TE-MPE Annual Meeting 15 December 2014 15/12/2014 TE-MPE Annual Meeting3.
Research Article Robust Stability, Stabilization, and...
12
Research Article Robust Stability, Stabilization, and ∞ Control of a Class of Nonlinear Discrete Time Stochastic Systems Tianliang Zhang, 1 Yu-Hong Wang, 1 Xiushan Jiang, 1 and Weihai Zhang 2 1 College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, Shandong 266510, China 2 College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, Shandong 266590, China Correspondence should be addressed to Weihai Zhang; w [email protected] Received 14 November 2015; Revised 20 March 2016; Accepted 31 March 2016 Academic Editor: Mingcong Deng Copyright © 2016 Tianliang Zhang et al. 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 studies robust stability, stabilization, and ∞ control for a class of nonlinear discrete time stochastic systems. Firstly, the easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs). en a robust ∞ state feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but also satisfies robust ∞ performance. Moreover, the previous results of the nonlinearly perturbed discrete stochastic system are generalized to the system with state, control, and external disturbance dependent noise simultaneously. Two numerical examples are given to illustrate the effectiveness of the proposed results. 1. Introduction Stochastic control has been one of the most important research topics in modern control theory. e study of stochastic stability can be traced back to the 1960s; see [1] and the recently well-known monographs [2, 3]. Stability is the first considered problem in system analysis and synthesis, while stabilization is to look for a controller to stabilize an unstable system. ∞ control is one of the most important robust control approaches, which aims to design the controller to restrain the external disturbance below a given level. We refer the reader to [4–9] for stability and stabilization of Itˆ o-type stochastic systems and [10–14] for stability and stabilization of discrete time stochastic systems. Stochastic ∞ control of Itˆ o-type systems starts from [15], which has been extensively studied in recent years; see [16– 20] and the references therein. Discrete time ∞ control with multiplicative noise can be found in [21–25]. Along with the development of computer technology, discrete time difference systems have attracted a great deal of attention, which have been studied extensively; see [26, 27]. e reason is twofold: Firstly, discrete time systems are ideal mathematical models in the study of satellite attitude control [28], mathematical finance [29], single degree of freedom inverted pendulums [21], and gene regulator networks [30]. Secondly, as said in [27], the study for discrete time systems has the advantage over continuous time differential systems from the perspective of computation; moreover, it presents a very good approach to study differential equations and functional differential equations. From the existing works on stability, stabilization, and ∞ control of discrete time stochastic systems with multi- plicative noise, we can find that, except for linear stochastic systems where perfect results have been obtained [22–25], few works are on the stability of the general nonlinear discrete time stochastic system [12] ( + 1) = ( () , () , ) , (0) = 0 (1) or the ∞ control of affine nonlinear discrete time stochastic system [21] ( + 1) = ( ()) + ( ()) () + ℎ ( ()) ] () + [ 1 ( ()) + 1 ( ()) () + ℎ 1 ( ())] () , () = ( ()) . (2) Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 5185784, 11 pages http://dx.doi.org/10.1155/2016/5185784
Transcript of Research Article Robust Stability, Stabilization, and...
Research ArticleRobust Stability Stabilization and119867
infinControl of a Class of
Nonlinear Discrete Time Stochastic Systems
Tianliang Zhang1 Yu-Hong Wang1 Xiushan Jiang1 and Weihai Zhang2
1College of Information and Control Engineering China University of Petroleum (East China) Qingdao Shandong 266510 China2College of Electrical Engineering andAutomation ShandongUniversity of Science andTechnology Qingdao Shandong 266590 China
Correspondence should be addressed to Weihai Zhang w hzhang163com
Received 14 November 2015 Revised 20 March 2016 Accepted 31 March 2016
Academic Editor Mingcong Deng
Copyright copy 2016 Tianliang Zhang et al 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 studies robust stability stabilization and119867infincontrol for a class of nonlinear discrete time stochastic systems Firstly the
easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs) Then arobust119867
infinstate feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but
also satisfies robust 119867infin
performance Moreover the previous results of the nonlinearly perturbed discrete stochastic system aregeneralized to the system with state control and external disturbance dependent noise simultaneously Two numerical examplesare given to illustrate the effectiveness of the proposed results
1 Introduction
Stochastic control has been one of the most importantresearch topics in modern control theory The study ofstochastic stability can be traced back to the 1960s see [1]and the recently well-known monographs [2 3] Stabilityis the first considered problem in system analysis andsynthesis while stabilization is to look for a controller tostabilize an unstable system 119867
infincontrol is one of the most
important robust control approaches which aims to designthe controller to restrain the external disturbance below agiven level We refer the reader to [4ndash9] for stability andstabilization of Ito-type stochastic systems and [10ndash14] forstability and stabilization of discrete time stochastic systemsStochastic 119867
infincontrol of Ito-type systems starts from [15]
which has been extensively studied in recent years see [16ndash20] and the references therein Discrete time119867
infincontrol with
multiplicative noise can be found in [21ndash25]Along with the development of computer technology
discrete time difference systems have attracted a great deal ofattention which have been studied extensively see [26 27]The reason is twofold Firstly discrete time systems are idealmathematical models in the study of satellite attitude control[28] mathematical finance [29] single degree of freedom
inverted pendulums [21] and gene regulator networks [30]Secondly as said in [27] the study for discrete time systemshas the advantage over continuous time differential systemsfrom the perspective of computation moreover it presentsa very good approach to study differential equations andfunctional differential equations
From the existing works on stability stabilization and119867infin
control of discrete time stochastic systems with multi-plicative noise we can find that except for linear stochasticsystemswhere perfect results have been obtained [22ndash25] fewworks are on the stability of the general nonlinear discretetime stochastic system [12]
119909 (119905 + 1) = 119891 (119909 (119905) 119908 (119905) 119905) 119909 (0) = 1199090
(1)
or the119867infincontrol of affine nonlinear discrete time stochastic
system [21]
119909 (119905 + 1)
= 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) + ℎ (119909 (119905)) ] (119905)
+ [1198911(119909 (119905)) + 119892
1(119909 (119905)) 119906 (119905) + ℎ
1(119909 (119905))] 119908 (119905)
119911 (119905) = 119871 (119909 (119905))
(2)
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5185784 11 pageshttpdxdoiorg10115520165185784
2 Mathematical Problems in Engineering
Up to now the results of the deterministic discrete timenonlinear 119867
infincontrol [31] have not been perfectly general-
ized to the above nonlinear stochastic systems For examplealthough some stability results in continuous time Ito systems[3] can be extended to nonlinear discrete stochastic systems[12] the corresponding criteria are not easily applied inpractice this is because the mathematical expectation of thetrajectory is involved in the preconditions In addition [21]tried to discuss a general nonlinear 119867
infincontrol of discrete
time stochastic systems but only the 119867infin
control of a classof norm bounded systems was perfectly solved based onlinear matrix inequality (LMI) approach As said in [32]the general119867
infincontrol of nonlinear discrete time stochastic
multiplicative noise systems remains unsolved We have toadmit such a fact that some research issues of discrete systemsare more difficult to solve than those of continuous timesystems For instance a stochastic maximum principle for Itosystems was obtained in 1990 [33] but a nonlinear discretetime maximum principle has just been presented in [34]
Recently [7 13] investigated the robust quadratic stabilityand feedback stabilization of a class of nonlinear continuetime and discrete time systems respectively where thenonlinear terms are quadratically bounded Such a nonlinearconstraint possesses great practical importance and has beenwidely used in many types of systems such as singularly per-turbed systems with uncertainties [35 36] neutral systemswith nonlinear perturbations [37] impulsive Takagi-Sugenofuzzy systems [38] and some time-delay systems [18] Itshould be pointed out that the small gain theorem can also beused to examine the robustness as done in [39] for the study ofthe simple adaptive control system within the framework ofthe small gain theorem In addition the robustness of a classof nonlinear feedback systems with unknown perturbationswas discussed based on the robust right coprime factorizationand passivity property [40] All these methods are expectedto play important roles in stochastic uncertain119867
infincontrol
This paper deals with a class of nonlinear uncertaindiscrete time stochastic systems for which the system statecontrol input and external disturbance depend on noisesimultaneously which was often called (119909 119906 V)-dependentnoise for short [24] and which mean that not only thesystem state as in [21] but also the control input and externaldisturbance are subject to random noise Hence our con-cerned models have more wide applications The considerednonlinear dynamic term is priorly unknown but belongsto a class of functions with a bounded energy level whichrepresent a kind of very important nonlinear functions andhas been studied by many researchers see for example[41] For such a class of nonlinear discrete time stochasticsystems the stochastic stability stabilization and119867
infincontrol
have been discussed respectively and easily testing criteriahave also been obtained What we have obtained extends theprevious works to more general models
The paper is organized as follows in Section 2 we givea description of the considered nonlinear stochastic sys-tems and define robust stochastic stability and stabilizationSection 3 contains our main results Section 31 presents arobust stability criterion which extends the result of [13] tomore general stochastic systems Section 32 gives a sufficient
condition for robust stabilization criterion Section 33 isabout 119867
infincontrol where an LMI-based sufficient condition
for the existence of a static state feedback 119867infin
controller isestablished All our main results are expressed in terms ofLMIs In Section 4 two examples are constructed to show theeffectiveness of our obtained results
For convenience the notations adopted in this paper areas follows1198721015840 is the transpose of the matrix119872 or vector119872 119872 ge 0
(119872 gt 0) 119872 is a positive semidefinite (positive definite)symmetric matrix 119868 is the identity matrix 119877119899 is the 119899-dimensional Euclidean space 119877119899times119898 is the space of all 119899 times119898 matrices with entries in 119877 119873 is the natural numberset that is 119873 represents 0 1 2 119873
119905denotes the set of
0 1 119905 1198972119908(119873 119877119899
) denotes the set of all nonanticipativesquare summable stochastic processes
119910 = 119910119899 119910119899
isin 1198712
(Ω 119877119899
) 119910119899is F119899minus1
measurable119899isin119873
(3)
The 1198972-norm of 119910 isin 1198972119908(119873 119877119899
) is defined by
100381710038171003817100381711991010038171003817100381710038171198972
119908(119873119877119899)= (
infin
sum
119899=0
1198641003817100381710038171003817119910119899
1003817100381710038171003817
2
)
12
(4)
Similarly 1198972119908(119873119879 119877119899
) and 1199101198972
119908(119873119877119899)can be defined
2 System Descriptions and Definitions
Consider the discrete stochastic iterative system described bythe following equation
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905)) + 119861119906 (119905)
+ (119862119909 (119905) + ℎ2(119905 119909 (119905)) + 119863119906 (119905)) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(5)
where 119909(119905) isin 119877119899 is the 119899-dimensional state vector and 119906(119905) isin119877119898 is the 119898-dimensional control input 119908(119905)
119905ge0is a se-
quence of one-dimensional independent white noise proc-esses defined on the complete filtered probability space (ΩF F
119905119905ge0P) whereF
119905= 120590119908(0) 119908(1) 119908(119905)Assume
that 119864[119908(119905)] = 0 119864[119908(119905)119908(119895)] = 120575119905119895 where 119864 stands for the
expectation operation and 120575119905119895is a Kronecker function defined
by 120575119905119895= 0 for 119905 = 119895 while 120575
119905119895= 1 for 119905 = 119895 Without loss of
generality 1199090is assumed to be determined The following is
assumed to hold throughout this paper
Assumption 1 The nonlinear functions ℎ1(119905 119909(119905)) and
ℎ2(119905 119909(119905)) describe parameter uncertainty of the system and
satisfy the following quadratic inequalities
ℎ1015840
1(119905 119909 (119905)) ℎ
1(119905 119909 (119905)) le 120572
2
11199091015840
(119905)1198671015840
11198671119909 (119905) (6)
ℎ1015840
2(119905 119909 (119905)) ℎ
2(119905 119909 (119905)) le 120572
2
21199091015840
(119905)1198671015840
21198672119909 (119905) (7)
for all 119905 isin 119873 where 120572119894is a constant related to the function ℎ
119894
for 119894 = 1 2119867119894is a constant matrix reflecting structure of ℎ
119894
Mathematical Problems in Engineering 3
We note that inequalities (6) and (7) can be written as amatrix form
[
[
[
119909
ℎ1
ℎ2
]
]
]
1015840
[
[
[
minus1205722
11198671015840
11198671minus 1205722
21198671015840
211986720 0
0 119868 0
0 0 119868
]
]
]
[
[
[
119909
ℎ1
ℎ2
]
]
]
le 0 (8)
System (5) is regarded as the generalized version of thesystem in [13 42] We note that in system (5) the systemstate control input and uncertain terms depend on noisesimultaneously which makes (1) more useful in describingmany practical phenomena
Definition 2 The unforced system (5) with 119906 = 0 is said to berobustly stochastically stable with margins 120572
1gt 0 and 120572
2gt 0
if there exists a constant 120575(1199090 1205721 1205722) such that
119864[
infin
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) (9)
Definition 2 implies 119864119909(119905)2 rarr 0
Definition 3 System (5) is said to be robustly stochasticallystabilizable if there exists a state feedback control law 119906(119905) =119870119909(119905) such that the closed-loop system
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ ((119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(10)
is robustly stochastically stable for all nonlinear functionsℎ119894(119905 119909(119905)) (119894 = 1 2) satisfying (6) and (7)
When there is the external disturbance ](sdot) in system (5)we consider the following nonlinear perturbed system
119909 (119905 + 1)
= 119860119909 (119905) + 1198600] (119905) + 119861119906 (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + 119863119906 (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(11)
where ](119905) isin 119877119902 and 119911(119905) isin 119877119901 are respectively thedisturbance signal and the controlled output ](119905) is assumedto belong to 1198972
119908(119873 119877119902
) so ](119905) is independent of 119908(119905)
Definition 4 (119867infin
control) For a given disturbance attenua-tion level 120574 gt 0 119906(119905) = 119870119909(119905) is an119867
infincontrol of system (11)
if
(i) system (11) is internally stochastically stabilizable for119906(119905) = 119870119909(119905) in the absence of ](119905) that is
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ [(119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))] 119908 (119905)
(12)
is robustly stochastically stable
(ii) The119867infinnorm of system (11) is less than 120574 gt 0 that is
119879 = sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
119911 (119905)1198972
119908(119873119877119901)
] (119905)1198972
119908(119873119877119902)
= sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
(suminfin
119905=0119864 119911 (119905)
2
)
12
(suminfin
119905=0119864 ] (119905)2)
12
lt 120574
(13)
3 Main Results
In this section we give ourmain results on stochastic stabilitystochastic stabilization and robust 119867
infincontrol via LMI-
based approach Firstly we introduce the following twolemmas which will be used in the proof of our main results
Lemma 5 (Schurrsquos lemma) For a real symmetric matrix 119878 =[
1198781111987812
119878119879
1211987822
] the following three conditions are equivalent
(i) 119878 lt 0
(ii) 11987811lt 0 119878
22minus 119878119879
12119878minus1
1111987812lt 0
(iii) 11987822lt 0 119878
11minus 11987812119878minus1
22119878119879
12lt 0
Lemma 6 For any real matrices 119880 1198731015840 = 119873 gt 0 and119882 withappropriate dimensions we have
1198801015840
119873119882+1198821015840
119873119880 le 1198801015840
119873119880 +1198821015840
119873119882 (14)
Proof Because 1198731015840 = 119873 gt 0 1198801015840119873119882 + 1198821015840
119873119880 =
(1198801015840
11987312
)(11987312
119882) + (1198821015840
11987312
)(11987312
119882) Inequality (14) is animmediate corollary of the well-known inequality
1198831015840
119884 + 1198841015840
119883 le 1198831015840
119883 + 1198841015840
119884 (15)
31 Robust Stability Criteria Consider the following un-forced stochastic discrete time system
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(16)
where ℎ1(119905 119909(119905))
119905ge0and ℎ
2(119905 119909(119905))
119905ge0satisfy (8) The
following theorem gives a sufficient condition of robuststochastic stability for system (16)
Theorem 7 System (16) with margins 1205721gt 0 and 120572
2gt
0 is said to be robustly stochastically stable if there exists
4 Mathematical Problems in Engineering
a symmetric positive definite matrix 119876 gt 0 and a scalar 120572 gt 0such that
[
[
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876 0
lowast minus
1
2
119876 0 0
lowast lowast minus
1
2
119876 0
lowast lowast lowast 119876 minus 120572119868
]
]
]
]
]
]
]
]
lt 0
(17)
Proof If (17) holds we set 119881(119909(119905)) = 1199091015840
(119905)119876119909(119905) as aLyapunov function candidate of system (16) where 0 lt 119876 lt120572119868 by (17) Note that 119909(119905) and 119908(119905) are independent so thedifference generator is
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 1199091015840
(119905) (1198601015840
119876119860 + 1198621015840
119876119862 minus 119876)119909 (119905)
+ 1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + 119909
1015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905))
+ ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
(18)
Applying Lemma 6 and inequalities (6)-(7) by 0 lt 119876 lt 120572119868we have
1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 1198601015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
le 1199091015840
(119905) (1198601015840
119876119860 + 1205722
11205721198671015840
119867)119909 (119905)
1199091015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) (1198621015840
119876119862 + 1205722
21205721198671015840
119867)119909 (119905)
ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
le 1199091015840
(119905) (1205722
11205721198671015840
119867 + 1205722
21205721198671015840
119867)119909 (119905)
(19)
Substituting (19) into (18) we achieve that
119864Δ119881 (119909 (119905)) le 119864 1199091015840
(119905) (21198601015840
119876119860 + 21198621015840
119876119862 minus 119876
+ 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867)119909 (119905)
(20)
By Schurrsquos complement
Ω fl 21198601015840119876119860 + 21198621015840119876119862 minus 119876 + 2120572211205721198671015840
119867 + 21205722
21205721198671015840
119867
lt 0
(21)
is equivalent to
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0 (22)
which holds by (17) We denote 120582max(Ω) and 120582min(Ω) to bethe largest and the minimum eigenvalues of the matrix Ωrespectively then (20) yields
119864Δ119881 (119909 (119905)) le 120582max (Ω) 119864 119909 (119905)2
(23)
Taking summation on both sides of the above inequality from119905 = 0 to 119905 = 119879 ge 0 we get
119864 [119881 (119909 (119879))] minus 119881 (1199090) = 119864[
119879
sum
119905=0
Δ119881 (119909 (119905))]
le 120582max (Ω) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)]
(24)
Therefore
120582min (minusΩ) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 119881 (1199090) (25)
which leads to
119864[
119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) fl
119881 (1199090)
120582min (minusΩ) (26)
Hence the robust stochastic stability of system (16) isobtained by (26) via letting 119879 rarr infin
Remark 8 FromTheorem7 if LMI (17) has feasible solutionsthen for any bounded parameters
1and
2on the uncertain
perturbation satisfying 1le 1205721and
2le 1205722 system (16) is
robustly stochastically stable with margins 1and
2
32 Robust Stabilization Criteria In this subsection a suffi-cient condition about robust stochastic stabilization via LMIwill be given
Theorem 9 System (5) with margins 1205721and 120572
2is robustly
stochastically stabilizable if there exist real matrices 119884 and119883 gt0 and a real scalar 120573 gt 0 such that
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 12057211198831198671015840
12057221198831198671015840
1198691015840
11198691015840
20
lowast minus
1
2
120573119868 0 0 0 0
lowast lowast minus
1
2
120573119868 0 0 0
lowast lowast lowast minus
1
2
119883 0 0
lowast lowast lowast lowast minus
1
2
119883 0
lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (27)
holds where
1198691= 119860119883 + 119861119884
1198692= 119862119883 + 119863119884
(28)
In this case 119906(119905) = 119870119909(119905) = 119884119883minus1
119909(119905) is a robustlystochastically stabilizing control law
Mathematical Problems in Engineering 5
Proof We consider synthesizing a state feedback controller119906(119905) = 119870119909(119905) to stabilize system (5) Substituting 119906(119905) = 119870119909(119905)into system (5) yields the closed-loop system described by
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(29)
where 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 By Theorem 7 system(29) is robustly stochastically stable if there exists a matrix119876 0 lt 119876 lt 120572119868 such that the following LMI
Ω fl[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 119860
1015840
119876 119862
1015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0
(30)
holds Let 119876minus1 = 119883 and pre- and postmultiply
diag [119883119883119883] (31)
on both sides of inequality (30) and it yields
[
[
[
[
[
[
minus119883 + 21205722
11205721198831198671015840
119867119883 + 21205722
21205721198831198671015840
119867119883 119883119860
1015840
119883119862
1015840
lowast minus
1
2
119883 0
lowast lowast minus
1
2
119883
]
]
]
]
]
]
lt 0
(32)
In order to transform (32) into a suitable LMI form we set120573 = 1120572 then 0 lt 119876 lt 120572119868 is equivalent to
120573119868 minus 119883 lt 0 120573 gt 0 (33)
Combining (33) with (32) and setting the gain matrix 119870 =119884119883minus1 LMI (27) is obtained The proof is completed
33 119867infinControl In this subsectionmain result about robust
119867infin
control will be given via LMI approach
Theorem 10 Consider system (11) with margins 1205721gt 0 and
1205722gt 0 For the given 120574 gt 0 if there exist real matrices 119883 gt 0
and 119884 and scalar 120573 gt 0 satisfying the following LMI
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 1198711015840
1205722
11198831198671015840
11205722
21198831198671015840
21198691015840
11198691015840
20 0 0 0 119869
1015840
11198691015840
20
lowast minus119868 0 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198601015840
01198621015840
01198721015840
11986001198620
0
lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (34)
where 1198691= 119860119883 + 119861119884 119869
2= 119862119883 + 119863119884 then system (11) is119867
infin
controllable and the robust119867infin
control law is 119906(119905) = 119870119909(119905) =119884119883minus1
119909(119905) for 119905 isin 119873
Proof When ](119905) = 0 byTheorem 9 system (11) is internallystabilizable via 119906(119905) = 119870119909(119905) = 119884119883minus1119909(119905) because LMI (34)implies LMI (27) Next we only need to show 119879 lt 120574
Take 119906(119905) = 119870119909(119905) and choose the Lyapunov function119881(119909(119905)) = 119909
1015840
(119905)119876119909(119905) where
0 lt 119876 lt 120572119868 120572 gt 0 (35)
for some 120572 gt 0 to be determined then for the system
119909 (119905 + 1) = 119860119909 (119905) + 1198600] (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(36)
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
2 Mathematical Problems in Engineering
Up to now the results of the deterministic discrete timenonlinear 119867
infincontrol [31] have not been perfectly general-
ized to the above nonlinear stochastic systems For examplealthough some stability results in continuous time Ito systems[3] can be extended to nonlinear discrete stochastic systems[12] the corresponding criteria are not easily applied inpractice this is because the mathematical expectation of thetrajectory is involved in the preconditions In addition [21]tried to discuss a general nonlinear 119867
infincontrol of discrete
time stochastic systems but only the 119867infin
control of a classof norm bounded systems was perfectly solved based onlinear matrix inequality (LMI) approach As said in [32]the general119867
infincontrol of nonlinear discrete time stochastic
multiplicative noise systems remains unsolved We have toadmit such a fact that some research issues of discrete systemsare more difficult to solve than those of continuous timesystems For instance a stochastic maximum principle for Itosystems was obtained in 1990 [33] but a nonlinear discretetime maximum principle has just been presented in [34]
Recently [7 13] investigated the robust quadratic stabilityand feedback stabilization of a class of nonlinear continuetime and discrete time systems respectively where thenonlinear terms are quadratically bounded Such a nonlinearconstraint possesses great practical importance and has beenwidely used in many types of systems such as singularly per-turbed systems with uncertainties [35 36] neutral systemswith nonlinear perturbations [37] impulsive Takagi-Sugenofuzzy systems [38] and some time-delay systems [18] Itshould be pointed out that the small gain theorem can also beused to examine the robustness as done in [39] for the study ofthe simple adaptive control system within the framework ofthe small gain theorem In addition the robustness of a classof nonlinear feedback systems with unknown perturbationswas discussed based on the robust right coprime factorizationand passivity property [40] All these methods are expectedto play important roles in stochastic uncertain119867
infincontrol
This paper deals with a class of nonlinear uncertaindiscrete time stochastic systems for which the system statecontrol input and external disturbance depend on noisesimultaneously which was often called (119909 119906 V)-dependentnoise for short [24] and which mean that not only thesystem state as in [21] but also the control input and externaldisturbance are subject to random noise Hence our con-cerned models have more wide applications The considerednonlinear dynamic term is priorly unknown but belongsto a class of functions with a bounded energy level whichrepresent a kind of very important nonlinear functions andhas been studied by many researchers see for example[41] For such a class of nonlinear discrete time stochasticsystems the stochastic stability stabilization and119867
infincontrol
have been discussed respectively and easily testing criteriahave also been obtained What we have obtained extends theprevious works to more general models
The paper is organized as follows in Section 2 we givea description of the considered nonlinear stochastic sys-tems and define robust stochastic stability and stabilizationSection 3 contains our main results Section 31 presents arobust stability criterion which extends the result of [13] tomore general stochastic systems Section 32 gives a sufficient
condition for robust stabilization criterion Section 33 isabout 119867
infincontrol where an LMI-based sufficient condition
for the existence of a static state feedback 119867infin
controller isestablished All our main results are expressed in terms ofLMIs In Section 4 two examples are constructed to show theeffectiveness of our obtained results
For convenience the notations adopted in this paper areas follows1198721015840 is the transpose of the matrix119872 or vector119872 119872 ge 0
(119872 gt 0) 119872 is a positive semidefinite (positive definite)symmetric matrix 119868 is the identity matrix 119877119899 is the 119899-dimensional Euclidean space 119877119899times119898 is the space of all 119899 times119898 matrices with entries in 119877 119873 is the natural numberset that is 119873 represents 0 1 2 119873
119905denotes the set of
0 1 119905 1198972119908(119873 119877119899
) denotes the set of all nonanticipativesquare summable stochastic processes
119910 = 119910119899 119910119899
isin 1198712
(Ω 119877119899
) 119910119899is F119899minus1
measurable119899isin119873
(3)
The 1198972-norm of 119910 isin 1198972119908(119873 119877119899
) is defined by
100381710038171003817100381711991010038171003817100381710038171198972
119908(119873119877119899)= (
infin
sum
119899=0
1198641003817100381710038171003817119910119899
1003817100381710038171003817
2
)
12
(4)
Similarly 1198972119908(119873119879 119877119899
) and 1199101198972
119908(119873119877119899)can be defined
2 System Descriptions and Definitions
Consider the discrete stochastic iterative system described bythe following equation
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905)) + 119861119906 (119905)
+ (119862119909 (119905) + ℎ2(119905 119909 (119905)) + 119863119906 (119905)) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(5)
where 119909(119905) isin 119877119899 is the 119899-dimensional state vector and 119906(119905) isin119877119898 is the 119898-dimensional control input 119908(119905)
119905ge0is a se-
quence of one-dimensional independent white noise proc-esses defined on the complete filtered probability space (ΩF F
119905119905ge0P) whereF
119905= 120590119908(0) 119908(1) 119908(119905)Assume
that 119864[119908(119905)] = 0 119864[119908(119905)119908(119895)] = 120575119905119895 where 119864 stands for the
expectation operation and 120575119905119895is a Kronecker function defined
by 120575119905119895= 0 for 119905 = 119895 while 120575
119905119895= 1 for 119905 = 119895 Without loss of
generality 1199090is assumed to be determined The following is
assumed to hold throughout this paper
Assumption 1 The nonlinear functions ℎ1(119905 119909(119905)) and
ℎ2(119905 119909(119905)) describe parameter uncertainty of the system and
satisfy the following quadratic inequalities
ℎ1015840
1(119905 119909 (119905)) ℎ
1(119905 119909 (119905)) le 120572
2
11199091015840
(119905)1198671015840
11198671119909 (119905) (6)
ℎ1015840
2(119905 119909 (119905)) ℎ
2(119905 119909 (119905)) le 120572
2
21199091015840
(119905)1198671015840
21198672119909 (119905) (7)
for all 119905 isin 119873 where 120572119894is a constant related to the function ℎ
119894
for 119894 = 1 2119867119894is a constant matrix reflecting structure of ℎ
119894
Mathematical Problems in Engineering 3
We note that inequalities (6) and (7) can be written as amatrix form
[
[
[
119909
ℎ1
ℎ2
]
]
]
1015840
[
[
[
minus1205722
11198671015840
11198671minus 1205722
21198671015840
211986720 0
0 119868 0
0 0 119868
]
]
]
[
[
[
119909
ℎ1
ℎ2
]
]
]
le 0 (8)
System (5) is regarded as the generalized version of thesystem in [13 42] We note that in system (5) the systemstate control input and uncertain terms depend on noisesimultaneously which makes (1) more useful in describingmany practical phenomena
Definition 2 The unforced system (5) with 119906 = 0 is said to berobustly stochastically stable with margins 120572
1gt 0 and 120572
2gt 0
if there exists a constant 120575(1199090 1205721 1205722) such that
119864[
infin
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) (9)
Definition 2 implies 119864119909(119905)2 rarr 0
Definition 3 System (5) is said to be robustly stochasticallystabilizable if there exists a state feedback control law 119906(119905) =119870119909(119905) such that the closed-loop system
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ ((119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(10)
is robustly stochastically stable for all nonlinear functionsℎ119894(119905 119909(119905)) (119894 = 1 2) satisfying (6) and (7)
When there is the external disturbance ](sdot) in system (5)we consider the following nonlinear perturbed system
119909 (119905 + 1)
= 119860119909 (119905) + 1198600] (119905) + 119861119906 (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + 119863119906 (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(11)
where ](119905) isin 119877119902 and 119911(119905) isin 119877119901 are respectively thedisturbance signal and the controlled output ](119905) is assumedto belong to 1198972
119908(119873 119877119902
) so ](119905) is independent of 119908(119905)
Definition 4 (119867infin
control) For a given disturbance attenua-tion level 120574 gt 0 119906(119905) = 119870119909(119905) is an119867
infincontrol of system (11)
if
(i) system (11) is internally stochastically stabilizable for119906(119905) = 119870119909(119905) in the absence of ](119905) that is
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ [(119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))] 119908 (119905)
(12)
is robustly stochastically stable
(ii) The119867infinnorm of system (11) is less than 120574 gt 0 that is
119879 = sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
119911 (119905)1198972
119908(119873119877119901)
] (119905)1198972
119908(119873119877119902)
= sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
(suminfin
119905=0119864 119911 (119905)
2
)
12
(suminfin
119905=0119864 ] (119905)2)
12
lt 120574
(13)
3 Main Results
In this section we give ourmain results on stochastic stabilitystochastic stabilization and robust 119867
infincontrol via LMI-
based approach Firstly we introduce the following twolemmas which will be used in the proof of our main results
Lemma 5 (Schurrsquos lemma) For a real symmetric matrix 119878 =[
1198781111987812
119878119879
1211987822
] the following three conditions are equivalent
(i) 119878 lt 0
(ii) 11987811lt 0 119878
22minus 119878119879
12119878minus1
1111987812lt 0
(iii) 11987822lt 0 119878
11minus 11987812119878minus1
22119878119879
12lt 0
Lemma 6 For any real matrices 119880 1198731015840 = 119873 gt 0 and119882 withappropriate dimensions we have
1198801015840
119873119882+1198821015840
119873119880 le 1198801015840
119873119880 +1198821015840
119873119882 (14)
Proof Because 1198731015840 = 119873 gt 0 1198801015840119873119882 + 1198821015840
119873119880 =
(1198801015840
11987312
)(11987312
119882) + (1198821015840
11987312
)(11987312
119882) Inequality (14) is animmediate corollary of the well-known inequality
1198831015840
119884 + 1198841015840
119883 le 1198831015840
119883 + 1198841015840
119884 (15)
31 Robust Stability Criteria Consider the following un-forced stochastic discrete time system
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(16)
where ℎ1(119905 119909(119905))
119905ge0and ℎ
2(119905 119909(119905))
119905ge0satisfy (8) The
following theorem gives a sufficient condition of robuststochastic stability for system (16)
Theorem 7 System (16) with margins 1205721gt 0 and 120572
2gt
0 is said to be robustly stochastically stable if there exists
4 Mathematical Problems in Engineering
a symmetric positive definite matrix 119876 gt 0 and a scalar 120572 gt 0such that
[
[
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876 0
lowast minus
1
2
119876 0 0
lowast lowast minus
1
2
119876 0
lowast lowast lowast 119876 minus 120572119868
]
]
]
]
]
]
]
]
lt 0
(17)
Proof If (17) holds we set 119881(119909(119905)) = 1199091015840
(119905)119876119909(119905) as aLyapunov function candidate of system (16) where 0 lt 119876 lt120572119868 by (17) Note that 119909(119905) and 119908(119905) are independent so thedifference generator is
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 1199091015840
(119905) (1198601015840
119876119860 + 1198621015840
119876119862 minus 119876)119909 (119905)
+ 1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + 119909
1015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905))
+ ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
(18)
Applying Lemma 6 and inequalities (6)-(7) by 0 lt 119876 lt 120572119868we have
1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 1198601015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
le 1199091015840
(119905) (1198601015840
119876119860 + 1205722
11205721198671015840
119867)119909 (119905)
1199091015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) (1198621015840
119876119862 + 1205722
21205721198671015840
119867)119909 (119905)
ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
le 1199091015840
(119905) (1205722
11205721198671015840
119867 + 1205722
21205721198671015840
119867)119909 (119905)
(19)
Substituting (19) into (18) we achieve that
119864Δ119881 (119909 (119905)) le 119864 1199091015840
(119905) (21198601015840
119876119860 + 21198621015840
119876119862 minus 119876
+ 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867)119909 (119905)
(20)
By Schurrsquos complement
Ω fl 21198601015840119876119860 + 21198621015840119876119862 minus 119876 + 2120572211205721198671015840
119867 + 21205722
21205721198671015840
119867
lt 0
(21)
is equivalent to
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0 (22)
which holds by (17) We denote 120582max(Ω) and 120582min(Ω) to bethe largest and the minimum eigenvalues of the matrix Ωrespectively then (20) yields
119864Δ119881 (119909 (119905)) le 120582max (Ω) 119864 119909 (119905)2
(23)
Taking summation on both sides of the above inequality from119905 = 0 to 119905 = 119879 ge 0 we get
119864 [119881 (119909 (119879))] minus 119881 (1199090) = 119864[
119879
sum
119905=0
Δ119881 (119909 (119905))]
le 120582max (Ω) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)]
(24)
Therefore
120582min (minusΩ) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 119881 (1199090) (25)
which leads to
119864[
119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) fl
119881 (1199090)
120582min (minusΩ) (26)
Hence the robust stochastic stability of system (16) isobtained by (26) via letting 119879 rarr infin
Remark 8 FromTheorem7 if LMI (17) has feasible solutionsthen for any bounded parameters
1and
2on the uncertain
perturbation satisfying 1le 1205721and
2le 1205722 system (16) is
robustly stochastically stable with margins 1and
2
32 Robust Stabilization Criteria In this subsection a suffi-cient condition about robust stochastic stabilization via LMIwill be given
Theorem 9 System (5) with margins 1205721and 120572
2is robustly
stochastically stabilizable if there exist real matrices 119884 and119883 gt0 and a real scalar 120573 gt 0 such that
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 12057211198831198671015840
12057221198831198671015840
1198691015840
11198691015840
20
lowast minus
1
2
120573119868 0 0 0 0
lowast lowast minus
1
2
120573119868 0 0 0
lowast lowast lowast minus
1
2
119883 0 0
lowast lowast lowast lowast minus
1
2
119883 0
lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (27)
holds where
1198691= 119860119883 + 119861119884
1198692= 119862119883 + 119863119884
(28)
In this case 119906(119905) = 119870119909(119905) = 119884119883minus1
119909(119905) is a robustlystochastically stabilizing control law
Mathematical Problems in Engineering 5
Proof We consider synthesizing a state feedback controller119906(119905) = 119870119909(119905) to stabilize system (5) Substituting 119906(119905) = 119870119909(119905)into system (5) yields the closed-loop system described by
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(29)
where 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 By Theorem 7 system(29) is robustly stochastically stable if there exists a matrix119876 0 lt 119876 lt 120572119868 such that the following LMI
Ω fl[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 119860
1015840
119876 119862
1015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0
(30)
holds Let 119876minus1 = 119883 and pre- and postmultiply
diag [119883119883119883] (31)
on both sides of inequality (30) and it yields
[
[
[
[
[
[
minus119883 + 21205722
11205721198831198671015840
119867119883 + 21205722
21205721198831198671015840
119867119883 119883119860
1015840
119883119862
1015840
lowast minus
1
2
119883 0
lowast lowast minus
1
2
119883
]
]
]
]
]
]
lt 0
(32)
In order to transform (32) into a suitable LMI form we set120573 = 1120572 then 0 lt 119876 lt 120572119868 is equivalent to
120573119868 minus 119883 lt 0 120573 gt 0 (33)
Combining (33) with (32) and setting the gain matrix 119870 =119884119883minus1 LMI (27) is obtained The proof is completed
33 119867infinControl In this subsectionmain result about robust
119867infin
control will be given via LMI approach
Theorem 10 Consider system (11) with margins 1205721gt 0 and
1205722gt 0 For the given 120574 gt 0 if there exist real matrices 119883 gt 0
and 119884 and scalar 120573 gt 0 satisfying the following LMI
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 1198711015840
1205722
11198831198671015840
11205722
21198831198671015840
21198691015840
11198691015840
20 0 0 0 119869
1015840
11198691015840
20
lowast minus119868 0 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198601015840
01198621015840
01198721015840
11986001198620
0
lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (34)
where 1198691= 119860119883 + 119861119884 119869
2= 119862119883 + 119863119884 then system (11) is119867
infin
controllable and the robust119867infin
control law is 119906(119905) = 119870119909(119905) =119884119883minus1
119909(119905) for 119905 isin 119873
Proof When ](119905) = 0 byTheorem 9 system (11) is internallystabilizable via 119906(119905) = 119870119909(119905) = 119884119883minus1119909(119905) because LMI (34)implies LMI (27) Next we only need to show 119879 lt 120574
Take 119906(119905) = 119870119909(119905) and choose the Lyapunov function119881(119909(119905)) = 119909
1015840
(119905)119876119909(119905) where
0 lt 119876 lt 120572119868 120572 gt 0 (35)
for some 120572 gt 0 to be determined then for the system
119909 (119905 + 1) = 119860119909 (119905) + 1198600] (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(36)
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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 3
We note that inequalities (6) and (7) can be written as amatrix form
[
[
[
119909
ℎ1
ℎ2
]
]
]
1015840
[
[
[
minus1205722
11198671015840
11198671minus 1205722
21198671015840
211986720 0
0 119868 0
0 0 119868
]
]
]
[
[
[
119909
ℎ1
ℎ2
]
]
]
le 0 (8)
System (5) is regarded as the generalized version of thesystem in [13 42] We note that in system (5) the systemstate control input and uncertain terms depend on noisesimultaneously which makes (1) more useful in describingmany practical phenomena
Definition 2 The unforced system (5) with 119906 = 0 is said to berobustly stochastically stable with margins 120572
1gt 0 and 120572
2gt 0
if there exists a constant 120575(1199090 1205721 1205722) such that
119864[
infin
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) (9)
Definition 2 implies 119864119909(119905)2 rarr 0
Definition 3 System (5) is said to be robustly stochasticallystabilizable if there exists a state feedback control law 119906(119905) =119870119909(119905) such that the closed-loop system
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ ((119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(10)
is robustly stochastically stable for all nonlinear functionsℎ119894(119905 119909(119905)) (119894 = 1 2) satisfying (6) and (7)
When there is the external disturbance ](sdot) in system (5)we consider the following nonlinear perturbed system
119909 (119905 + 1)
= 119860119909 (119905) + 1198600] (119905) + 119861119906 (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + 119863119906 (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(11)
where ](119905) isin 119877119902 and 119911(119905) isin 119877119901 are respectively thedisturbance signal and the controlled output ](119905) is assumedto belong to 1198972
119908(119873 119877119902
) so ](119905) is independent of 119908(119905)
Definition 4 (119867infin
control) For a given disturbance attenua-tion level 120574 gt 0 119906(119905) = 119870119909(119905) is an119867
infincontrol of system (11)
if
(i) system (11) is internally stochastically stabilizable for119906(119905) = 119870119909(119905) in the absence of ](119905) that is
119909 (119905 + 1) = (119860 + 119861119870) 119909 (119905) + ℎ1(119905 119909 (119905))
+ [(119862 + 119863119870) 119909 (119905) + ℎ2(119905 119909 (119905))] 119908 (119905)
(12)
is robustly stochastically stable
(ii) The119867infinnorm of system (11) is less than 120574 gt 0 that is
119879 = sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
119911 (119905)1198972
119908(119873119877119901)
] (119905)1198972
119908(119873119877119902)
= sup]isin1198972119908(119873119877119902)119906(119905)=119870119909(119905)] =0119909
0=0
(suminfin
119905=0119864 119911 (119905)
2
)
12
(suminfin
119905=0119864 ] (119905)2)
12
lt 120574
(13)
3 Main Results
In this section we give ourmain results on stochastic stabilitystochastic stabilization and robust 119867
infincontrol via LMI-
based approach Firstly we introduce the following twolemmas which will be used in the proof of our main results
Lemma 5 (Schurrsquos lemma) For a real symmetric matrix 119878 =[
1198781111987812
119878119879
1211987822
] the following three conditions are equivalent
(i) 119878 lt 0
(ii) 11987811lt 0 119878
22minus 119878119879
12119878minus1
1111987812lt 0
(iii) 11987822lt 0 119878
11minus 11987812119878minus1
22119878119879
12lt 0
Lemma 6 For any real matrices 119880 1198731015840 = 119873 gt 0 and119882 withappropriate dimensions we have
1198801015840
119873119882+1198821015840
119873119880 le 1198801015840
119873119880 +1198821015840
119873119882 (14)
Proof Because 1198731015840 = 119873 gt 0 1198801015840119873119882 + 1198821015840
119873119880 =
(1198801015840
11987312
)(11987312
119882) + (1198821015840
11987312
)(11987312
119882) Inequality (14) is animmediate corollary of the well-known inequality
1198831015840
119884 + 1198841015840
119883 le 1198831015840
119883 + 1198841015840
119884 (15)
31 Robust Stability Criteria Consider the following un-forced stochastic discrete time system
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(16)
where ℎ1(119905 119909(119905))
119905ge0and ℎ
2(119905 119909(119905))
119905ge0satisfy (8) The
following theorem gives a sufficient condition of robuststochastic stability for system (16)
Theorem 7 System (16) with margins 1205721gt 0 and 120572
2gt
0 is said to be robustly stochastically stable if there exists
4 Mathematical Problems in Engineering
a symmetric positive definite matrix 119876 gt 0 and a scalar 120572 gt 0such that
[
[
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876 0
lowast minus
1
2
119876 0 0
lowast lowast minus
1
2
119876 0
lowast lowast lowast 119876 minus 120572119868
]
]
]
]
]
]
]
]
lt 0
(17)
Proof If (17) holds we set 119881(119909(119905)) = 1199091015840
(119905)119876119909(119905) as aLyapunov function candidate of system (16) where 0 lt 119876 lt120572119868 by (17) Note that 119909(119905) and 119908(119905) are independent so thedifference generator is
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 1199091015840
(119905) (1198601015840
119876119860 + 1198621015840
119876119862 minus 119876)119909 (119905)
+ 1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + 119909
1015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905))
+ ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
(18)
Applying Lemma 6 and inequalities (6)-(7) by 0 lt 119876 lt 120572119868we have
1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 1198601015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
le 1199091015840
(119905) (1198601015840
119876119860 + 1205722
11205721198671015840
119867)119909 (119905)
1199091015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) (1198621015840
119876119862 + 1205722
21205721198671015840
119867)119909 (119905)
ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
le 1199091015840
(119905) (1205722
11205721198671015840
119867 + 1205722
21205721198671015840
119867)119909 (119905)
(19)
Substituting (19) into (18) we achieve that
119864Δ119881 (119909 (119905)) le 119864 1199091015840
(119905) (21198601015840
119876119860 + 21198621015840
119876119862 minus 119876
+ 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867)119909 (119905)
(20)
By Schurrsquos complement
Ω fl 21198601015840119876119860 + 21198621015840119876119862 minus 119876 + 2120572211205721198671015840
119867 + 21205722
21205721198671015840
119867
lt 0
(21)
is equivalent to
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0 (22)
which holds by (17) We denote 120582max(Ω) and 120582min(Ω) to bethe largest and the minimum eigenvalues of the matrix Ωrespectively then (20) yields
119864Δ119881 (119909 (119905)) le 120582max (Ω) 119864 119909 (119905)2
(23)
Taking summation on both sides of the above inequality from119905 = 0 to 119905 = 119879 ge 0 we get
119864 [119881 (119909 (119879))] minus 119881 (1199090) = 119864[
119879
sum
119905=0
Δ119881 (119909 (119905))]
le 120582max (Ω) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)]
(24)
Therefore
120582min (minusΩ) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 119881 (1199090) (25)
which leads to
119864[
119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) fl
119881 (1199090)
120582min (minusΩ) (26)
Hence the robust stochastic stability of system (16) isobtained by (26) via letting 119879 rarr infin
Remark 8 FromTheorem7 if LMI (17) has feasible solutionsthen for any bounded parameters
1and
2on the uncertain
perturbation satisfying 1le 1205721and
2le 1205722 system (16) is
robustly stochastically stable with margins 1and
2
32 Robust Stabilization Criteria In this subsection a suffi-cient condition about robust stochastic stabilization via LMIwill be given
Theorem 9 System (5) with margins 1205721and 120572
2is robustly
stochastically stabilizable if there exist real matrices 119884 and119883 gt0 and a real scalar 120573 gt 0 such that
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 12057211198831198671015840
12057221198831198671015840
1198691015840
11198691015840
20
lowast minus
1
2
120573119868 0 0 0 0
lowast lowast minus
1
2
120573119868 0 0 0
lowast lowast lowast minus
1
2
119883 0 0
lowast lowast lowast lowast minus
1
2
119883 0
lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (27)
holds where
1198691= 119860119883 + 119861119884
1198692= 119862119883 + 119863119884
(28)
In this case 119906(119905) = 119870119909(119905) = 119884119883minus1
119909(119905) is a robustlystochastically stabilizing control law
Mathematical Problems in Engineering 5
Proof We consider synthesizing a state feedback controller119906(119905) = 119870119909(119905) to stabilize system (5) Substituting 119906(119905) = 119870119909(119905)into system (5) yields the closed-loop system described by
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(29)
where 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 By Theorem 7 system(29) is robustly stochastically stable if there exists a matrix119876 0 lt 119876 lt 120572119868 such that the following LMI
Ω fl[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 119860
1015840
119876 119862
1015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0
(30)
holds Let 119876minus1 = 119883 and pre- and postmultiply
diag [119883119883119883] (31)
on both sides of inequality (30) and it yields
[
[
[
[
[
[
minus119883 + 21205722
11205721198831198671015840
119867119883 + 21205722
21205721198831198671015840
119867119883 119883119860
1015840
119883119862
1015840
lowast minus
1
2
119883 0
lowast lowast minus
1
2
119883
]
]
]
]
]
]
lt 0
(32)
In order to transform (32) into a suitable LMI form we set120573 = 1120572 then 0 lt 119876 lt 120572119868 is equivalent to
120573119868 minus 119883 lt 0 120573 gt 0 (33)
Combining (33) with (32) and setting the gain matrix 119870 =119884119883minus1 LMI (27) is obtained The proof is completed
33 119867infinControl In this subsectionmain result about robust
119867infin
control will be given via LMI approach
Theorem 10 Consider system (11) with margins 1205721gt 0 and
1205722gt 0 For the given 120574 gt 0 if there exist real matrices 119883 gt 0
and 119884 and scalar 120573 gt 0 satisfying the following LMI
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 1198711015840
1205722
11198831198671015840
11205722
21198831198671015840
21198691015840
11198691015840
20 0 0 0 119869
1015840
11198691015840
20
lowast minus119868 0 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198601015840
01198621015840
01198721015840
11986001198620
0
lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (34)
where 1198691= 119860119883 + 119861119884 119869
2= 119862119883 + 119863119884 then system (11) is119867
infin
controllable and the robust119867infin
control law is 119906(119905) = 119870119909(119905) =119884119883minus1
119909(119905) for 119905 isin 119873
Proof When ](119905) = 0 byTheorem 9 system (11) is internallystabilizable via 119906(119905) = 119870119909(119905) = 119884119883minus1119909(119905) because LMI (34)implies LMI (27) Next we only need to show 119879 lt 120574
Take 119906(119905) = 119870119909(119905) and choose the Lyapunov function119881(119909(119905)) = 119909
1015840
(119905)119876119909(119905) where
0 lt 119876 lt 120572119868 120572 gt 0 (35)
for some 120572 gt 0 to be determined then for the system
119909 (119905 + 1) = 119860119909 (119905) + 1198600] (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(36)
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
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
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
4 Mathematical Problems in Engineering
a symmetric positive definite matrix 119876 gt 0 and a scalar 120572 gt 0such that
[
[
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876 0
lowast minus
1
2
119876 0 0
lowast lowast minus
1
2
119876 0
lowast lowast lowast 119876 minus 120572119868
]
]
]
]
]
]
]
]
lt 0
(17)
Proof If (17) holds we set 119881(119909(119905)) = 1199091015840
(119905)119876119909(119905) as aLyapunov function candidate of system (16) where 0 lt 119876 lt120572119868 by (17) Note that 119909(119905) and 119908(119905) are independent so thedifference generator is
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 1199091015840
(119905) (1198601015840
119876119860 + 1198621015840
119876119862 minus 119876)119909 (119905)
+ 1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + 119909
1015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905))
+ ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
(18)
Applying Lemma 6 and inequalities (6)-(7) by 0 lt 119876 lt 120572119868we have
1199091015840
(119905) 1198601015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 1198601015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
le 1199091015840
(119905) (1198601015840
119876119860 + 1205722
11205721198671015840
119867)119909 (119905)
1199091015840
(119905) 1198621015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) (1198621015840
119876119862 + 1205722
21205721198671015840
119867)119909 (119905)
ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
le 1199091015840
(119905) (1205722
11205721198671015840
119867 + 1205722
21205721198671015840
119867)119909 (119905)
(19)
Substituting (19) into (18) we achieve that
119864Δ119881 (119909 (119905)) le 119864 1199091015840
(119905) (21198601015840
119876119860 + 21198621015840
119876119862 minus 119876
+ 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867)119909 (119905)
(20)
By Schurrsquos complement
Ω fl 21198601015840119876119860 + 21198621015840119876119862 minus 119876 + 2120572211205721198671015840
119867 + 21205722
21205721198671015840
119867
lt 0
(21)
is equivalent to
[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 1198601015840
119876 1198621015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0 (22)
which holds by (17) We denote 120582max(Ω) and 120582min(Ω) to bethe largest and the minimum eigenvalues of the matrix Ωrespectively then (20) yields
119864Δ119881 (119909 (119905)) le 120582max (Ω) 119864 119909 (119905)2
(23)
Taking summation on both sides of the above inequality from119905 = 0 to 119905 = 119879 ge 0 we get
119864 [119881 (119909 (119879))] minus 119881 (1199090) = 119864[
119879
sum
119905=0
Δ119881 (119909 (119905))]
le 120582max (Ω) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)]
(24)
Therefore
120582min (minusΩ) 119864[119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 119881 (1199090) (25)
which leads to
119864[
119879
sum
119905=0
1199091015840
(119905) 119909 (119905)] le 120575 (1199090 1205721 1205722) fl
119881 (1199090)
120582min (minusΩ) (26)
Hence the robust stochastic stability of system (16) isobtained by (26) via letting 119879 rarr infin
Remark 8 FromTheorem7 if LMI (17) has feasible solutionsthen for any bounded parameters
1and
2on the uncertain
perturbation satisfying 1le 1205721and
2le 1205722 system (16) is
robustly stochastically stable with margins 1and
2
32 Robust Stabilization Criteria In this subsection a suffi-cient condition about robust stochastic stabilization via LMIwill be given
Theorem 9 System (5) with margins 1205721and 120572
2is robustly
stochastically stabilizable if there exist real matrices 119884 and119883 gt0 and a real scalar 120573 gt 0 such that
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 12057211198831198671015840
12057221198831198671015840
1198691015840
11198691015840
20
lowast minus
1
2
120573119868 0 0 0 0
lowast lowast minus
1
2
120573119868 0 0 0
lowast lowast lowast minus
1
2
119883 0 0
lowast lowast lowast lowast minus
1
2
119883 0
lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (27)
holds where
1198691= 119860119883 + 119861119884
1198692= 119862119883 + 119863119884
(28)
In this case 119906(119905) = 119870119909(119905) = 119884119883minus1
119909(119905) is a robustlystochastically stabilizing control law
Mathematical Problems in Engineering 5
Proof We consider synthesizing a state feedback controller119906(119905) = 119870119909(119905) to stabilize system (5) Substituting 119906(119905) = 119870119909(119905)into system (5) yields the closed-loop system described by
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(29)
where 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 By Theorem 7 system(29) is robustly stochastically stable if there exists a matrix119876 0 lt 119876 lt 120572119868 such that the following LMI
Ω fl[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 119860
1015840
119876 119862
1015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0
(30)
holds Let 119876minus1 = 119883 and pre- and postmultiply
diag [119883119883119883] (31)
on both sides of inequality (30) and it yields
[
[
[
[
[
[
minus119883 + 21205722
11205721198831198671015840
119867119883 + 21205722
21205721198831198671015840
119867119883 119883119860
1015840
119883119862
1015840
lowast minus
1
2
119883 0
lowast lowast minus
1
2
119883
]
]
]
]
]
]
lt 0
(32)
In order to transform (32) into a suitable LMI form we set120573 = 1120572 then 0 lt 119876 lt 120572119868 is equivalent to
120573119868 minus 119883 lt 0 120573 gt 0 (33)
Combining (33) with (32) and setting the gain matrix 119870 =119884119883minus1 LMI (27) is obtained The proof is completed
33 119867infinControl In this subsectionmain result about robust
119867infin
control will be given via LMI approach
Theorem 10 Consider system (11) with margins 1205721gt 0 and
1205722gt 0 For the given 120574 gt 0 if there exist real matrices 119883 gt 0
and 119884 and scalar 120573 gt 0 satisfying the following LMI
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 1198711015840
1205722
11198831198671015840
11205722
21198831198671015840
21198691015840
11198691015840
20 0 0 0 119869
1015840
11198691015840
20
lowast minus119868 0 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198601015840
01198621015840
01198721015840
11986001198620
0
lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (34)
where 1198691= 119860119883 + 119861119884 119869
2= 119862119883 + 119863119884 then system (11) is119867
infin
controllable and the robust119867infin
control law is 119906(119905) = 119870119909(119905) =119884119883minus1
119909(119905) for 119905 isin 119873
Proof When ](119905) = 0 byTheorem 9 system (11) is internallystabilizable via 119906(119905) = 119870119909(119905) = 119884119883minus1119909(119905) because LMI (34)implies LMI (27) Next we only need to show 119879 lt 120574
Take 119906(119905) = 119870119909(119905) and choose the Lyapunov function119881(119909(119905)) = 119909
1015840
(119905)119876119909(119905) where
0 lt 119876 lt 120572119868 120572 gt 0 (35)
for some 120572 gt 0 to be determined then for the system
119909 (119905 + 1) = 119860119909 (119905) + 1198600] (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(36)
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
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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 5
Proof We consider synthesizing a state feedback controller119906(119905) = 119870119909(119905) to stabilize system (5) Substituting 119906(119905) = 119870119909(119905)into system (5) yields the closed-loop system described by
119909 (119905 + 1) = 119860119909 (119905) + ℎ1(119905 119909 (119905))
+ (119862119909 (119905) + ℎ2(119905 119909 (119905))) 119908 (119905)
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(29)
where 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 By Theorem 7 system(29) is robustly stochastically stable if there exists a matrix119876 0 lt 119876 lt 120572119868 such that the following LMI
Ω fl[
[
[
[
[
[
minus119876 + 21205722
11205721198671015840
119867 + 21205722
21205721198671015840
119867 119860
1015840
119876 119862
1015840
119876
lowast minus
1
2
119876 0
lowast lowast minus
1
2
119876
]
]
]
]
]
]
lt 0
(30)
holds Let 119876minus1 = 119883 and pre- and postmultiply
diag [119883119883119883] (31)
on both sides of inequality (30) and it yields
[
[
[
[
[
[
minus119883 + 21205722
11205721198831198671015840
119867119883 + 21205722
21205721198831198671015840
119867119883 119883119860
1015840
119883119862
1015840
lowast minus
1
2
119883 0
lowast lowast minus
1
2
119883
]
]
]
]
]
]
lt 0
(32)
In order to transform (32) into a suitable LMI form we set120573 = 1120572 then 0 lt 119876 lt 120572119868 is equivalent to
120573119868 minus 119883 lt 0 120573 gt 0 (33)
Combining (33) with (32) and setting the gain matrix 119870 =119884119883minus1 LMI (27) is obtained The proof is completed
33 119867infinControl In this subsectionmain result about robust
119867infin
control will be given via LMI approach
Theorem 10 Consider system (11) with margins 1205721gt 0 and
1205722gt 0 For the given 120574 gt 0 if there exist real matrices 119883 gt 0
and 119884 and scalar 120573 gt 0 satisfying the following LMI
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119883 1198711015840
1205722
11198831198671015840
11205722
21198831198671015840
21198691015840
11198691015840
20 0 0 0 119869
1015840
11198691015840
20
lowast minus119868 0 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3
120573119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119883 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198601015840
01198621015840
01198721015840
11986001198620
0
lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119883 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast 120573119868 minus 119883
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (34)
where 1198691= 119860119883 + 119861119884 119869
2= 119862119883 + 119863119884 then system (11) is119867
infin
controllable and the robust119867infin
control law is 119906(119905) = 119870119909(119905) =119884119883minus1
119909(119905) for 119905 isin 119873
Proof When ](119905) = 0 byTheorem 9 system (11) is internallystabilizable via 119906(119905) = 119870119909(119905) = 119884119883minus1119909(119905) because LMI (34)implies LMI (27) Next we only need to show 119879 lt 120574
Take 119906(119905) = 119870119909(119905) and choose the Lyapunov function119881(119909(119905)) = 119909
1015840
(119905)119876119909(119905) where
0 lt 119876 lt 120572119868 120572 gt 0 (35)
for some 120572 gt 0 to be determined then for the system
119909 (119905 + 1) = 119860119909 (119905) + 1198600] (119905) + ℎ
1(119905 119909 (119905))
+ (119862119909 (119905) + 1198620] (119905) + ℎ
2(119905 119909 (119905))) 119908 (119905)
119911 (119905) = [
119871119909 (119905)
119872] (119905)]
119909 (0) = 1199090isin 119877119899
119905 isin 119873
(36)
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
with 119860 = 119860 + 119861119870 and 119862 = 119862 + 119863119870 we have with 119909(119905) and](119905) independent of 119908(119905) in mind that
119864Δ119881 (119909 (119905)) = 119864 [119881 (119909 (119905 + 1)) minus 119881 (119909 (119905))]
= 119864 [1199091015840
(119905 + 1)119876119909 (119905 + 1) minus 1199091015840
(119905) 119876119909 (119905)]
= 119864 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) (119860101584001198761198600+ 1198621015840
01198761198620) ] (119905)
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
(37)
Set 1199090= 0 and then for any ](119905) isin 1198972
119908(119873 119877119901
)
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)
= 119864
119879
sum
119905=0
[1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + Δ119881 (119905)] minus 1199091015840 (119879)119876119909 (119879)
le 119864
119879
sum
119905=0
1199091015840
(119905) 1198711015840
119871119909 (119905) + ]1015840 (119905)1198721015840119872] (119905)
minus 1205742]1015840 (119905) ] (119905) + ]1015840 (119905) (1198601015840
01198761198600+ 1198621015840
01198761198620) ] (119905)
+ 1199091015840
(119905) [119860
1015840
119876119860 + 119862
1015840
119876119862 minus 119876]119909 (119905)
+ 1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + 119909
1015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905))
+ 1199091015840
(119905) [1198601198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ℎ1015840
1(119905 119909 (119905)) 119876119860119909 (119905) + ℎ
1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
+ ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ℎ1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
+ ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862
0] (119905)
+ ]1015840 (119905) 11986010158400119876ℎ1(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
+ ]1015840 (119905) [11986010158400119876119860 + 119862
1015840
0119876119862] 119909 (119905)
(38)
Using Lemma 6 and setting 1199091015840(119905)1198601015840 = 1198801015840 ℎ1(119905 119909(119905)) = 119882
and119873 = 119876 gt 0 we have
1199091015840
(119905) 119860
1015840
119876ℎ1(119905 119909 (119905)) + ℎ
1015840
1(119905 119909 (119905)) 119876119860119909 (119905)
le 1199091015840
(119905) 119860
1015840
119876119860119909 (119905) + ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905))
(39)
Similarly the following inequalities can also be obtained
1199091015840
(119905) 119862
1015840
119876ℎ2(119905 119909 (119905)) + ℎ
1015840
2(119905 119909 (119905)) 119876119862119909 (119905)
le 1199091015840
(119905) 119862
1015840
119876119862119909 (119905) + ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905))
ℎ1015840
1(119905 119909 (119905)) 119876119860
0] (119905) + ]1015840 (119905) 1198601015840
0119876ℎ1(119905 119909 (119905))
le ℎ1015840
1(119905 119909 (119905)) 119876ℎ
1(119905 119909 (119905)) + ]1015840 (119905) 1198601015840
01198761198600] (119905)
ℎ1015840
2(119905 119909 (119905)) 119876119862
0] (119905) + ]1015840 (119905) 1198621015840
0119876ℎ2(119905 119909 (119905))
le ℎ1015840
2(119905 119909 (119905)) 119876ℎ
2(119905 119909 (119905)) + ]1015840 (119905) 1198621015840
01198761198620] (119905)
(40)
Substituting (39)-(40) into inequality (38) and considering(35) it yields
119911 (119905)2
1198972
119908(119873119879119877119901)minus 1205742
] (119905)21198972
119908(119873119879119877119902)le 119864
119879
sum
119905=0
1199091015840
(119905) [minus119876
+ 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672] 119909 (119905) + 119909
1015840
(119905) [119860
1015840
1198761198600+ 119862
1015840
1198761198620] ] (119905)
+ ]1015840 (119905) [1198600119876119860 + 119862
0119876119862] 119909
1015840
(119905) + ] (119905) (2119860101584001198761198600
+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872) ] (119905) = 119864119879
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
sdot Ξ [
119909 (119905)
] (119905)]
(41)
where
Ξ fl [Ξ11Ξ12
lowast Ξ22
] (42)
with
Ξ11= minus119876 + 2119860
1015840
119876119860 + 2119862
1015840
119876119862 + 1198711015840
119871 + 31205722
11205721198671015840
11198671
+ 31205722
21205721198671015840
21198672
Ξ12= 119860
1015840
1198761198600+ 119862
1015840
1198761198620
Ξ22= 21198601015840
01198761198600+ 21198621015840
01198761198620minus 1205742
119868 +1198721015840
119872
(43)
Let 119879 rarr infin in (41) then we have
119911 (119905)2
1198972
119908(119873119877119901)minus 1205742
] (119905)21198972
119908(119873119877119902)
le 119864
infin
sum
119905=0
[
119909 (119905)
] (119905)]
1015840
Ξ[
119909 (119905)
] (119905)]
(44)
Mathematical Problems in Engineering 7
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
It is easy to see that if Ξ lt 0 then 119879 lt 120574 for system (11)Next we give an LMI sufficient condition for Ξ lt 0 Noticethat
Ξ = [
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
lowast Ξ22minus 1198601015840
01198761198600
]
+ [
119860
1015840
1198601015840
0
]119876 [119860 + 119861119870 1198600] lt 0
lArrrArr
[
[
[
[
Ξ11minus 119860
1015840
119876119860 119862
1015840
1198761198620
119860
1015840
lowast Ξ22minus 1198601015840
011987611986001198600
lowast lowast minus119876minus1
]
]
]
]
lt 0
lArrrArr
[
[
[
[
[
[
[
Ξ110 119860
1015840
119862
1015840
lowast Ξ2211986001198620
lowast lowast minus119876minus1
0
lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
lt 0
(45)
where
Ξ11= Ξ11minus 119860
1015840
119876119860 minus 119862
1015840
119876119862
Ξ22= Ξ22minus 1198601015840
01198761198600minus 1198621015840
01198761198620
(46)
Using Lemma 5 Ξ lt 0 is equivalent to
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119876 1198711015840
1205722
11198671015840
11205722
21198671015840
2119860
1015840
119876 119862
1015840
119876 0 0 0 0 119860
1015840
119862
1015840
lowast minus119868 0 0 0 0 0 0 0 0 0 0
lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0 0
lowast lowast lowast minus
1
3120572
119868 0 0 0 0 0 0 0 0
lowast lowast lowast lowast minus119876 0 0 0 0 0 0 0
lowast lowast lowast lowast lowast minus119876 0 0 0 0 0 0
lowast lowast lowast lowast lowast lowast minus1205742
119868 1198600119876 1198620119876 119872
1015840
11986001198620
lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast minus119876 0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119876minus1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
lt 0 (47)
It is obvious that seeking 119867infin
gain matrix 119870 needs to solveLMIs (47) and119876minus120572119868 lt 0 Setting119876minus1 = 119883 and 120573 = 1120572 andpre- and postmultiplying
diag [119883 119868 119868 119868 119883119883 119868 119883119883 119868 119868 119868] (48)on both sides of (47) and considering (35) (34) is obtainedimmediately The proof is completed
4 Numerical Examples
This section presents two numerical examples to demonstratethe validity of our main results described above
Example 1 Consider system (5) with parameters as
119860 = [
08 03
04 09
]
119862 = [
03 02
04 02
]
1198671= 1198672= [
1 0
0 1
]
119861 = 119863 = [
1
1
]
119909 (0) = [
1199091(0)
1199092(0)
] = [
50
minus50
]
(49)
For the unforced system (16) with 1205721= 1205722= 03 its
corresponding state locus diagram is made in Figure 1 FromFigure 1 it is easy to see that the status values are of seriousdivergences through 50 iterationsHence the unforced systemis not stable
To design a feedback controller such that the closed-loopsystem is stochastically stable usingMatlab LMI Toolbox we
8 Mathematical Problems in Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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 Engineeringx1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus2000
0
2000
4000
6000
8000
10000
Figure 1 State trajectories of the autonomous system with 120572 = 03
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 2 State trajectories of the closed-loop system with 120572 = 03
find that a symmetric positive definitematrix119883 a realmatrix119884 and a scalar 120573 given by
119883 = [
3318 0177
0177 1920
]
119884 = [minus1587 minus0930]
120573 = 1868
(50)
solve LMI (27) So we get the feedback gain 119870 =
[minus0455 minus0443] Submitting
119906 (119905) = [minus0455 minus0443] [
1199091(119905)
1199092(119905)
] (51)
into system (5) the state trajectories of the closed-loop systemare shown as in Figure 2
From Figure 2 one can find that the controlled systemachieves stability using the proposed controller Meanwhile
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
0
1000
2000
3000
Figure 3 State trajectories of the autonomous system with 120572 = 01
in the case 120572 le 03 the controlled system maintainsstabilization
In order to show the robustness we use different valuesof 120572 in Example 1 below We reset 120572 = 01 and 120572 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4 respectively By comparing Figures 13 and 4 intuitively speaking the more value 120572 takesthe more divergence the autonomous system (5) has Byusing Theorem 9 we can get that under the conditionof 120572 = 01 and 120572 = 1 the corresponding controllersare 119906120572=01(119905) = [minus04750 minus04000] [
1199091(119905)
1199092(119905)] and 119906
120572=1(119905) =
[minus04742 minus04016] [1199091(119905)
1199092(119905)] respectively Substitute 119906
120572=01(119905)
and 119906120572=1(119905) into system (5) in turn and the corresponding
closed-loop system is shown in Figures 5 and 6 respectivelyFrom the simulation results we can see that the larger
the value 120572 takes the slower the system converges Thisobservation is reasonable because the larger uncertainty thesystem has the stronger the robustness of controller requires
Example 2 Consider system (11) with parameters as
1198620= [
02 0
0 01
]
1198600= [
03 0
0 03
]
119871 = [
02 0
0 02
]
119872 = [
02 0
0 02
]
1198671= 1198672= [
08 0
0 08
]
(52)
Mathematical Problems in Engineering 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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 9
times104
minus8
minus6
minus4
minus2
0
2
4
6
8
10
x1andx2
5 10 15 20 25 30 35 40 45 500
t (sec)
Figure 4 State trajectories of the autonomous system with 120572 = 1
minus60
minus40
minus20
0
20
40
60
x1andx2
2 4 6 8 10 12 14 16 18 200
t (sec)
Figure 5 State trajectories of the closed-loop system with 120572 = 01
x1andx2
t (sec)50454035302520151050
minus500
minus400
minus300
minus200
minus100
0
100
200
300
400
500
Figure 6 State trajectories of the closed-loop system with 120572 = 1
t (sec)302520151050
x1andx2
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
Figure 7 State trajectories of the closed-loop system with 120572 = 03
119860 119861 119862119863 have the same values as in Example 1 Setting 119867infin
norm bound 120574 = 06 and 1205721= 1205722= 03 a group of the
solutions for LMI (34) are
119883 = [
55257 3206
3206 31482
]
119884 = [minus26412 minus15475]
120573 = 30596
(53)
and the119867infin
controller is
119906 (119905) = 119870119909 (119905) = 119884119883minus1
119909 (119905)
= [minus0452 minus0446] [
1199091(119905)
1199092(119905)
]
(54)
The simulation result about Theorem 10 is described inFigure 7This further verifies the effectiveness ofTheorem 10
In order to give a comparison with 120572 = 03 we set 120572 = 01and 120572 = 1 By using Theorem 10 the 119867
infincontroller for 120572 =
01 is 119906(119905) = [minus04737 minus04076] [ 1199091(119905)1199092(119905)] with the simulation
result given in Figure 8 When 120572 = 1 LMI (34) is infeasiblethat is in this case system (11) is not119867
infincontrollable
5 Conclusion
This paper has discussed robust stability stabilization and119867infin
control of a class of nonlinear discrete time stochasticsystems with system state control input and external distur-bance dependent noise Sufficient conditions for stochasticstability stabilization and robust119867
infincontrol law have been
respectively given in terms of LMIs Two examples have alsobeen supplied to show the effectiveness of our main results
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
10 Mathematical Problems in Engineering
5 10 15 20 25 300
t (sec)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
50
x1andx2
Figure 8 State trajectories of the closed-loop system with 120572 = 01
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61573227) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina theNatural Science Foundation of Shandong Provinceof China (no 2013ZRE28089) SDUST Research Fund (no2015TDJH105) State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (noLAPS16011) and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos YCX2015048 andYCX2015051)
References
[1] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967
[2] R Z Hasrsquominskii Stochastic Stability of Differential EquationsSijtjoff and Noordhoff Alphen Netherlands 1980
[3] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 2nd edition 2007
[4] M Ait Rami and X Y Zhou ldquoLinearmatrix inequalities Riccatiequations and indefinite stochastic linear quadratic controlsrdquoIEEE Transactions on Automatic Control vol 45 no 6 pp 1131ndash1143 2000
[5] Z Lin J Liu Y Lin and W Zhang ldquoNonlinear stochasticpassivity feedback equivalence and global stabilizationrdquo Inter-national Journal of Robust and Nonlinear Control vol 22 no 9pp 999ndash1018 2012
[6] F Li and Y Liu ldquoGlobal stability and stabilization of moregeneral stochastic nonlinear systemsrdquo Journal of MathematicalAnalysis and Applications vol 413 no 2 pp 841ndash855 2014
[7] S Sathananthan M J Knap and L H Keel ldquoRobust stabilityand stabilization of a class of nonlinear Ito-type stochasticsystems via linear matrix inequalitiesrdquo Stochastic Analysis andApplications vol 31 no 2 pp 235ndash249 2013
[8] Y Xia E-K Boukas P Shi and J Zhang ldquoStability and stabiliza-tion of continuous-time singular hybrid systemsrdquo Automaticavol 45 no 6 pp 1504ndash1509 2009
[9] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[10] T Hou W Zhang and B-S Chen ldquoStudy on general stabilityand stabilizability of linear discrete-time stochastic systemsrdquoAsian Journal of Control vol 13 no 6 pp 977ndash987 2011
[11] C S Kubrusly and O L Costa ldquoMean square stability condi-tions for discrete stochastic bilinear systemsrdquo IEEE Transactionson Automatic Control vol 30 no 11 pp 1082ndash1087 1985
[12] Y Li W Zhang and X Liu ldquoStability of nonlinear stochasticdiscrete-time systemsrdquo Journal of Applied Mathematics vol2013 Article ID 356746 8 pages 2013
[13] S Sathananthan M J Knap A Strong and L H Keel ldquoRobuststability and stabilization of a class of nonlinear discrete timestochastic systems an LMI approachrdquoAppliedMathematics andComputation vol 219 no 4 pp 1988ndash1997 2012
[14] W ZhangW Zheng and B S Chen ldquoDetectability observabil-ity and Lyapunov-type theorems of linear discrete time-varyingstochastic systems with multiplicative noiserdquo httparxivorgabs150904379
[15] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM
Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998
[16] B S Chen andWZhang ldquoStochastic1198672119867infincontrol with state-
dependent noiserdquo IEEE Transactions on Automatic Control vol49 no 1 pp 45ndash57 2004
[17] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Linear Stochastic Systems SpringerNewYork NY USA 2006
[18] H Li andY Shi ldquoState-feedback119867infincontrol for stochastic time-
delay nonlinear systems with state and disturbance-dependentnoiserdquo International Journal of Control vol 85 no 10 pp 1515ndash1531 2012
[19] I R Petersen V AUgrinovskii andAV SavkinRobust controldesign using H
infin-methods Springer New York NY USA 2000
[20] W Zhang and B-S Chen ldquoState feedback119867infincontrol for a class
of nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006
[21] N Berman and U Shaked ldquo119867infin
control for discrete-timenonlinear stochastic systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 6 pp 1041ndash1046 2006
[22] V Dragan T Morozan and A-M Stoica Mathematical Meth-ods in Robust Control of Discrete-Time Linear Stochastic SystemsSpringer New York NY USA 2010
[23] A El Bouhtouri D Hinrichsen and A J Pritchard ldquo119867infin-
type control for discrete-time stochastic systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 9 no 13 pp 923ndash948 1999
[24] THouW Zhang andHMa ldquoInfinite horizon1198672119867infinoptimal
control for discrete-time Markov jump systems with (x u v)-dependent noiserdquo Journal of Global Optimization vol 57 no 4pp 1245ndash1262 2013
[25] W Zhang Y Huang and L Xie ldquoInfinite horizon stochastic1198672119867infincontrol for discrete-time systems with state and distur-
bance dependent noiserdquo Automatica vol 44 no 9 pp 2306ndash2316 2008
[26] S Elaydi An Introduction to Difference Equation Springer NewYork NY USA 2005
[27] J P LaSalle The Stability and Control of Discrete ProcessesSpringer New York NY USA 1986
Mathematical Problems in Engineering 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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 11
[28] N Athanasopoulos M Lazar C Bohm and F Allgower ldquoOnstability and stabilization of periodic discrete-time systemswithan application to satellite attitude controlrdquo Automatica vol 50no 12 pp 3190ndash3196 2014
[29] O L V Costa and A de Oliveira ldquoOptimal meanndashvariancecontrol for discrete-time linear systems with Markovian jumpsand multiplicative noisesrdquo Automatica vol 48 no 2 pp 304ndash315 2012
[30] Q Ma S Xu Y Zou and J Lu ldquoRobust stability for discrete-time stochastic genetic regulatory networksrdquo Nonlinear Analy-sis Real World Applications vol 12 no 5 pp 2586ndash2595 2011
[31] W Lin and C I Byrnes ldquo119867infincontrol of discrete-time nonlinear
systemsrdquo IEEE Transactions on Automatic Control vol 41 no 4pp 494ndash510 1996
[32] T Zhang Y H Wang X Jiang and W Zhang ldquoA Nash gameapproach to stochastic 119867
2119867infin
control overview and furtherresearch topicsrdquo in Proceedings of the 34th Chinese ControlConference pp 2848ndash2853 Hangzhou China July 2015
[33] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[34] X Lin andWZhang ldquoAmaximumprinciple for optimal controlof discrete-time stochastic systems with multiplicative noiserdquoIEEE Transactions on Automatic Control vol 60 no 4 pp 1121ndash1126 2015
[35] Z Du Q Zhang and L Liu ldquoNew delay-dependent robuststability of discrete singular systems with time-varying delayrdquoAsian Journal of Control vol 13 no 1 pp 136ndash147 2011
[36] K-S Park and J-T Lim ldquoRobust stability of non-standard non-linear singularly perturbed discrete systems with uncertaintiesrdquoInternational Journal of Systems Science vol 45 no 3 pp 616ndash624 2014
[37] S Lakshmanan T Senthilkumar and P BalasubramaniamldquoImproved results on robust stability of neutral systemswith mixed time-varying delays and nonlinear perturbationsrdquoApplied Mathematical Modelling vol 35 no 11 pp 5355ndash53682011
[38] X Zhang CWang D Li X Zhou and D Yang ldquoRobust stabil-ity of impulsive Takagi-Sugeno fuzzy systems with parametricuncertaintiesrdquo Information Sciences vol 181 no 23 pp 5278ndash5290 2011
[39] S Shah Z Iwai I Mizumoto and M Deng ldquoSimple adaptivecontrol of processes with time-delayrdquo Journal of Process Controlvol 7 no 6 pp 439ndash449 1997
[40] M Deng and N Bu ldquoRobust control for nonlinear systemsusing passivity-based robust right coprime factorizationrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2599ndash2604 2012
[41] Z Y Gao and N U Ahmed ldquoFeedback stabilizability ofnonlinear stochastic systems with state-dependent noiserdquo Inter-national Journal of Control vol 45 no 2 pp 729ndash737 1987
[42] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
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
