942isin 119877119898119894 The matrices 119878
119894= [1198781198941 1198781198942
1198781198943 1198781198944]
119866119894= [1198661198941
1198661198942] 119883119894= [1198831198941 1198831198942] 119861119894 = [
0
1198611198942] 119860119894119895= [1198601198941198951 1198601198941198952
1198601198941198953 1198601198941198954]
119863119894119895= [1198631198941198951
1198631198941198952] and 119864
119894119895= [1198641198941198951 1198641198941198952] are known matrices of
appropriate dimensions The matrices Δ119894(V119894 119905) and Δ
119894119895(V119895 119905)
are unknown but bounded as Δ119894(V119894 119905) le 1 and Δ
119894119895(V119895 119905) le
1 If the matrix 1198781198941is stable then sum119871
119894=1V1198941(119905) is bounded by
sum119871
119894=1120601119894(119905) for all time where 120601
119894(119905) is the solution of
120601119894 (119905) = 119894120601119894 (119905) + 119896119894
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
119894 = 1 2 119871
(32)
in which 119894
= 119896119894[11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 +
11986311989511989411198641198951198941)] + 120582
119894lt 0 119896
119894gt 0 120582
119894is the maximum
eigenvalue of the matrix 1198781198941
Proof of Lemma 17 We are now in the position to proveLemma 17 From (31) it is obvious that
V1198941 (119905) = (119878
1198941+ 1198661198941Δ1198941198831198941) V1198941+ (1198781198942+ 1198661198941Δ1198941198831198942) V1198942
+
119871
sum
119895=1
119895 =119894
[(1198601198941198951+ 1198631198941198951Δ1198941198951198641198941198951) V1198951
+ (1198601198941198952+ 1198631198941198951Δ1198941198951198641198941198952) V1198952]
(33)
According to (33) we can obtain
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
1003817100381710038171003817exp (1198781198941119905)1003817100381710038171003817
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
+ int
119905
0
1003817100381710038171003817exp (1198781198941 (119905 minus 120591))1003817100381710038171003817
sdot
[[[
[
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(34)
The stable matrix 1198781198941implies that exp(119878
1198941119905) le 119896
119894exp(120582
119894119905)
for some 119896119894gt 0 and the inequality (34) can be rewritten as
1003817100381710038171003817V11989411003817100381710038171003817 le 119896119894 exp (120582119894119905)
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (120582
119894 (119905 minus 120591))
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(35)
For the above inequality we multiply both sides by the termexp(minus120582
119894119905)
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905)
le 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817 + (
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 ]119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(36)
Let 119904119894(119905) represent the right side of the inequality (36)
119904119894 (119905) = 119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot [10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+ (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817] 119889120591
+ int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
]]]
]
119889120591
(37)
Hence by taking the time derivative of 119904119894(119905) we can obtain
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
= (10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198951
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198951
10038171003817100381710038171003817)10038171003817100381710038171003817V1198951
10038171003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198941198952
10038171003817100381710038171003817+100381710038171003817100381710038171198631198941198951
10038171003817100381710038171003817
100381710038171003817100381710038171198641198941198952
10038171003817100381710038171003817)10038171003817100381710038171003817V1198952
10038171003817100381710038171003817
(38)
Then by taking the summation 119894 = 1 2 119871 to both sides of(38) we have
119871
sum
119894=1
1
119896119894
exp (120582119894119905)119889
119889119905119904119894 (119905)
=
119871
sum
119894=1
(10038171003817100381710038171198781198942
1003817100381710038171003817 +10038171003817100381710038171198661198941
1003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989411003817100381710038171003817
1003817100381710038171003817V11989411003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)1003817100381710038171003817V1198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198941
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198941
10038171003817100381710038171003817)1003817100381710038171003817V1198941
1003817100381710038171003817
(39)
For the above equation we multiply both sides by the term119896119894exp(minus120582
119894119905) Since V
1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) one can get that
119871
sum
119894=1
119889
119889119905119904119894 (119905) le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817
+
119871
sum
119894=1
119896119894119904119894 (119905)
(40)
where 119896119894= 119896119894(11986611989411198831198941 + sum
119871
119895=1119895 =119894(1198601198951198941 + 119863
11989511989411198641198951198941))
Wemultiply the term exp(minus119896119894119905) to both sides of the inequality
(40) and then
119871
sum
119894=1
119889
119889119905[119904119894 (119905) exp (minus119896119894119905)]
le
119871
sum
119894=1
119896119894exp (minus120582
119894119905)
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
1003817100381710038171003817V11989421003817100381710038171003817 exp (minus119896119894119905)
(41)
Integrating inequality (41) on both sides we obtain
119871
sum
119894=1
119904119894 (119905) le
119871
sum
119894=1
119896119894
1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp (119896119894119905)
+
119871
sum
119894=1
int
119905
0
119896119894exp (minus120582
119894120591)
sdot
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 exp (minus119896119894120591) 119889120591
exp (119896119894119905)
(42)
Since V1198941(119905) exp(minus120582
119894119905) le 119904119894(119905) we can show that
119871
sum
119894=1
1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le
119871
sum
119894=1
120601119894 (0) exp ((119896119894 + 120582119894) 119905)
+ int
119905
0
119896119894exp [(119896
119894+ 120582119894) (119905 minus 120591)]
times
[[[
[
100381710038171003817100381711987811989421003817100381710038171003817 +
100381710038171003817100381711986611989411003817100381710038171003817
100381710038171003817100381711988311989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198601198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198631198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198641198951198942
10038171003817100381710038171003817)
]]]
]
sdot1003817100381710038171003817V1198942
1003817100381710038171003817 119889120591
=
119871
sum
119894=1
120601119894 (119905) if 120601
119894 (0) ge 1198961198941003817100381710038171003817V1198941 (0)
1003817100381710038171003817 gt 0
(43)
where the time function 120601119894(t) satisfies (32) Hence we can
see that sum119871119894=1120601119894(119905) ge sum
119871
119894=1V1198941(119905) for all time if 120601
119894(0) is
sufficiently large
Remark 18 It is obvious that the time function 120601119894(119905) is only
dependent on the state variable V1198942 Therefore the term
sum119871
119894=1V1198941 is bounded by a function of state variable V
1198942 This
feature is useful in the design of a controller which only usesoutput variables
33 Decentralized Output Feedback Single-Phase SlidingModeController Design In the last section we proved that thesliding motion (13) is asymptotically stable We furtherestablished Lemma 17 Now by applying this lemma wedesign a decentralized output feedback controller to keep thesystem states to stay in the sliding surface for all time Thisis achieved when the following two conditions are satisfied(1) reaching time is equal to zero (120590
119894(119909119894(0) 0) = 0) (2) the
reaching conditions are satisfied by the Lyapunov function119881(120590119894(119909119894(119905) 119905)) gt 0 and (120590
119894(119909119894(119905) 119905)) lt 0 holds for all 119905 ge 0
Sliding surface (7) allows for the first condition to be metIn order to prove the second condition is also satisfied thesingle-phase sliding mode controller is selected to be
119906119894 (119905) = minus (11987011989421198611198942)
minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(44)
where
120581119894=10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989411003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)
120581119894=
[[[
[
100381710038171003817100381711987011989421003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817 + 119887119894
100381710038171003817100381711986111989421003817100381710038171003817
100381710038171003817100381711988211989421003817100381710038171003817)
+
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
120581119894= 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)
+ 120572119894+ 119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
[1198821198941 1198821198942] = 119879119894minus1
(45)
and the scalar 120572119894gt 0 and 120578
119894(119905) is the solution of
120578119894 (119905) = 119894120578119894 (119905)
+ 119896119894
[[[
[
1003817100381710038171003817119860 11989421003817100381710038171003817 +
100381710038171003817100381711986311989411003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817
+
119871
sum
119895=1
119895 =119894
(100381710038171003817100381710038171198671198951198942
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198941
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
]]]
]
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
(46)
in which 119894
= 119896119894[11986311989411198641198941 + sum
119871
119895=1119895 =119894(1198671198951198941 +
11987211989511989411198731198951198941)] + 120582
119894lt 0 119896
119894gt 0 and 120582
119894is the maximum
eigenvalue of the matrix 1198601198941 It should be pointed out that
controller (44) uses only output variablesNow we can establish the following theorem
Theorem 19 Suppose that LMI (14) has a feasible solution119875119894gt 0 and positive constants 120593
119894 120576119894 120592119894 120593119895 120593119895 120588119894 Consider
the closed loop of the system (1) with the above-decentralizedoutput feedback controller (44) where the sliding surface isgiven by (7) Then the system states stay on the sliding surfacefor all time
Proof of Theorem 19 Now we are going to proveTheorem 19Let us consider the following Lyapunov function
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
10038171003817100381710038171205901198941003817100381710038171003817
(47)
By differentiating (47) along the trajectories of (7) we canobtain
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
119894
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
(11987011989421198942+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905))
(48)
From (4) it is clear that
1198942= (1198601198943+ 11986311989421198651198941198641198941) 1199111198941+ 1198611198942(119906119894+ 120585119894)
+ (1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+
119871
sum
119895=1
119895 =119894
[(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
+ (1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952]
(49)
According to (48) and (49) we have
=
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
[1198701198942(1198601198943+ 11986311989421198651198941198641198941) 1199111198941
+ 1198701198942(1198601198944+ 11986311989421198651198941198641198942) 1199111198942
+ 120573119894119870119894119862minus1
1198942119910119894 (0) exp (minus120573119894119905)]
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(119906119894+ 120585119894(119909119894 119905))
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198954+11987211989411989521198651198941198951198731198941198952) 1199111198952
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
1198701198942(1198671198941198953+11987211989411989521198651198941198951198731198941198951) 1199111198951
(50)
Using (50) and property 119860119861 le 119860119861 we can generate
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
10038171003817100381710038171205851198941003817100381710038171003817
(51)
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Since
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198954
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198952
10038171003817100381710038171003817)100381710038171003817100381710038171199111198952
10038171003817100381710038171003817
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
=
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381711987011989421003817100381710038171003817 (100381710038171003817100381710038171198671198941198953
10038171003817100381710038171003817+100381710038171003817100381710038171198721198941198952
10038171003817100381710038171003817
100381710038171003817100381710038171198731198941198951
10038171003817100381710038171003817)100381710038171003817100381710038171199111198951
10038171003817100381710038171003817
(52)
the above inequality can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817
(53)
In addition 119909119894= 119879minus1
119894119911119894 where 119879
119894
minus1= [1198821198941 1198821198942] 119911119894 = [
11991111989411199111198942]
and using Assumption 6 we have
1003817100381710038171003817120585119894 (119909119894 119905)1003817100381710038171003817 le 119888119894 + 119887119894
1003817100381710038171003817119909119894 (119905)1003817100381710038171003817
= 119888119894+ 119887119894(10038171003817100381710038171198821198941
1003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)
(54)
Substituting (54) into (53) we obtain
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)10038171003817100381710038171199111198942
1003817100381710038171003817
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817)10038171003817100381710038171199111198941
1003817100381710038171003817
+
119871
sum
119894=1
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 [119888119894 + 119887119894 (
100381710038171003817100381711988211989411003817100381710038171003817
100381710038171003817100381711991111989411003817100381710038171003817 +
100381710038171003817100381711988211989421003817100381710038171003817
100381710038171003817100381711991111989421003817100381710038171003817)]
(55)
Equation (8) implies that
100381710038171003817100381711991111989421003817100381710038171003817 le
10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817 (56)
Now for the design of a controller using only output variableswe apply Lemma 17 to the system (4) Let V
1198941= 1199111198941 V1198942= 1199111198942
1198781198941= 1198601198941 1198661198941= 1198631198941 1198661198942= 1198631198942 1198831198941= 1198641198941 1198831198942= 1198641198942 Δ119894= 119865119894
1198781198942= 1198601198942 Δ119894119895= 119865119894119895 1198601198941198951
= 1198671198941198951 1198601198941198952
= 1198671198941198952 1198631198941198951
= 1198721198941198951
1198631198941198952
= 1198721198941198952 1198641198941198951
= 1198731198941198951 and 119864
1198941198952= 1198731198941198952 Then from (32)
(46) and (56) we obtain
120601119894 (119905) = 120578119894 (119905) (57)
119871
sum
119894=1
100381710038171003817100381711991111989411003817100381710038171003817 le
119871
sum
119894=1
120578119894 (119905) (58)
Using (56) and (58) the inequality (55) can be rewritten as
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942119906119894
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
(59)
Inserting the control law (44) into the right-hand side of (59)yields
le
119871
sum
119894=1
[10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198943
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989411003817100381710038171003817) 120578119894
+10038171003817100381710038171198701198942
1003817100381710038171003817 (1003817100381710038171003817119860 1198944
1003817100381710038171003817 +10038171003817100381710038171198631198942
1003817100381710038171003817
100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+ 120573119894
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
1003817100381710038171003817119910119894 (0)1003817100381710038171003817 exp (minus120573119894119905)]
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198943
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198941
10038171003817100381710038171003817) 120578119894
+
119871
sum
119894=1
119871
sum
119895=1
119895 =119894
100381710038171003817100381710038171198701198952
10038171003817100381710038171003817(100381710038171003817100381710038171198671198951198944
10038171003817100381710038171003817+100381710038171003817100381710038171198721198951198942
10038171003817100381710038171003817
100381710038171003817100381710038171198731198951198942
10038171003817100381710038171003817)
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817
+
119871
sum
119894=1
119887119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941
1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942
1003817100381710038171003817
sdot10038171003817100381710038171003817119870minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171003817119870119894119862minus1
1198942
10038171003817100381710038171003817
10038171003817100381710038171199101198941003817100381710038171003817)
+
119871
sum
119894=1
119888119894
100381710038171003817100381711987011989421003817100381710038171003817
100381710038171003817100381711986111989421003817100381710038171003817
minus
119871
sum
119894=1
120590119879
119894
10038171003817100381710038171205901198941003817100381710038171003817
11987011989421198611198942(11987011989421198611198942)minus1(120581119894+ 120581119894
10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894120578119894)
120590119894
10038171003817100381710038171205901198941003817100381710038171003817
(60)
Therefore
le minus
119871
sum
119894=1
120572119894lt 0 (61)
It is then clear that (120590119894(119909119894(119905) 119905)) lt 0 Since 120590
119894(119909119894(0) 0) = 0
suppose that there exists a time 119905 gt 0 such that
119881 (120590119894(119909119894 (119905) 119905)) =
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (119905) 119905)1003817100381710038171003817 le 119881 (120590119894 (119909119894 (0) 0))
=
119871
sum
119894=1
1003817100381710038171003817120590119894 (119909119894 (0) 0)1003817100381710038171003817 = 0
(62)
Thus the identities
120590119894(119909119894 (119905) 119905) = 119894 (119909119894 (119905) 119905) = 0 (63)
hold for all 119905 ge 0 that is there is no reaching phase and thesystem states remain on the sliding mode for all time 119905 ge 0Thus the proof is completed
Remark 20 From sliding mode control theory Theorems 14and 19 together show that the sliding surface (7) with thedecentralized output feedback control law (44) guarantees thefollowing (1) at any initial value the system states remainon the sliding surface for all time 119905 ge 0 and (2) thecomplex interconnected system (1) in the sliding mode isasymptotically stable
Remark 21 Unlike the existing related work such as [13ndash25]the stability of interconnected system (1) can be assured forall time
Remark 22 In contrast to other SMC approaches such asthose presented in [1 7ndash12] the proposed method canbe applied to complex interconnected systems where onlyoutput information is available
Remark 23 It is obvious that this approach uses the outputinformation completely in the sliding surface and controllerdesign Therefore conservatism is reduced and robustness isenhanced
4 Numerical Examples
To verify the effectiveness of the proposed decentralizedoutput feedback SMC law we apply our single-phase SMC toa mismatched uncertain interconnected system composed of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
two third-order subsystems which is modified from [30] asfollows
1= ([
[
minus8 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198601)1199091+ [
[
0
0
1
]
]
(1199061+ 1205851(1199091 119905))
+ ([
[
1 0 0
0 0 1
0 1 0
]
]
+ Δ11986712)1199092
(64)
2= ([
[
minus6 0 1
0 minus7 1
1 0 0
]
]
+ Δ1198602)1199092+ [
[
0
0
1
]
]
(1199062+ 1205852(1199092 119905))
+ ([
[
1 0 0
0 1 0
0 1 0
]
]
+ Δ11986721)1199091
(65)
119910119894= [
1 1 0
0 0 1] 119909119894 119894 = 1 2 (66)
where 1199091= [119909111199091211990913
] isin 1198773 1199092= [119909211199092211990923
] isin 1198773 1199061isin 1198771
1199101= [1199101111991012] isin 119877
2 1199062isin 1198771 and 119910
2= [1199102111991022] isin 119877
2 Themismatched uncertainties in the state matrix are assumedto satisfy Δ119860
1= [01 01 0]
1198791198651[01 0 0] and Δ119860
2=
[0 01 01]1198791198652[0 0 1] with
1198651= 09 sin (1199092
1111990913+ 119905 times 119909
12+ 11990913+ 119905 times 119909
1111990912)
1198652= 09 sin (119909
2111990923+ 1199092
2311990922+ 119905 times 119909
22+ 1199092111990922)
(67)
The mismatched interconnections are given byΔ11986712
= [01 0 01]11987911986512[01 0 01] and Δ119867
21=
[01 01 01]11987911986521[0 01 01] with
11986512= 08 sin (119909
2211990923+ 11990922+ 119905 times 119909
2111990922)
11986521= 07 sin (119909
1111990913+ 119905 times 119909
12+ 119909111199091211990913)
(68)
The exogenous disturbances are given as follows1205851(1199091 119905)) le 12 + 13119909
1 and 120585
2(1199092 119905)) le 2 + 21119909
2
For this work the following parameters are given asfollows 120572
1= 004 120572
2= 03 120573
1= 61 120573
2= 108 120593
1= 09
1205932= 04 120593
1= 08 120593
2= 07 120593
1= 05 120593
2= 06 120593
1= 11
1205932= 12 120575
1= 03 120575
2= 04 120575
1= 01 120575
2= 02 120575
1= 08
1205752= 04 120576
1= 120593minus1
1+120575minus1
1+120575minus1
1= 54924 120576
2= 120593minus1
2+120575minus1
2+120575minus1
2=
58333 1205921= 120593minus1
1+ 120593minus1
1= 23611 120592
2= 120593minus1
2+ 120593minus1
2= 39286
1205881= 120593minus1
1+120575minus1
1= 12 120588
2= 120593minus1
2+120575minus1
2= 66667 119896
1= 1002 and
1198962= 1 The initial conditions for two subsystems are selected
to be 1199091(0) = [minus11 9 35]
119879 and 1199092(0) = [minus9 10 minus1]
119879respectively According to the algorithm given in [30] thecoordinate transformations are given by 119879
1= 1198792= [1 0 0
1 1 0
0 0 1]
By solving LMI (14) we find a feasible solution 119875119894 119894 = 1 2
as follows 1198751= [01361 minus00107
minus00107 01534] and 119875
2= [02598 00051
00051 00673]
The matrix 1198701198942 119894 = 1 2 is given as 119870
12= 000833
11987022= 00113548 From (7) the single-phase sliding surface
for the systems (64) and (65) is designed as
1205901= [0 0 00013] 119909
1minus 00047 exp (minus61119905) = 0
1205902= [0 0 00114] 119909
2minus 00114 exp (minus108119905) = 0
(69)
Then from Theorem 14 we know that the sliding motionof the systems (64) and (65) associated with the slidingsurfaces 120590
1and 120590
2are asymptotically stable From (44) the
decentralized output feedback controller for the systems (64)and (65) are given as
1199061 (119905) = minus 7502281
sdot [00416 + 00285 exp (minus61119905)
+ 0001810038171003817100381710038171199101
1003817100381710038171003817 + 002041205781 (119905)]1205901
100381710038171003817100381712059011003817100381710038171003817
1199062 (119905) = minus 880679
sdot [03227 + 01226 exp (minus108119905)
+ 002510038171003817100381710038171199102
1003817100381710038171003817 + 005181205782 (119905)]1205902
100381710038171003817100381712059021003817100381710038171003817
(70)
where the time functions 1205781(119905) and 120578
2(119905) are the solution of
1205781(119905) = minus6613120578
1(119905) + 2258119910
1 and 120578
2(119905) = minus57120578
2(119905) +
3351199102 respectively
From Theorem 19 the system states stay on the slidingsurface from beginning to endThis is to say that the stabilityof systems (64) and (65) is guaranteed for all time
Remark 24 In the example above the mismatched uncer-tainties in the state matrix of the systems (64) and (65) arenonlinear and time-variable and the mismatched intercon-nections are also nonlinear and time-variable as shown in(67) and (68) Thus the stability of systems (64) and (65) ismore difficult to ensure than that of [17 30] Therefore theapproaches given in [17 30] are not applicable here FromFigures 3 and 4 we can see that the sliding mode exists forall time Even though the mismatched uncertainties in thestatematrix and interconnections of the systems (64) and (65)are nonlinear and time-variable the systems still exhibit goodperformance with low control energy as seen in Figures 1 25 and 6
5 Conclusion
In this paper a single-phase SMC law is presented for decen-tralized robust stability of complex interconnected systemsfrom the beginning to the end It is proved that the proposedsingle-phase SMC guaranteed the robustness of complexinterconnected system throughout an entire response of thesystem starting from the initial time instance One of thekey features of the single-phase SMC scheme is that reachingtime which is required in most of the existing two phasesof SMC approaches to stabilize the complex interconnectedsystems is removed As a consequence the proposed single-phase SMC law can be applied to complex interconnected
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x11
x12
x13
Figure 1 Time responses of states 11990911(solid) 119909
12(dashed) and 119909
13
(dotted)
0 1 1505 2
0
5
10
Time (s)
Mag
nitu
de
minus5
minus10
x21
x22
x23
Figure 2 Time responses of states 11990921(solid) 119909
22(dashed) and 119909
23
(dotted)
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 3 Sliding surface 1205901
0 1 1505 2Time (s)
Mag
nitu
de
00001
0
minus00001
Figure 4 Sliding surface 1205902
0 1 1505 2Time (s)
Mag
nitu
de30
20
10
0
minus10
minus20
minus30
Figure 5 Control input 1199061
0 1 1505 2Time (s)
Mag
nitu
de
15
10
5
0
minus5
minus10
minus15
Figure 6 Control input 1199062
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
systems which is not always achievable in the traditionalSMC design for complex interconnected systems using onlyoutput variables
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the financial support pro-vided by the National Science Council in Taiwan (NSC 102-2632-E-212-001-MY3)
References
[1] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007
[2] H H Choi ldquoAn explicit formula of linear sliding surfacesfor a class of uncertain dynamic systems with mismatcheduncertaintiesrdquo Automatica vol 34 no 8 pp 1015ndash1020 1998
[3] S H Zak and S Hui ldquoOn variable structure output feedbackcontrollers for uncertain dynamic systemsrdquo IEEE Transactionson Automation Control vol 38 no 10 pp 1509ndash1512 1993
[4] C M Kwan ldquoOn variable structure output feedback con-trollersrdquo IEEE Transactions on Automatic Control vol 41 no11 pp 1691ndash1693 1996
[5] Y-W Tsai K-HMai andK-K Shyu ldquoSlidingmode control forunmatched uncertain systems with totally invariant propertyand exponential stabilityrdquo Journal of the Chinese Institute ofEngineers Transactions of the Chinese Institute of EngineersSeries A vol 29 no 1 pp 179ndash183 2006
[6] R J Mantz H de Battista and P Puleston ldquoA new approach toreaching mode of VSS using trajectory planningrdquo Automaticavol 37 no 5 pp 763ndash767 2001
[7] A Bartoszewicz and A Nowacka-Leverton ldquoSMC without thereaching phasemdashthe switching plane design for the third-ordersystemrdquo IET Control Theory and Applications vol 1 no 5 pp1461ndash1470 2007
[8] J Ackermann and V Utkin ldquoSlidingmode control design basedon Ackermannrsquos formulardquo IEEE Transactions on AutomaticControl vol 43 no 2 pp 234ndash237 1998
[9] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[10] Q Gao L Liu G Feng Y Wang and J Qiu ldquoUniversal fuzzyintegral sliding-mode controllers based on T-S fuzzy modelsrdquoIEEE Transactions on Fuzzy Systems vol 22 no 2 pp 350ndash3622014
[11] D Ginoya P D Shendge and S B Phadke ldquoSliding modecontrol for mismatched uncertain systems using an extendeddisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 61 no 4 pp 1983ndash1992 2014
[12] M Liu L Zhang P Shi and H R Karimi ldquoRobust con-trol of stochastic systems against bounded disturbances withapplication to flight controlrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1504ndash1515 2014
[13] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystemsmdasha new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001
[14] K K Shyu W J Liu and K C Hsu ldquoDecentralised variablestructure control of uncertain large-scale systems containing adead-zonerdquo IEE Proceedings Control Theory and Applicationsvol 150 no 5 pp 467ndash475 2003
[15] K-K Shyu W-J Liu and K-C Hsu ldquoDesign of large-scaletime-delayed systems with dead-zone input via variable struc-ture controlrdquo Automatica vol 41 no 7 pp 1239ndash1246 2005
[16] C-WChung andYChang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory ampApplications vol 5 no 1 pp 221ndash230 2011
[17] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010
[18] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012
[19] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[20] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 115 no 3 pp 551ndash554 1993
[21] K-K Shyu and J-J Yan ldquoVariable-structure model followingadaptive control for systems with time-varying delayrdquo ControlTheory and Advanced Technology vol 10 no 3 pp 513ndash5211994
[22] K-C Hsu ldquoDecentralized variable-structure control designfor uncertain large-scale systems with series nonlinearitiesrdquoInternational Journal of Control vol 68 no 6 pp 1231ndash12401997
[23] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998
[24] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003
[25] C-JWang and J-S Chiou ldquoOn delay independent stabilizationanalysis for a class of switched large-scale time-delay systemsrdquoAbstract and Applied Analysis vol 2013 Article ID 849578 10pages 2013
[26] Y Cui K Liu Y Zhao and X Wang ldquoRobust119867infincontrol for a
class of uncertain switched fuzzy time-delay systems based onT-S modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 234612 8 pages 2013
[27] P P Khargonekar I R Petersen and K Zhou ldquoRobust stabi-lization of uncertain linear systems quadratic stabilizability andHinfin control theoryrdquo IEEE Transactions on Automatic Controlvol 35 no 3 pp 356ndash361 1990
[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[29] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003
[30] X-G Yan C Edwards and S K Spurgeon ldquoDecentralisedrobust sliding mode control for a class of nonlinear intercon-nected systems by static output feedbackrdquo Automatica vol 40no 4 pp 613ndash620 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
