Research Article Robust Fault Reconstruction in Discrete...

Research ArticleRobust Fault Reconstruction in Discrete-TimeLipschitz Nonlinear Systems via Euler-ApproximateProportional Integral Observers

Qingxian Jia1 Wen Chen2 Yingchun Zhang13 and Yu Jiang1

1Research Center of Satellite Technology Harbin Institute of Technology Harbin 150001 China2Division of Engineering Technology Wayne State University Detroit MI 48202 USA3Shenzhen Aerospace Dongfanghong HIT Satellite Company Ltd Shenzhen 518057 China

Correspondence should be addressed to Yingchun Zhang zhanghiteducn

Received 8 August 2014 Accepted 15 October 2014

Academic Editor Ke Zhang

Copyright copy 2015 Qingxian Jia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of observer-based robust fault reconstruction for a class of nonlinear sampled-data systems is investigated A discrete-time Lipschitz nonlinear system is first established and its Euler-approximate model is described then an Euler-approximateproportional integral observer (EPIO) is constructed such that simultaneous reconstruction of system states and actuator faultsare guaranteed The presented EPIO possesses the disturbance-decoupling ability because its architecture is similar to that of anonlinear unknown input observer The robust stability of the EPIO and convergence of fault-reconstructing errors are provedusing Lyapunov stability theory together with 119867

infintechniques The design of the EPIO is reformulated into convex optimization

problem involving linear matrix inequalities (LMIs) such that its gain matrices can be conveniently calculated using standard LMItools In addition to guarantee the implementation of the EPIO on the exact model sufficient conditions of its semiglobal practicalconvergence are provided explicitly Finally a single-link flexible robot is employed to verify the effectiveness of the proposed fault-reconstructing method

1 Introduction

Increasing complexity of modern engineering systems andhigher demand for system performance particularly insafety-critical systems will correspondingly raise the prob-ability of system faults andor failures As a response to therequirement for system safety reliability and survivabilityfault diagnosis of dynamic systems has been an attractivesubject during the past few decades In general fault-diagnostic module consists of three essential tasks faultdetection isolation and reconstruction (also known as faultestimation or fault identification) [1ndash3] In literature faultdetection and isolation (FDI) schemes are considered tobe the most important and are of main focuses never-theless fault reconstruction is an indispensable componentin active fault-tolerant control (FTC) systems where faultscan be effectively accommodated utilizing reconfigurable

fault-tolerant controller with reconstructed fault informationin real time

During the past decade observer-based methods forfault reconstruction have been increasingly attracting theattention of many researchers and different categories offault reconstruction observers have been developed such asadaptive observers [4 5] sliding mode observers (SMOs)[6 7] learning observers [8 9] and proportional integralobservers (PIOs) [10 11] to name just a few It is worthnoting thatmost of existing observers aremainly applicable tocontinuous-time systems On the other hand it is well knownthat fault reconstruction for nonlinear discrete-time systemsis considerably practical and challenging because the nonlin-earities inherently exist in nature and most continuous-timesystems are implemented digitally in practical applicationsUnder the assumption that nonlinear discrete-time modelsare accurate numerous fault-reconstructing strategies have

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 741702 14 pageshttpdxdoiorg1011552015741702

2 Mathematical Problems in Engineering

been reported [12ndash15] However the availability of the exactmodels originated from discretization of a continuous-timeplant is usually unrealistic A practical solution to thisdifficult situation is to employ approximate discrete-timemodels especially Euler-approximate models instead ofexact models

More recently many researchers have paid much atten-tion to observer design and observer-based fault recon-struction for nonlinear discrete-time systems with Euler-approximate models Reference [16] proposes a generalframework for nonlinear observer design via the approximatediscrete models For Lipschitz nonlinear systems the authorsin [17] suggest a robust 119867

infinobserver whose main advan-

tage is that maximum admissible Lipschitz constant androbustness can be solved using LMI optimization techniquesBased on [16 17] a Euler-approximate observer (EAO) ispresented for fault reconstruction and active FTC in [18]however the robustness is not well guaranteed For nonlinearnetworked control systems an SMO-based fault reconstruc-tion approach is investigated in [19] where the chatteringphenomenon resulting from the signum function in theSMO is inevitable In [20] a Euler-approximate unknowninput observer- (UIO-) based robust fault detection strategyis developed However to the best of our knowledge fewresults have been reported on observer-based robust faultreconstruction for nonlinear discrete-time systems via Euler-approximate models In addition how to solve the errorproblem caused by the approximate models such that theobserver designed under Euler-approximate models can beimplemented on the exactmodels has still been an interestingand challenging issue All of these motivate us to pursue thisinvestigation

The main objective of this work is to design and ana-lyze an observer-based robust fault-reconstructing strategyfor a class of nonlinear discrete-time systems using Euler-approximate models First a sampled-data nonlinear systemsatisfying the Lipschitz condition is formulated into a Euler-approximate model then on the basis of our previous resultsin [20] a discrete-time Euler-approximate proportional inte-gral observer (EPIO) is constructed to simultaneously recon-struct system states and actuator faults The EPIO has anarchitecture similar to that of a nonlinear UIO Comparedwith the EAO proposed in [18] the designed EPIO is able topartially decouple external disturbances and is robust to thereminding part of external disturbances and measurementnoises As a result the accuracy of the EPIO-based faultreconstruction can be guaranteed In addition to guaranteeimplementation of the EPIO on the exact models sufficientconditions for semiglobal practical convergence which isdefined in [16] are explicitly provided Besides systematicobserver synthesis with an119867

infintechnique is effectively solved

using LMI optimization techniques Simulation results ona single-link flexible robot are also presented to verify theeffectiveness of the proposed fault-reconstructing method

The rest of this paper is organized as follows In Section 2an Euler-approximate model of a Lipschitz nonlinear systemis described andmain problems are formulated In Section 3a discrete-time EPIO is constructed and robust stability andsemiglobal practical convergence of the proposed observer

are analyzed Simulation studies are reported in Section 4and conclusions are drawn in the last section

2 System Description andProblem Formulation

A continuous-time nonlinear system subject to actuatorfaults external disturbances and measurement noises isdescribed as

(119905) = 119860119909 (119905) + Φ (119909 (119905) 119905) + 119861119906 (119905) + 11986311198891(119905) + 119865119891 (119905)

(1a)

119910 (119905) = 119862119909 (119905) + 11986321198892(119905) (1b)

where 119909(119905) isin R119899 119906(119905) isin R119898 and 119910(119905) isin R119901 representingsystem state vector control input vector and measurableoutput vector respectively Variable 119891(119905) denotes actuatorfaults with 119891(119905) isin R119898 Vectors 119889

1isin R119889 and 119889

2isin R119904

representing external disturbances and measurement noisesrespectively Without loss of generality matrices 119863

1and 119862

are assumed to be of full column rank and of full row rankrespectively The symbol Φ(119909(119905) 119905) is a continuous nonlinearfunction that is assumed to satisfy the Lipschitz condition (atleast local) that is Φ(119909(119905) 119905) minusΦ(119909(119905) 119905) le 119871

119892119909(119905) minus119909(119905)

where 119871119892is a Lipschitz constant

Herein system control input 119906(119905) is taken to be a piece-wise constant signal as 119906(119896) during the sampling intervals[119896120591 (119896 + 1)120591] with a zero-order hold where 120591 is the samplingperiod Therefore the Euler-approximate discrete model ofcontinuous-time system ((1a) (1b)) can be formulated as

119909 (119896 + 1) = (119868119899+ 120591119860) 119909 (119896) + 120591Φ (119909 (119896) 119896) + 120591119861119906 (119896)

+ 12059111986311198891(119896) + 120591119865119891 (119896)

(2a)

119910 (119896) = 119862119909 (119896) + 11986321198892(119896) (2b)

where discrete nonlinear term Φ(119909(119896) 119896) still satisfiesLipschitz constraint namely

Φ (119909 (119896) 119896) minus Φ (119909 (119896) 119896) le 119871119892119909 (119896) minus 119909 (119896) (3)

Before finishing this section the following lemma isprovided for proof of theorems in the latter section

Lemma 1 For any positive scalar 120575 there exists a positive-definite symmetric matrix119873 such that the following inequalityholds

2119886119879

119873119887 le 120575119886119879

119873119886 +1

120575119887119879

119873119887 forall (119886 119887) isin R119899

(4)

Throughout the paper 119860 gt 0 (119860 lt 0) denotes that 119860is positive (negative) definite 120582min(119860) and 120582max(119860) are theminimum and maximum eigenvalues of 119860 120590(119860) representsthe maximum singular value of matrix 119860 sdot and sdot

infin

represent Euclidean norm and infinity norm of a vector ormatrix lowast means symmetric term 119868

119899is an identity matrix

of size 119899 0119899is a zero matrix of size 119899 dagger represents a

pseudoinverse of a matrix and C denotes the set of allcomplex numbers

Mathematical Problems in Engineering 3

3 Fault Reconstruction via Euler-ApproximateProportional Integral Observers

In this section we will construct a discrete-time EPIO toachieve robust actuator-fault reconstruction then robust sta-bility and semiglobal practical convergence of the presentedEPIO will be discussed in detail

31 Design of a Euler-Approximate Proportional IntegralObserver In order to perform fault reconstruction for actu-ators in Lipschitz nonlinear system ((1a) (1b)) a discrete-time nonlinear EPIO is established for the Euler-approximatemodel ((2a) (2b)) as follows

119911 (119896 + 1) = 119879 (119868119899+ 120591119860) 119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 120591119879119861119906 (119896)

+ 119871 (119910 (119896) minus 119910 (119896)) + 120591119879119865119891 (119896)

(5a)

119909 (119896) = 119911 (119896) + 119867119910 (119896) (5b)

119910 (119896) = 119862119909 (119896) (5c)

119891 (119896 + 1) = 119891 (119896) + 119870 (119910 (119896) minus 119910 (119896)) (5d)

where 119911(119896) isin R119899 119909(119896) isin R119899 and 119910(119896) isin R119901they represent the observer state vector the estimated statevector and measurement output vector respectively Faultreconstruction signal 119891(119896) is recursively updated by bothitself and output estimation errors at the sampling time 119896minus 1Here 119879 119867 119871 and 119870 are gain matrices with appropriatedimensions to be determined later

Defining 119890119909(119896) = 119909(119896) minus 119909(119896) 119890

119910(119896) = 119910(119896) minus 119910(119896) and

119890119891(119896) = 119891(119896)minus119891(119896) and letting119879+119867119862 = 119868

119899 estimation error

dynamics can be easily obtained as

119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 12059111987911986311198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(6a)

119890119910(119896) = 119862119890

119909(119896) + 119863

21198892(119896) (6b)

where ΔΦ(119896) = Φ(119909(119896) 119896) minus Φ(119909(119896) 119896)Since the nonlinear function Φ(119909(119896) 119896) satisfies the

Lipschitz condition then

ΔΦ119879

(119896) ΔΦ (119896) minus 1198712

119892119890119879

119909(119896) 119890119909(119896) le 0 (7)

Inspired by disturbance-decoupling principle of UIOs [2021] we will design gain matrix 119879 such that state estimationerror can be partially decoupled from external disturbancesAs such state error dynamics (6a) can be reorganized as

119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 120591119879[11986311

11986312]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198631

times [119889119879

11(119896) 119889

119879

12(119896)]119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198891(119896)

minus 11987111986321198892(119896) minus 119867119863

21198892(119896 + 1) + 120591119879119865119890

119891(119896)

(8)

where 11988911

isin R119902 and 119902 is the maximum number of columnsof the matrix 119863

1for which the condition rank(119862119863

11) =

rank(11986311) is satisfied Matrix119863

11is composed of 119902 columns

of the matrix 1198631while matrix 119863

12represents the remaining

part of the matrix1198631

Matrix 119879 is selected such that the condition 11987911986311

= 0holds Herein we define matrix 119864 as [0

11990211986312] Hence state

error dynamics (6a) can be further manipulated as

119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 1205911198791198641198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(9)

Considering (5d) fault-reconstructing errors 119890119891(119896) can

be expressed in the following equation119890119891(119896 + 1) = 119890

119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) + Δ119891 (119896) (10)

where fault variation vector Δ119891(119896) = 119891(119896 + 1) minus 119891(119896) Ifconstant actuator faults are considered in system ((1a) (1b))then Δ119891(119896) equiv 0 Therefore one obtains

119890119891(119896 + 1) = 119890

119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) (11)

Remark 2 Equations (8) and (9) display that the designedEPIO is only partially decoupled from external disturbance1198891(119896) It is because either the number of measurement

outputs is not greater than that of external disturbances orfault-reconstructing ability of the presented EPIO should beguaranteed when coupling problem exists between exter-nal disturbances and actuator faults To tackle the abovedisturbance-decoupling problem gain matrices 119879 and 119867

should be designed such that matrix equations 119879 + 119867119862 =

119868119899 and 119879119863

11= 0 are satisfied simultaneously Augmented

matrix equation composing of these two constraints can bewritten as

[119879 119867] [11986811989911986311

119862 0] = [119868

1198990] (12)

Therefore a general solution of (12) can be determined by

[119879 119867] = [1198681198990] [

11986811989911986311

119862 0]

dagger

(13)

In addition to guarantee actuator-fault detectability by theproposed EPIO the constraint rank(119879119865) = rank(119865) shouldbe also satisfied

According to (9) and (10) the following augmented errordynamics can be constructed as

[119890119909(119896 + 1)

119890119891(119896 + 1)

]

= [119879 (119868119899+ 120591119860) minus 119871119862 120591119879119865

minus119870119862 119868119898

] [119890119909(119896)

119890119891(119896)

] + 120591119879 [ΔΦ (119896)

0]

+ [120591119879119864 minus119871119863

2minus1198671198632

0

0 minus1198701198632

0 119868119898

]

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

(14)


Denote

119890 (119896) = [119890119909(119896)

119890119891(119896)

] 120596 (119896) =

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

119879 = [119879 0

0 119868119898

] 119860 = [119868119899+ 120591119860 120591119865

0 119868119898

]

ΔΦ (119896) = [ΔΦ (119896)

0] 119863

1= [

minus120591119879119864 0 1198671198632

0

0 0 0 minus119868119898

]

119871 = [119871

119870] 119862 = [119862 0] 119863

2= [0 minus119863

20 0]

(15)

Further augmented error dynamics (14) can be reorga-nized into a compact form

119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896) + 120591119879ΔΦ (119896)

+ (1198711198632minus 1198631) 120596 (119896)

119890119910(119896) = 119862119890 (119896) minus 119863

2120596 (119896)

(16)

Besides fault reconstruction error 119890119891(119896) = 119862

119891119890(119896) where

119862119891= [0 119868

119898]

For nonlinear function ΔΦ(119896) the following inequalityholds

Γ = ΔΦ119879

(119896) ΔΦ (119896) minus 119890119879

(119896) 119871119879

119892119871119892119890 (119896) le 0 (17)

where 119871119892= [1198711198921198681198990

0 0119898

]It is noticed that augmented error system (16) can be

treated as an observation-error equation of the followingnonlinear system

119909 (119896 + 1) = 119879119860119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 1198631120596 (119896)

119910 (119896) = 119862119909 (119896) minus 1198632120596 (119896)

(18)

where

119909 (119896) = [119909119879

(119896) 119891119879

(119896)]119879

Φ (119909 (119896) 119896) = [Φ119879

(119909 (119896) 119896) 0]119879

(19)

An augmented state observer can be established for (18) asfollows

(119896 + 1) = 119879119860 (119896) + 120591119879Φ ( (119896) 119896) + 119871 (119910 (119896) minus 119910 (119896))

119910 (119896) = 119862 (119896)

(20)

Remark 3 If both the nonlinear term ΔΦ(119896) and augmenteddisturbance 120596(119896) are not considered in (16) we can obtain

119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896)

119890119910(119896) = 119862119890 (119896)

(21)

The linear error dynamics (21) is asymptotically stable if andonly if the pair (119879119860 119862) is detectable It is equivalent to

rank [119911119868119899+119898 minus 119879119860119862

] = 119899 + 119898 forall119911 isin C |119911| ge 1 (22)

and one obtains

rank [119911119868119899+119898 minus 119879119860119862

]

= rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

(23)

(1) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119868119899 minus 119879 (119868119899 + 120591119860) minus120591119879119865

119862 0]

= rank [119868119899 minus (119868119899 minus 119867119862) (119868119899 + 120591119860) minus120591119879119865

119862 0]

= rank [119860 119879119865

119862 0]

(24)

(2) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119911119868119899 minus 119879 (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 minus 119867119862) (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 + 120591119860)119862

] + 119898

(25)

Thus the pair (119879119860 119862) is detectable if and only if

rank [119860 119879119865

119862 0] = 119899 + 119898

rank [119911119868119899 minus (119868119899 + 120591119860)119862

] = 119899

(26)

when 119911 = minus1 and |119911| gt 1Therefore the above two conditionscan be regarded as necessary conditions for existence of theproposed nonlinear EPIO ((5a) (5b) (5c) and (5d))

Remark 4 Equations (14)ndash(20) imply that the design of theproposed nonlinear EPIO is now converted into the analysisof robust stability of the augmented error dynamics (16)


that is the design of robust observer (20) for the augmentednonlinear system (18)Therefore to guarantee the robustnessof the proposed EPIO against the remaining part of externaldisturbance 119889

1(119896) and measurement noise 119889

2(119896) we shall

design augmented matrix 119871 which is composed of twounknown matrices 119871 and 119870 such that the robust stabilityof the augmented error dynamics (16) is guaranteed with aprescribed119867

infinperformance specification

32 Robust Stability Analysis of the Euler-Approximate Pro-portional Integral Observer In this subsection we will foucson the robust stability analysis of the proposed nonlinearEPIO ((5a) (5b) (5c) and (5d)) Equation (14) shows that dif-ferent components of the augmented disturbance 120596(119896) havedifferent influences on fault reconstruction performance Toguarantee better robustness of the designed nonlinear EPIOagainst external disturbance 119889

1(119896) measurement noise 119889

2(119896)

and fault variation vector Δ119891(119896) multiple attenuation levelsin an 119867

infinperformance specification will be adopted in the

EPIO synthesis In what follows a theorem is presented tocharacterize the robust stability of the proposed EPIO ((5a)(5b) (5c) and (5d))

Theorem 5 Consider the Euler-approximate discrete model((2a) (2b)) For a given positive scalar 120574

119894 119894 = 1 2 3 4 if there

exist a positive-definite symmetric matrix 119875 isin R(119899+119898)times(119899+119898) amatrix 119884 isin R(119899+119898)times119901 and positive scalars 120575

119894 119894 = 1 2 3 4

such that

[[

[

minus119875 radic1 + 1205751(119875119879119860 minus 119884119862) 120591radic1 + 120575

2119875119879

lowast minus119875 + 1205753119871119879

119892119871119892

0

lowast lowast minus1205753119868119899+119898

]]

]

lt 0 (27)

[[[[[[[[[[

[

minus119875 119875119879119860 minus 119884119862 120591119875119879 120591119875119879119864 minus1198841198632minus119875119863ℎ

119875 119868119898

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0


0 0 0 0

lowast lowast lowast minus1205742

1119868119889

0 0 0

lowast lowast lowast lowast minus1205742

2119868119904

0 0

lowast lowast lowast lowast lowast minus1205742

3119868119904

0

lowast lowast lowast lowast lowast 0 minus1205742

4119868119898

]]]]]]]]]]

]

lt 0 (28)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] 119863ℎ= [1198671198632

0] and 119868

119898= [0

119868119898]

then the proposed nonlinear EPIO ((5a) (5b) (5c) and (5d))can realize uniform ultimate boundedness of state-estimatingerror 119890

119909(119896) and fault-reconstructing error 119890

119891(119896) with an 119867

infin

performance criterion10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891 (119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

+ 1205742

4

1003817100381710038171003817Δ119891(119896)1003817100381710038171003817

2

(29)

Moreover augmented matrix 119871 can be determined by 119871 =

119875minus1

119884

Proof Consider the Lyapunov function candidate 119881(119896) =

119890119879

(119896)119875119890(119896)The external disturbance 1198891(119896) andmeasurement

noise 1198892(119896) are temporarily dropped then the difference

Δ119881(119896) = 119881(119896 + 1) minus 119881(119896) is derived as

Δ119881 (119896) = [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(30)

where 119877 = (119879119860 minus 119871119862)119879

119875(119879119860 minus 119871119862) minus 119875

Considering Lemma 1 the following inequalities hold

2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

le 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

le 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(31)

where 1205751gt 0 and 120575

2gt 0

Substituting (17) (31) into (30) leads to

Δ119881 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

minus 1205753Γ + 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)


+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879

119875(119879119860 minus 119871119862) minus 119875 and 1205753gt 0

If Ω lt 0 then we have

Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)

where 120598 = 120582min(minusΩ) and ] = (1 + (11205751) +

(11205752))120582max(119875)Δ119891(119896)infin

Further the conditionΩ lt 0 can be rewritten as

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)

By the Scular complement lemma [22] (34) is equivalent to(27)

Therefore if (27) holds then Δ119881(119896) lt 0 for 120598119890(119896)2 gt ]which means that the trajectory of 119890(119896) that is outside ofthe set Φ = 119890(119896) | 119890(119896)

2

le ]120598 will converge to theset Φ according to the Lyapunov stability theory Thereforethe uniform ultimate boundedness of the state-estimatingerror 119890

119909(119896) and the fault-reconstructing error 119890

119891(119896) can be

guaranteedTo guarantee that the proposed nonlinear EPIO ((5a)

(5b) (5c) and (5d)) is robust to 1198891(119896) 119889

2(119896) and Δ119891(119896) an

119867infin

performance index function is chosen as [23]

119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)

Under zero-initial conditions the following is obtained

119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)

Considering external disturbance 1198891(119896) and measure-

ment noise 1198892(119896) it follows from (17) (30) and (36) that

Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36

lowast lowast lowast Π44

Π45

Π46

lowast lowast lowast lowast Π55

Π56

lowast lowast lowast lowast lowast Π66

]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898

Further the inequality Π lt 0 can be transformed into

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0

lowast lowast minus1205742

1119868119889

0 0 0


2119868119904

0 0


3119868119904

0

lowast lowast lowast lowast 0 minus1205742

4119868119898

]]]]]]]]

]

lt 0

(39)

Using the Schur complement lemma again (39) is equiv-alent to (28) which means that for all nonzero 119889

1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one

obtains 119869 lt 0 Therefore we can conclude from (35) that the119867infinperformance criterion (29) is satisfiedThis completes the

proof

In Theorem 5 multiple attenuation levels in the119867infin

per-formance criterion (29) are prescribed for better restrictingeach component of the augmented disturbance 120596(119896) on faultreconstruction performance If only a single119867

infinattenuation

level is considered for the augmented disturbance 120596(119896)the following corollary can be readily obtained based onTheorem 5

Corollary 6 Consider the Euler-approximate discrete model((2a) (2b)) For a given positive scalar 120574 if there exist a positive-definite symmetric matrix 119875 isin R(119899+119898)times(119899+119898) a matrix 119884 isin

R(119899+119898)times119901 and positive scalars 120575119894 119894 = 1 2 3 4 satisfying (27)

and

[[[[

[

minus119875 119875119879119860 minus 119884119862 120591119875119879 1198841198632minus 119875119863

1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)

then the proposed nonlinear EPIO ((5a) (5b) (5c) and(5d)) can realize the uniform ultimate boundedness of state-estimating error 119890


119891(119896)with

an 119867infin

performance criterion 119890119891(119896) le 120574120596(119896) Moreover

augmented matrix 119871 can be determined by 119871 = 119875minus1

119884

Proof The proof of Corollary 6 is similar to that ofTheorem 5 it is thus omitted here

Remark 7 InCorollary 6 the119867infinperformance criterionwith

an attenuation level 120574 is adopted such that the designednonlinear EPIO is robust to the augmented disturbance120596(119896) However this design inevitably results in conservatismsuch that the EPIO has a limited robust performance Toobtain less conservative result in the EPIO design the 119867

infin

criterion with attenuation levels 120574119894 forall119894 isin 1 2 3 4 is

adopted in Theorem 5 such that the impact of 1198891(119896) 119889

2(119896)

and Δ119891(119896) on 119890119891(119896) can be effectively attenuated Therefore

Theorem 5 is more flexible than Corollary 6 Compared withCorollary 6 one may obtain smaller 119867

infinattenuation levels

usingTheorem 5 that is 120574 ge 120574119894 forall119894 isin 1 2 3 4 Additionally

to guarantee accurate reconstruction of time-varying faultsespecially fast-varying faults we can select a sufficiently smallattenuation level 120574

4to restrict variation vector Δ119891(119896) at the

expense of the robustness against external disturbance 1198891(119896)

and measurement noise 1198892(119896)


Considering constant actuator faults in the system ((1a)(1b)) the results proposed in Theorem 5 and in Corollary 6can be readily particularized in the following corollaries

Corollary 8 Consider the Euler-approximate discrete model((2a) (2b)) For given positive scalars 120574

1 1205742 and 120574

3 if there

exist a positive-definite symmetric matrix 119875 isin R(119899+119898)times(119899+119898)a matrix 119884 isin R(119899+119898)times119901 and a positive scalar 120575 such that thefollowing LMI holds

[[[[[[[[

[

minus119875 119875119879119860 minus 119884119862 120591119875119879 120591119875119879119864 minus1198841198632minus119875119863

ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then

the proposed nonlinear EPIO ((5a) (5b) (5c) and (5d)) canasymptotically reconstruct constant actuator faults and fault-reconstructing error 119890

119891(119896) satisfies the following inequality

10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884

Corollary 9 Consider the Euler-approximate model ((2a)(2b)) For a given positive scalar 120574 if there exist a positive-definite symmetric matrix 119875 isin R(119899+119898)times(119899+119898) a matrix 119884 isin

R(119899+119898)times119901 and a positive scalar 120575 such that the following LMIholds

[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=

[0 minus11986320] then the proposed nonlinear EPIO ((5a) (5b)

(5c) and (5d)) can asymptotically reconstruct constantactuator faults and fault reconstruction error 119890

119891(119896)

satisfies that 119890119891(119896) le 120574120596

119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879

Moreover augmented matrix 119871

can be determined by 119871 = 119875minus1

119884

Remark 10 Theorem 5 and Corollary 6 can guarantee theuniform ultimate boundedness of state estimation error 119890

119909(119896)

and fault reconstruction error 119890119891(119896) such that the designed

EPIO can achieve accurate reconstruction of constant andtime-varying actuator faults If constant actuator faults areconsidered in the system 1 then one obtains Δ119891(119896) equiv 0

Therefore using Corollaries 8 and 9 the designed EPIO canasymptotically reconstruct constant actuator faults

Remark 11 Since the Lipschitz condition (32) is introducedin the proof of Theorem 5 there may be no feasible solutionfor (27) and (28) especially for a large Lipschitz constantHowever the condition (17) is not overly restrictive becausethe Lipschitz constant might be reduced via coordinatetransformation techniques as discussed in [14 17 24] Formultiobjective observer design the increased number of LMIconstraints inevitably results in conservatism It is worthnoting that slack matrix variable technique can be adoptedto reduce this conservatism the interested readers can bereferred to [14 25 26] for detailed information

According to Theorem 5 the design procedure of theproposed EPIO will be summarized as follows

Step 1 Find themaximumnumber of columns 119902 ofmatrix1198631

for which the condition rank(11986211986311) = rank(119863

11) is satisfied

Step 2 Solve (13) to obtain 119879 and 119867 and verify thatrank(119879119865) = rank(119865)

Step 3 Choose appropriate scalars 120575119894 119894 = 1 2 3 4 then

based on (27) and (40)119867infinperformance level 120574 is computed

using the Solvermincx in the LMI toolbox of Matlab

Step 4 Choose 120574119894le 120574 119894 = 1 2 3 4 then solving (27)

and (28) using the Solver feasp in the LMI toolbox of Matlabmatrices 119875 and 119884 can be obtained

Step 5 Compute 119871 = 119875minus1

119884 then obtain gain matrices 119871 and119870

Step 6 Construct the EPIO based on the above calculatedmatrices

Using the above theorem and corollaries we can designobserver gainmatrices such that the stability and convergenceof the state estimation error and fault reconstruction errorbetween the nonlinear EPIO and the Euler-approximatemodel are guaranteed However it might be impossible forthe EPIO designed under the Euler-approximate model tobe implemented on the exact system models To solve thisproblem the EPIO will be required to satisfy semiglobalpractical convergence property introduced by [16] such thatthe EPIO can track actuator faults with satisfactory accuracyThis is what we shall address in next subsection

33 Semiglobal Practical Convergence Analysis In this sub-section we will derive sufficient conditions for existenceof observer gain matrices and the sampling interval 120591 toachieve semiglobal practical convergence under the Euler-approximate model without considering external distur-bances andmeasurement noisesThe definition of semiglobalpractical convergence [16] is given as follows

Definition 12 An observer is said to be semiglobal practicalin the sampling period 120591 if there exists a class-KL function


120573(sdot sdot) such that for any119863 gt 119889 gt 0 and compact setsX sub R119899U sub R119898 we can find a number 120591lowast gt 0with the property thatfor all 120591 isin (0 120591lowast]

119909 (0) minus 119909 (0) le 119863

119909 (119896) isin X 119906 (119896) isin U forall119896 ge 0

(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)

In reference to themethodproposed by [16] the followingtheorem presents the semiglobal practical convergence of theEPIO ((5a) (5b) (5c) and (5d))

Theorem 13 The proposed nonlinear EPIO under the Euler-approximate model ((2a) (2b)) is semiglobal practical in thesampling period 120591 if there exist a positive scalar 120590

1gt 1 a fixed

positive-definite symmetric matrix 1198761isin R119899times119899 and a positive-

definite symmetric matrix 1198751isin R119899times119899 such that

1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)

Since the convergence of the proposed nonlinear EPIO ((5a)(5b) (5c) and (5d)) to the Euler-approximate model hasbeen established using the aforementioned theorem andcorollaries then one obtains

exist119872 isin (0infin) 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)

With the help of Lemma 1 the following inequalities canbe easily derived

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)

where 1205982gt 0 and

2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0

Substituting (52)ndash(54) into (51) leads to

Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3


According to (46) and (47) (55) can be reorganized andsimplified

Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)

In addition the following functions are defined

1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)

Therefore the following results can be derived

1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)

where 1205721(sdot) 1205722(sdot) 1205723(sdot) and 120588

0(sdot) are in class-K

infinand

]0(sdot) ]1(sdot) ]2(sdot) and ]

3(sdot) are nondecreasing functions

Based on the theorem and corollaries discussed inSubsection 32 it can be known that fault reconstructionerror 119890

119891(119896) are uniformly bounded herein they are con-

sidered as the external disturbances According to [17 27]the Euler-approximate model is consistent with the exactdiscrete-time model Conditions (50) (58) together displaythat all the conditions for semiglobal practical convergence asrequired by [16] are satisfied hence the designed nonlinearEPIO is semiglobal practical convergence in the samplingtime in 120591 This completes the proof

Motivated by [18] we will present another sufficient con-ditions for semiglobal practical convergence of the proposednonlinear EPIO ((5a) (5b) (5c) and (5d))

Theorem 14 The proposed nonlinear EPIO ((5a) (5b) (5c)and (5d)) designed under the Euler-approximate model ((2a)(2b)) is semiglobal practical in the sampling period 120591 if there

exist positive-definite symmetric matrices 1198752isin R119899times119899 and 119876

2isin

R119899times119899 a matrix 119871 isin R119899times119901 and a positive scalar 1205984 such that

(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)

Proof The proof of Theorem 14 is similar to that ofTheorem 2 in [18] it is thus omitted here

Remark 15 Theorem 5 and Corollaries 6ndash9 imply thatobserver gain matrices 119871 and 119870 can be conveniently cal-culated using standard LMI optimization technique fur-thermore using Theorem 13 or Theorem 14 whether or notsemiglobal practical convergence property of the designedEPIO can be guaranteed can be checked If yes the EPIOdesigned under Euler-approximate models can be imple-mented on the exact discrete-time models

4 Simulation Studies

In this section a single-link robot with a flexible jointborrowed from [28] is employed to illustrate the effectivenessof the proposed fault-reconstructingmethodThe consideredsystem dynamics is described in a form of (1a) (1b) with

119860 =

[[[

[

0 1 0 0

minus486 minus125 486 0

0 0 0 1

195 0 minus195 minus1325

]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)

where system states are the motor position link positionand velocitiesMeasurement outputs are the first three systemstates In this work we consider actuator faults that usuallyoccur in input channels Therefore it is reasonable to assumethat fault distribution matrix 119865 = 119861

The Lipschitz constant 119871119892for the system ((1a) (1b))

is selected as 0333 and the sampling period 120591 is selectedas 001 s Therefore an Euler-approximate model of theconsidered system ((1a) (1b)) can be established for theflexible-joint robot


In order to achieve robust actuator-fault reconstructionobserver gain matrices 119879 and119867 are chosen as follows

119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)

Letting 120575119894= 01 forall119894 = 1 2 3 4 and solving (27) and

(40) in Corollary 6 using the Solvermincx in the LMI toolboxof Matlab one can obtain the 119867

infinattenuation level 120574 =

20230 Further choosing 1205741= 05 120574

2= 20230 120574

3= 20230

and 1205744= 20230 and solving (27) and (28) in Theorem 5

using the LMI toolbox of Matlab a feasible solution for gainmatrices 119871 and119870 can be obtained as

119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)

Therefore the EPIO designed using Theorem 5 has betterrobustness against external disturbances and measurementnoises

Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be

obtained by solving (46)

1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)

Further choosing 1205984= 001 and solving (59) lead to

1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[

35024 minus00044 minus00268 minus00077

minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast

= 06162 s which is greater than 001 s We cannow check that the designed observer gain matrices and thesampling period 001 s canmake theTheorems 13 and 14 holdHence it is concluded that the designedEPIO ((5a) (5b) (5c)and (5d)) is semiglobal practical convergent in the samplingperiod 001 s

According to Corollary 9 let 120575 = 01 then (43) can besolved to obtain 120574 = 13007 Further choosing 120574

1= 03 120574

2=

13007 and 1205743= 13007 and solving (41) in Corollary 8 leads

to

119871 =

[[[

[

minus00167 minus01111 minus00104

minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5

System state x1(t)Estimated state x1(k)

t (s)

minus005

Mot

or p

ositi

onx1

(rad

)

Figure 1 Actual system state 1199091(119905) and its estimate

Using Theorems 13 and 14 it is easily checked that theEPIO designed using Corollary 8 is also semiglobal practicalconvergent in 001 s

In the simulation the control input signal is assignedas 119906(119896) = 005 sin 5119896 the initial values of the estimatedstates are assumed as 119909(0) = [minus01 05 minus 01 02]

119879external disturbance 119889


2(119896) are

chosen as 0001 sin(10119896) and a Gaussian-distributed randomsignal with zero mean and variance of 10minus5 respectivelyAdditionally actuator faults are assumed to be changes of busvoltage and two fault scenarios are employed as

(1) a constant fault that is described as

119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)

(2) a time-varying fault that is described as

119891 (119896) =

005 sin (5119896120591) + 005 sin (3119896120591)+ 006 sin (2119896120591) + 02 119896120591 ge 2 s0 others

(68)

Using the designed EPIO via Theorem 5 the estimatedsystem states in fault-free conditions are exhibited in Fig-ures 1 2 3 and 4 where the estimated states can trackthe original states of system ((1a) (1b)) with satisfactoryaccuracy Further reconstruction signals for the constantand time-varying faults obtained by the EPIO designedusing Theorem 5 are demonstrated in Figures 5 and 6 whileconstant-fault reconstruction signal by the EPIO designedusing Corollary 8 is illustrated in Figure 7 The above figuresreveal that the proposed EPIO can accurately reconstructactuator faults All of simulation results claim that the EPIO-based fault-reconstructing method is effective for Lipschitznonlinear systems with Euler-approximate models


0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions

In this paper a robust fault-reconstructing scheme for Lip-schitz nonlinear systems with Euler-approximate models isinvestigated A new discrete-time EPIO is constructed toachieve robust actuator-fault reconstruction An importantadvantage of the newly designed EPIO is its disturbance-decoupling ability such that the accuracy of the reconstructedfaults can be guaranteed To ensure convergence of theestimated states to the system states of the continuous-time model semiglobal practical convergence of the EPIOis proved Another advantage of the proposed approach isthat a systematic observer synthesis with 119867

infintechniques

is effectively solved using a standard LMI tool Simulated

0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)

Constant faultFault reconstruction signal

t (s)

minus02

Figure 5 Reconstruction signal of a constant fault via the EPIOdesigned usingTheorem 5

results on a flexible-joint robot have clearly verified the effec-tiveness of the proposed fault-reconstruction method Withmismatching errors caused by approximate discrete modelsand reconstructed fault signals active FTC for Lipschitznonlinear systemswith Euler-approximatemodels will be ourfuture research target

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)

Time-varying faultFault reconstruction signal

0

02

04

minus02

t (s)

Figure 6 Reconstruction signal of a time-varying fault via the EPIOdesigned usingTheorem 5

0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)

Figure 7 Reconstruction signal of a constant fault via the EPIOdesigned using Corollary 8

Acknowledgments

This work is partially supported by National Natural ScienceFoundation of China (Grant no 61203185) andOpen Fundingfor National Defense Key Subject Laboratory of Micro- andSmall Spacecraft Technology (Grant no 20090450126)

References

[1] S X Ding Model-Based Fault Diagnosis Techniques-DesignSchemes Algorithms and Tools Springer 2008

[2] J Chen and R Patton Robust Model-Based Fault Diagnosis forDynamic Systems Springer London UK 2012

[3] I Hwang S Kim Y Kim and C E Seah ldquoA survey offault detection isolation and reconfiguration methodsrdquo IEEETransactions on Control Systems Technology vol 18 no 3 pp636ndash653 2010

[4] K Zhang B Jiang and V Cocquempot ldquoAdaptive observer-based fast fault estimationrdquo International Journal of ControlAutomation and Systems vol 6 no 3 pp 320ndash326 2008

[5] J H Fan Y M Zhang and Z Q Zheng ldquoAdaptive observer-based integrated fault diagnosis and fault-tolerant control sys-tems against actuator faults and saturationrdquo Journal of DynamicSystems Measurement and Control vol 135 no 4 Article ID041008 13 pages 2013

[6] H Alwi C Edwards and A Marcos ldquoFault reconstructionusing a LPV sliding mode observer for a class of LPV systemsrdquoJournal of the Franklin Institute vol 349 no 2 pp 510ndash530 2012

[7] M Liu and P Shi ldquoSensor fault estimation and tolerant controlfor Ito stochastic systems with a descriptor sliding modeapproachrdquo Automatica vol 49 no 5 pp 1242ndash1250 2013

[8] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006

[9] Q X Jia W Chen Y C Zhang and X Q Chen ldquoRobustfault reconstruction via learning observers in linear parameter-varying systems subject to loss of actuator effectivenessrdquo IETControl Theory and Applications vol 8 no 1 pp 42ndash50 2014

[10] D Koenig ldquoUnknown input proportional multiple-integralobserver design for linear descriptor systems application tostate and fault estimationrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 212ndash217 2005

[11] H Hamdi M Rodrigues C Mechmeche D Theilliol and NB Braiek ldquoFault detection and isolation in linear parameter-varying descriptor systems via proportional integral observerrdquoInternational Journal of Adaptive Control and Signal Processingvol 26 no 3 pp 224ndash240 2012

[12] F Caccavale F Pierri and L Villani ldquoAdaptive observer forfault diagnosis in nonlinear discrete-time systemsrdquo Journal ofDynamic Systems Measurement and Control vol 130 no 2Article ID 021005 2008

[13] X D ZhangMM Polycarpou and T Parisini ldquoFault diagnosisof a class of nonlinear uncertain systems with Lipschitz nonlin-earities using adaptive estimationrdquo Automatica vol 46 no 2pp 290ndash299 2010

[14] K Zhang B Jiang and P Shi ldquoObserver-based integratedrobust fault estimation and accommodation design for discrete-time systemsrdquo International Journal of Control vol 83 no 6 pp1167ndash1181 2010

[15] S M Tabatabaeipour and T Bak ldquoRobust observer-basedfault estimation and accommodation of discrete-time piecewiselinear systemsrdquo Journal of the Franklin Institute vol 351 no 1pp 277ndash295 2014

[16] M Arcak and D Nesic ldquoA framework for nonlinear sampled-data observer design via approximate discrete-time models andemulationrdquo Automatica vol 40 no 11 pp 1931ndash1938 2004

[17] MAbbaszadeh andH JMarquez ldquoRobust119867infinobserver design

for sampled-data Lipschitz nonlinear systems with exact andEuler approximate modelsrdquo Automatica vol 44 no 3 pp 799ndash806 2008

[18] Z Mao B Jiang and P Shi ldquoFault-tolerant control for a classof nonlinear sampled-data systems via a Euler approximateobserverrdquo Automatica vol 46 no 11 pp 1852ndash1859 2010


[19] B Jiang P Shi and Z Mao ldquoSliding mode observer-based faultestimation for nonlinear networked control systemsrdquo CircuitsSystems and Signal Processing vol 30 no 11 pp 1ndash16 2011

[20] Q X Jia Y C Zhang W Chen and Y H Geng ldquoDesignof novel observers for robust fault detection in discrete-timeLipschitz nonlinear systems with Euler-approximate modelsrdquoInternational Journal of Automation and Control vol 8 no 3pp 191ndash210 2014

[21] F Amato C Cosentino M Mattei and G Paviglianiti ldquoAdirectfunctional redundancy scheme for fault detection andisolation on an aircraftrdquo Aerospace Science and Technology vol10 no 4 pp 338ndash345 2006

[22] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and ControlTheory PhiladelphiaPa USA Society for Industrial and Applied Mathematics(SIAM) Philadelphia PA 1994

[23] ZWangM Rodrigues DTheilliol and Y Shen ldquoActuator faultestimation observer design for discrete-time linear parameter-varying descriptor systemsrdquo International Journal of AdaptiveControl and Signal Processing 2014

[24] F Zhu and Z Han ldquoA note on observers for Lipschitz nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no10 pp 1751ndash1754 2002

[25] W Xie ldquoMulti-objective 119867infin120572-stability controller synthesis of

LTI systemsrdquo IET Control Theory and Applications vol 2 no 1pp 51ndash55 2008

[26] K Zhang B Jiang P Shi and J Xu ldquoMulti-constrained faultestimation observer design with finite frequency specificationsfor continuous-time systemsrdquo International Journal of Controlvol 87 no 8 pp 1635ndash1645 2014

[27] D Nesic A R Teel and P V Kokotovic ldquoSufficient conditionsfor stabilization of sampled-data nonlinear systems via discrete-time approximationsrdquo Systems and Control Letters vol 38 no4-5 pp 259ndash270 1999

[28] K Vijayaraghavan R Rajamani and J Bokor ldquoQuantitativefault estimation for a class of non-linear systemsrdquo InternationalJournal of Control vol 80 no 1 pp 64ndash74 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


been reported [12ndash15] However the availability of the exactmodels originated from discretization of a continuous-timeplant is usually unrealistic A practical solution to thisdifficult situation is to employ approximate discrete-timemodels especially Euler-approximate models instead ofexact models

More recently many researchers have paid much atten-tion to observer design and observer-based fault recon-struction for nonlinear discrete-time systems with Euler-approximate models Reference [16] proposes a generalframework for nonlinear observer design via the approximatediscrete models For Lipschitz nonlinear systems the authorsin [17] suggest a robust 119867

infinobserver whose main advan-

tage is that maximum admissible Lipschitz constant androbustness can be solved using LMI optimization techniquesBased on [16 17] a Euler-approximate observer (EAO) ispresented for fault reconstruction and active FTC in [18]however the robustness is not well guaranteed For nonlinearnetworked control systems an SMO-based fault reconstruc-tion approach is investigated in [19] where the chatteringphenomenon resulting from the signum function in theSMO is inevitable In [20] a Euler-approximate unknowninput observer- (UIO-) based robust fault detection strategyis developed However to the best of our knowledge fewresults have been reported on observer-based robust faultreconstruction for nonlinear discrete-time systems via Euler-approximate models In addition how to solve the errorproblem caused by the approximate models such that theobserver designed under Euler-approximate models can beimplemented on the exactmodels has still been an interestingand challenging issue All of these motivate us to pursue thisinvestigation

The main objective of this work is to design and ana-lyze an observer-based robust fault-reconstructing strategyfor a class of nonlinear discrete-time systems using Euler-approximate models First a sampled-data nonlinear systemsatisfying the Lipschitz condition is formulated into a Euler-approximate model then on the basis of our previous resultsin [20] a discrete-time Euler-approximate proportional inte-gral observer (EPIO) is constructed to simultaneously recon-struct system states and actuator faults The EPIO has anarchitecture similar to that of a nonlinear UIO Comparedwith the EAO proposed in [18] the designed EPIO is able topartially decouple external disturbances and is robust to thereminding part of external disturbances and measurementnoises As a result the accuracy of the EPIO-based faultreconstruction can be guaranteed In addition to guaranteeimplementation of the EPIO on the exact models sufficientconditions for semiglobal practical convergence which isdefined in [16] are explicitly provided Besides systematicobserver synthesis with an119867

infintechnique is effectively solved

using LMI optimization techniques Simulation results ona single-link flexible robot are also presented to verify theeffectiveness of the proposed fault-reconstructing method

The rest of this paper is organized as follows In Section 2an Euler-approximate model of a Lipschitz nonlinear systemis described andmain problems are formulated In Section 3a discrete-time EPIO is constructed and robust stability andsemiglobal practical convergence of the proposed observer

are analyzed Simulation studies are reported in Section 4and conclusions are drawn in the last section

2 System Description andProblem Formulation

A continuous-time nonlinear system subject to actuatorfaults external disturbances and measurement noises isdescribed as

(119905) = 119860119909 (119905) + Φ (119909 (119905) 119905) + 119861119906 (119905) + 11986311198891(119905) + 119865119891 (119905)

(1a)

119910 (119905) = 119862119909 (119905) + 11986321198892(119905) (1b)

where 119909(119905) isin R119899 119906(119905) isin R119898 and 119910(119905) isin R119901 representingsystem state vector control input vector and measurableoutput vector respectively Variable 119891(119905) denotes actuatorfaults with 119891(119905) isin R119898 Vectors 119889

1isin R119889 and 119889

2isin R119904

representing external disturbances and measurement noisesrespectively Without loss of generality matrices 119863

1and 119862

are assumed to be of full column rank and of full row rankrespectively The symbol Φ(119909(119905) 119905) is a continuous nonlinearfunction that is assumed to satisfy the Lipschitz condition (atleast local) that is Φ(119909(119905) 119905) minusΦ(119909(119905) 119905) le 119871

119892119909(119905) minus119909(119905)

where 119871119892is a Lipschitz constant

Herein system control input 119906(119905) is taken to be a piece-wise constant signal as 119906(119896) during the sampling intervals[119896120591 (119896 + 1)120591] with a zero-order hold where 120591 is the samplingperiod Therefore the Euler-approximate discrete model ofcontinuous-time system ((1a) (1b)) can be formulated as

119909 (119896 + 1) = (119868119899+ 120591119860) 119909 (119896) + 120591Φ (119909 (119896) 119896) + 120591119861119906 (119896)

+ 12059111986311198891(119896) + 120591119865119891 (119896)

(2a)

119910 (119896) = 119862119909 (119896) + 11986321198892(119896) (2b)

where discrete nonlinear term Φ(119909(119896) 119896) still satisfiesLipschitz constraint namely

Φ (119909 (119896) 119896) minus Φ (119909 (119896) 119896) le 119871119892119909 (119896) minus 119909 (119896) (3)

Before finishing this section the following lemma isprovided for proof of theorems in the latter section

Lemma 1 For any positive scalar 120575 there exists a positive-definite symmetric matrix119873 such that the following inequalityholds

2119886119879

119873119887 le 120575119886119879

119873119886 +1

120575119887119879

119873119887 forall (119886 119887) isin R119899

(4)

Throughout the paper 119860 gt 0 (119860 lt 0) denotes that 119860is positive (negative) definite 120582min(119860) and 120582max(119860) are theminimum and maximum eigenvalues of 119860 120590(119860) representsthe maximum singular value of matrix 119860 sdot and sdot

infin

represent Euclidean norm and infinity norm of a vector ormatrix lowast means symmetric term 119868

119899is an identity matrix

of size 119899 0119899is a zero matrix of size 119899 dagger represents a

pseudoinverse of a matrix and C denotes the set of allcomplex numbers





119911 (119896 + 1) = 119879 (119868119899+ 120591119860) 119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 120591119879119861119906 (119896)

+ 119871 (119910 (119896) minus 119910 (119896)) + 120591119879119865119891 (119896)

(5a)

119909 (119896) = 119911 (119896) + 119867119910 (119896) (5b)

119910 (119896) = 119862119909 (119896) (5c)

119891 (119896 + 1) = 119891 (119896) + 119870 (119910 (119896) minus 119910 (119896)) (5d)


Defining 119890119909(119896) = 119909(119896) minus 119909(119896) 119890

119910(119896) = 119910(119896) minus 119910(119896) and




119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 12059111987911986311198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(6a)

119890119910(119896) = 119862119890

119909(119896) + 119863

21198892(119896) (6b)



ΔΦ119879

(119896) ΔΦ (119896) minus 1198712

119892119890119879

119909(119896) 119890119909(119896) le 0 (7)


119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 120591119879[11986311

11986312]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198631

times [119889119879

11(119896) 119889

119879

12(119896)]119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198891(119896)

minus 11987111986321198892(119896) minus 119867119863

21198892(119896 + 1) + 120591119879119865119890

119891(119896)

(8)

where 11988911



11) =








11990211986312] Hence state


119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 1205911198791198641198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(9)



119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) + Δ119891 (119896) (10)


119890119891(119896 + 1) = 119890

119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) (11)




119868119899 and 119879119863



[119879 119867] [11986811989911986311

119862 0] = [119868

1198990] (12)


[119879 119867] = [1198681198990] [

11986811989911986311

119862 0]

dagger

(13)



[119890119909(119896 + 1)

119890119891(119896 + 1)

]

= [119879 (119868119899+ 120591119860) minus 119871119862 120591119879119865

minus119870119862 119868119898

] [119890119909(119896)

119890119891(119896)

] + 120591119879 [ΔΦ (119896)

0]

+ [120591119879119864 minus119871119863

2minus1198671198632

0

0 minus1198701198632

0 119868119898

]

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

(14)


Denote

119890 (119896) = [119890119909(119896)

119890119891(119896)

] 120596 (119896) =

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

119879 = [119879 0

0 119868119898

] 119860 = [119868119899+ 120591119860 120591119865

0 119868119898

]

ΔΦ (119896) = [ΔΦ (119896)

0] 119863

1= [

minus120591119879119864 0 1198671198632

0

0 0 0 minus119868119898

]

119871 = [119871

119870] 119862 = [119862 0] 119863

2= [0 minus119863

20 0]

(15)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896) + 120591119879ΔΦ (119896)

+ (1198711198632minus 1198631) 120596 (119896)

119890119910(119896) = 119862119890 (119896) minus 119863

2120596 (119896)

(16)


119891119890(119896) where

119862119891= [0 119868

119898]


Γ = ΔΦ119879

(119896) ΔΦ (119896) minus 119890119879

(119896) 119871119879

119892119871119892119890 (119896) le 0 (17)

where 119871119892= [1198711198921198681198990

0 0119898



119909 (119896 + 1) = 119879119860119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 1198631120596 (119896)

119910 (119896) = 119862119909 (119896) minus 1198632120596 (119896)

(18)

where

119909 (119896) = [119909119879

(119896) 119891119879

(119896)]119879

Φ (119909 (119896) 119896) = [Φ119879

(119909 (119896) 119896) 0]119879

(19)


(119896 + 1) = 119879119860 (119896) + 120591119879Φ ( (119896) 119896) + 119871 (119910 (119896) minus 119910 (119896))

119910 (119896) = 119862 (119896)

(20)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896)

119890119910(119896) = 119862119890 (119896)

(21)


rank [119911119868119899+119898 minus 119879119860119862

] = 119899 + 119898 forall119911 isin C |119911| ge 1 (22)

and one obtains

rank [119911119868119899+119898 minus 119879119860119862

]

= rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

(23)

(1) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119868119899 minus 119879 (119868119899 + 120591119860) minus120591119879119865

119862 0]


119862 0]

= rank [119860 119879119865

119862 0]

(24)

(2) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119911119868119899 minus 119879 (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 minus 119867119862) (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 + 120591119860)119862

] + 119898

(25)


rank [119860 119879119865

119862 0] = 119899 + 119898

rank [119911119868119899 minus (119868119899 + 120591119860)119862

] = 119899

(26)






2(119896) we shall





2(119896)





119894 119894 = 1 2 3 4 if there


119894 119894 = 1 2 3 4

such that

[[

[


2119875119879

lowast minus119875 + 1205753119871119879

119892119871119892

0


]]

]

lt 0 (27)

[[[[[[[[[[

[


119875 119868119898

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0


0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]]]

]

lt 0 (28)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] 119863ℎ= [1198671198632

0] and 119868

119898= [0

119868119898]



119891(119896) with an 119867

infin


10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891 (119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

+ 1205742

4

1003817100381710038171003817Δ119891(119896)1003817100381710038171003817

2

(29)


119875minus1

119884


119890119879




Δ119881 (119896) = [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(30)

where 119877 = (119879119860 minus 119871119862)119879

119875(119879119860 minus 119871119862) minus 119875


2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

le 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

le 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(31)

where 1205751gt 0 and 120575

2gt 0


Δ119881 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

minus 1205753Γ + 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)


+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879



Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)


(11205752))120582max(119875)Δ119891(119896)infin


[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)



2



119891(119896) can be



2(119896) and Δ119891(119896) an

119867infin


119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)


119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)



Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119911 (119896 + 1) = 119879 (119868119899+ 120591119860) 119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 120591119879119861119906 (119896)

+ 119871 (119910 (119896) minus 119910 (119896)) + 120591119879119865119891 (119896)

(5a)

119909 (119896) = 119911 (119896) + 119867119910 (119896) (5b)

119910 (119896) = 119862119909 (119896) (5c)

119891 (119896 + 1) = 119891 (119896) + 119870 (119910 (119896) minus 119910 (119896)) (5d)


Defining 119890119909(119896) = 119909(119896) minus 119909(119896) 119890

119910(119896) = 119910(119896) minus 119910(119896) and




119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 12059111987911986311198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(6a)

119890119910(119896) = 119862119890

119909(119896) + 119863

21198892(119896) (6b)



ΔΦ119879

(119896) ΔΦ (119896) minus 1198712

119892119890119879

119909(119896) 119890119909(119896) le 0 (7)


119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 120591119879[11986311

11986312]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198631

times [119889119879

11(119896) 119889

119879

12(119896)]119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198891(119896)

minus 11987111986321198892(119896) minus 119867119863

21198892(119896 + 1) + 120591119879119865119890

119891(119896)

(8)

where 11988911



11) =








11990211986312] Hence state


119890119909(119896 + 1) = (119879 (119868

119899+ 120591119860) minus 119871119862) 119890

119909(119896) + 120591119879ΔΦ (119896)

+ 1205911198791198641198891(119896) minus 119871119863

21198892(119896)

minus 11986711986321198892(119896 + 1) + 120591119879119865119890

119891(119896)

(9)



119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) + Δ119891 (119896) (10)


119890119891(119896 + 1) = 119890

119891(119896) minus 119870119862119890

119909(119896) minus 119870119863

21198892(119896) (11)




119868119899 and 119879119863



[119879 119867] [11986811989911986311

119862 0] = [119868

1198990] (12)


[119879 119867] = [1198681198990] [

11986811989911986311

119862 0]

dagger

(13)



[119890119909(119896 + 1)

119890119891(119896 + 1)

]

= [119879 (119868119899+ 120591119860) minus 119871119862 120591119879119865

minus119870119862 119868119898

] [119890119909(119896)

119890119891(119896)

] + 120591119879 [ΔΦ (119896)

0]

+ [120591119879119864 minus119871119863

2minus1198671198632

0

0 minus1198701198632

0 119868119898

]

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

(14)


Denote

119890 (119896) = [119890119909(119896)

119890119891(119896)

] 120596 (119896) =

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

119879 = [119879 0

0 119868119898

] 119860 = [119868119899+ 120591119860 120591119865

0 119868119898

]

ΔΦ (119896) = [ΔΦ (119896)

0] 119863

1= [

minus120591119879119864 0 1198671198632

0

0 0 0 minus119868119898

]

119871 = [119871

119870] 119862 = [119862 0] 119863

2= [0 minus119863

20 0]

(15)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896) + 120591119879ΔΦ (119896)

+ (1198711198632minus 1198631) 120596 (119896)

119890119910(119896) = 119862119890 (119896) minus 119863

2120596 (119896)

(16)


119891119890(119896) where

119862119891= [0 119868

119898]


Γ = ΔΦ119879

(119896) ΔΦ (119896) minus 119890119879

(119896) 119871119879

119892119871119892119890 (119896) le 0 (17)

where 119871119892= [1198711198921198681198990

0 0119898



119909 (119896 + 1) = 119879119860119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 1198631120596 (119896)

119910 (119896) = 119862119909 (119896) minus 1198632120596 (119896)

(18)

where

119909 (119896) = [119909119879

(119896) 119891119879

(119896)]119879

Φ (119909 (119896) 119896) = [Φ119879

(119909 (119896) 119896) 0]119879

(19)


(119896 + 1) = 119879119860 (119896) + 120591119879Φ ( (119896) 119896) + 119871 (119910 (119896) minus 119910 (119896))

119910 (119896) = 119862 (119896)

(20)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896)

119890119910(119896) = 119862119890 (119896)

(21)


rank [119911119868119899+119898 minus 119879119860119862

] = 119899 + 119898 forall119911 isin C |119911| ge 1 (22)

and one obtains

rank [119911119868119899+119898 minus 119879119860119862

]

= rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

(23)

(1) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119868119899 minus 119879 (119868119899 + 120591119860) minus120591119879119865

119862 0]


119862 0]

= rank [119860 119879119865

119862 0]

(24)

(2) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119911119868119899 minus 119879 (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 minus 119867119862) (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 + 120591119860)119862

] + 119898

(25)


rank [119860 119879119865

119862 0] = 119899 + 119898

rank [119911119868119899 minus (119868119899 + 120591119860)119862

] = 119899

(26)






2(119896) we shall





2(119896)





119894 119894 = 1 2 3 4 if there


119894 119894 = 1 2 3 4

such that

[[

[


2119875119879

lowast minus119875 + 1205753119871119879

119892119871119892

0


]]

]

lt 0 (27)

[[[[[[[[[[

[


119875 119868119898

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0


0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]]]

]

lt 0 (28)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] 119863ℎ= [1198671198632

0] and 119868

119898= [0

119868119898]



119891(119896) with an 119867

infin


10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891 (119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

+ 1205742

4

1003817100381710038171003817Δ119891(119896)1003817100381710038171003817

2

(29)


119875minus1

119884


119890119879




Δ119881 (119896) = [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(30)

where 119877 = (119879119860 minus 119871119862)119879

119875(119879119860 minus 119871119862) minus 119875


2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

le 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

le 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(31)

where 1205751gt 0 and 120575

2gt 0


Δ119881 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

minus 1205753Γ + 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)


+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879



Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)


(11205752))120582max(119875)Δ119891(119896)infin


[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)



2



119891(119896) can be



2(119896) and Δ119891(119896) an

119867infin


119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)


119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)



Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Denote

119890 (119896) = [119890119909(119896)

119890119891(119896)

] 120596 (119896) =

[[[

[

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]

]

119879 = [119879 0

0 119868119898

] 119860 = [119868119899+ 120591119860 120591119865

0 119868119898

]

ΔΦ (119896) = [ΔΦ (119896)

0] 119863

1= [

minus120591119879119864 0 1198671198632

0

0 0 0 minus119868119898

]

119871 = [119871

119870] 119862 = [119862 0] 119863

2= [0 minus119863

20 0]

(15)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896) + 120591119879ΔΦ (119896)

+ (1198711198632minus 1198631) 120596 (119896)

119890119910(119896) = 119862119890 (119896) minus 119863

2120596 (119896)

(16)


119891119890(119896) where

119862119891= [0 119868

119898]


Γ = ΔΦ119879

(119896) ΔΦ (119896) minus 119890119879

(119896) 119871119879

119892119871119892119890 (119896) le 0 (17)

where 119871119892= [1198711198921198681198990

0 0119898



119909 (119896 + 1) = 119879119860119909 (119896) + 120591119879Φ (119909 (119896) 119896) + 1198631120596 (119896)

119910 (119896) = 119862119909 (119896) minus 1198632120596 (119896)

(18)

where

119909 (119896) = [119909119879

(119896) 119891119879

(119896)]119879

Φ (119909 (119896) 119896) = [Φ119879

(119909 (119896) 119896) 0]119879

(19)


(119896 + 1) = 119879119860 (119896) + 120591119879Φ ( (119896) 119896) + 119871 (119910 (119896) minus 119910 (119896))

119910 (119896) = 119862 (119896)

(20)


119890 (119896 + 1) = (119879119860 minus 119871119862) 119890 (119896)

119890119910(119896) = 119862119890 (119896)

(21)


rank [119911119868119899+119898 minus 119879119860119862

] = 119899 + 119898 forall119911 isin C |119911| ge 1 (22)

and one obtains

rank [119911119868119899+119898 minus 119879119860119862

]

= rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

(23)

(1) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119868119899 minus 119879 (119868119899 + 120591119860) minus120591119879119865

119862 0]


119862 0]

= rank [119860 119879119865

119862 0]

(24)

(2) If 119911 = 1

rank[

[

119911119868119899minus 119879 (119868

119899+ 120591119860) minus120591119879119865

0 (119911 minus 1) 119868119898

119862 0

]

]

= rank [119911119868119899 minus 119879 (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 minus 119867119862) (119868119899 + 120591119860)119862

] + 119898

= rank [119911119868119899 minus (119868119899 + 120591119860)119862

] + 119898

(25)


rank [119860 119879119865

119862 0] = 119899 + 119898

rank [119911119868119899 minus (119868119899 + 120591119860)119862

] = 119899

(26)






2(119896) we shall





2(119896)





119894 119894 = 1 2 3 4 if there


119894 119894 = 1 2 3 4

such that

[[

[


2119875119879

lowast minus119875 + 1205753119871119879

119892119871119892

0


]]

]

lt 0 (27)

[[[[[[[[[[

[


119875 119868119898

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0


0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]]]

]

lt 0 (28)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] 119863ℎ= [1198671198632

0] and 119868

119898= [0

119868119898]



119891(119896) with an 119867

infin


10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891 (119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

+ 1205742

4

1003817100381710038171003817Δ119891(119896)1003817100381710038171003817

2

(29)


119875minus1

119884


119890119879




Δ119881 (119896) = [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(30)

where 119877 = (119879119860 minus 119871119862)119879

119875(119879119860 minus 119871119862) minus 119875


2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

le 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

le 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(31)

where 1205751gt 0 and 120575

2gt 0


Δ119881 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

minus 1205753Γ + 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)


+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879



Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)


(11205752))120582max(119875)Δ119891(119896)infin


[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)



2



119891(119896) can be



2(119896) and Δ119891(119896) an

119867infin


119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)


119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)



Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(119896) we shall





2(119896)





119894 119894 = 1 2 3 4 if there


119894 119894 = 1 2 3 4

such that

[[

[


2119875119879

lowast minus119875 + 1205753119871119879

119892119871119892

0


]]

]

lt 0 (27)

[[[[[[[[[[

[


119875 119868119898

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0


0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]]]

]

lt 0 (28)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] 119863ℎ= [1198671198632

0] and 119868

119898= [0

119868119898]



119891(119896) with an 119867

infin


10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891 (119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

+ 1205742

4

1003817100381710038171003817Δ119891(119896)1003817100381710038171003817

2

(29)


119875minus1

119884


119890119879




Δ119881 (119896) = [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(30)

where 119877 = (119879119860 minus 119871119862)119879

119875(119879119860 minus 119871119862) minus 119875


2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

le 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896)

le 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(31)

where 1205751gt 0 and 120575

2gt 0


Δ119881 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879

][119890 (119896)

ΔΦ (119896)]

minus 1205753Γ + 1205751119890119879

(119896) (119879119860 minus 119871119862)119879

119875 (119879119860 minus 119871119862) 119890 (119896)

+1

1205751

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)


+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879



Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)


(11205752))120582max(119875)Δ119891(119896)infin


[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)



2



119891(119896) can be



2(119896) and Δ119891(119896) an

119867infin


119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)


119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)



Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 12057521205912

ΔΦ119879

(119896) 119879119879

119875119879ΔΦ (119896)

+1

1205752

Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896) + Δ119891

119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[1198771+ 1205753119871119879

119892119871119892

120591 (119879119860 minus 119871119862)119879

119875119879

lowast (1 + 1205752) 1205912

119879119879

119875119879 minus 1205753119868119899+119898

]

times [119890 (119896)

ΔΦ (119896)] + (1 +

1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

le 120585119879

(119896)Ω120585 (119896) + (1 +1

1205751

+1

1205752

)Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

(32)

where 120585(119896) = [119890(119896)

ΔΦ(119896)] and Ω = [

1198771+1205753119871

119879

119892119871119892120591(119879119860minus119871119862)

119879119875119879

lowast (1+1205752)1205912119879

119879

119875119879minus1205753119868119899+119898

]

with 1198771= (1 + 120575

1)(119879119860 minus 119871119862)

119879



Δ119881 (119896) le minus120598 119890 (119896)2

+ ] (33)


(11205752))120582max(119875)Δ119891(119896)infin


[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]119875minus1

[radic1 + 120575

1(119879119860 minus 119871119862)

119879

119875

120591radic1 + 1205752119879119879

119875

]

119879

+ [minus119875 + 120575

3119871119879

119892119871119892

0

lowast minus1205753119868119899+119898

] lt 0

(34)



2



119891(119896) can be



2(119896) and Δ119891(119896) an

119867infin


119869 =

infin

sum

119896=0

[119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(35)


119869 le

infin

sum

119896=0

[Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896)

minus 1205742

2119889119879

2(119896) 1198892(119896) minus 120574

2

3119889119879

2(119896 + 1)

times 1198892(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)]

(36)



Δ119881 (119896) + 119890119879

119891(119896) 119890119891(119896) minus 120574

2

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1) minus 120574

2

4Δ119891119879

(119896) Δ119891 (119896)

le [119890 (119896)

ΔΦ (119896)]

119879

[119877 + 1205754119871119879

119892119871119892120591 (119879119860 minus 119871119862)

119879

119875119879

lowast 1205912

119879119879

119875119879 minus 1205754119868119899+119898

][119890 (119896)

ΔΦ (119896)]

+ 2120591119890119879

(119896) (119879119860 minus 119871119862)119879

1198751198791198641198891(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

11987511987111986321198892(119896)

minus 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875119867119863ℎ1198892(119896 + 1)

+ 2119890119879

(119896) (119879119860 minus 119871119862)119879

119875 119868119898Δ119891 (119896)

+ 21205912

ΔΦ119879

(119896) 119879119879

1198751198791198641198891(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

11987511987111986321198892(119896)

minus 2120591ΔΦ119879

(119896) 119879119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591ΔΦ119879

(119896) 119879119879

119875 119868119898Δ119891 (119896) + 120591

2

119889119879

1(119896) (119879119864)

119879

1198751198791198641198891(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

11987511987111986321198892(119896)

minus 2120591119889119879

1(119896) (119879119864)

119879

119875119867119863ℎ1198892(119896 + 1)

+ 2120591119889119879

1(119896) (119879119864)

119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896) (119871119863

2)119879

11987511987111986321198892(119896)

+ 2119889119879

2(119896) (119871119863

2)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896) (119871119863

2)119879

119875 119868119898Δ119891 (119896)

+ 119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875119867119863ℎ1198892(119896 + 1)

minus 2119889119879

2(119896 + 1) (119867119863

ℎ)119879

119875 119868119898Δ119891 (119896)

+ Δ119891119879

(119896) 119868119879

119898119875 119868119898Δ119891 (119896)

minus 1205742

1119889119879

1(119896) 1198891(119896) minus 120574

2

2119889119879

2(119896) 1198892(119896)

minus 1205742

3119889119879

2(119896 + 1) 119889

2(119896 + 1)

minus 1205742

4Δ119891119879

(119896) Δ119891 (119896)

=

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

119879

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

times

[[[[[[[

[

119890 (119896)

ΔΦ (119896)

1198891(119896)

1198892(119896)

1198892(119896 + 1)

Δ119891 (119896)

]]]]]]]

]

= 120577119879

(119896)Π120577 (119896)

(37)


where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120577 (119896)

= [119890119879

(119896) ΔΦ119879

(119896) 119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1) Δ119891

119879

(119896)]119879

Π =

[[[[[[[

[

Π11

Π12

Π13

Π14

Π15

Π16

lowast Π22

Π23

Π24

Π25

Π26

lowast lowast Π33

Π34

Π35

Π36


Π45

Π46


Π56


]]]]]]]

]

(38)

with Π11

= 119877 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891 Π12

= 120591(119879119860 minus 119871119862)119879

119875119879Π13

= 120591(119879119860 minus 119871119862)119879

119875119879119864 Π14

= minus(119879119860 minus 119871119862)119879

1198751198711198632

Π15= minus(119879119860 minus 119871119862)

119879

119875119867119863ℎ Π16= (119879119860 minus 119871119862)

119879

119875 119868119898 Π22=

1205912

119879119879

119875119879 minus 1205754119868119899+119898

Π23

= 1205912

119879119879

119875119879119864 Π24

= minus120591119879119879

1198751198711198632

Π25= minus120591119879

119879

119875119867119863ℎ Π26= 120591119879119879

119875 119868119898 Π33= 1205912

(119879 119864)119879

119875119879119864 minus

1205742

1119868119889 Π34

= minus120591(119879119864)119879

1198751198711198632 Π35

= minus120591(119879119864)119879

119875119867119863ℎ

Π36

= 120591(119879119864)119879

119875 119868119898 Π44

= (1198711198632)119879

1198751198711198632minus 1205742

2119868119904 Π45

=

(1198711198632)119879

119875119867119863ℎ Π46= minus(119871119863

2)119879

119875 119868119898 Π55= (119867119863

ℎ)119879

119875119867119863ℎminus

1205742

3119868119904 Π56= minus(119867119863

ℎ)119879

119875 119868119898 and Π

66= 119868119879

119898119875 119868119898minus 1205742

4119868119898


[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119875minus1

[[[[[[[[[[[

[

(119879119860 minus 119871119862)119879

119875

120591119879119879

119875

120591 (119879119864)119879

119875

119871119879

119875

(119867119863ℎ)119879

119875

119868119879

119898119875

]]]]]]]]]]]

]

119879

+

[[[[[[[[

[

minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0 0

lowast minus1205754119868119899+119898

0 0 0 0


1119868119889

0 0 0


2119868119904

0 0


3119868119904

0


4119868119898

]]]]]]]]

]

lt 0

(39)


1(119896) isin

1198712[0infin) 119889

2(119896) isin 119871

2[0infin) and Δ119891(119896) isin 119871

2[0infin) one


proof



infinattenuation




and

[[[[

[


1

lowast minus119875 + 1205754119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889+119898

]]]]

]

lt 0

(40)



119891(119896)with

an 119867infin



119884




infin



2(119896)












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 1205742 and 120574

3 if there


[[[[[[[[

[


ℎ

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0 0 0


0 0 0


1119868119889

0 0


2119868119904

0


3119868119904

]]]]]]]]

]

lt 0

(41)

where 119871119892= [1198711198921198681198990

0 0119898

] 119864 = [119864

0] and 119863

ℎ= [1198671198632

0] then



10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

le 1205742

1

10038171003817100381710038171198891(119896)1003817100381710038171003817

2

+ (1205742

2+ 1205742

3)10038171003817100381710038171198892(119896)

1003817100381710038171003817

2

(42)


119875minus1

119884



[[[[[

[


13

lowast minus119875 + 120575119871119879

119892119871119892+ 119862119879

119891119862119891

0 0


0


1198682119904+119889

]]]]]

]

lt 0

(43)

where 119871119892

= [1198711198921198681198990

0 0119898

] 11986313

= [minus120591119879119864 0 119867119863

2

0 0 0] and 119863

23=



119891(119896)


119889(119896) where 120596

119889(119896) =

[119889119879

1(119896) 119889

119879

2(119896) 119889

119879

2(119896 + 1)]

119879



119884


119909(119896)








11) is satisfied















119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 (0) minus 119909 (0) le 119863


(44)

imply

119909 (119896) minus 119909 (119896) le 120573 (119909 (0) minus 119909 (0) 119896120591) + 119889 (45)



1gt 1 a fixed



1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1= minus1198761

(46)

120582min (1198761) = 120591 (47)


119890119879

119909(119896)1198751119890119909(119896) then

1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817

=10038171003817100381710038171003817119890119879

119909(1198961) 1198751119890119909(1198961) minus 119890119879

119909(1198962) 1198751119890119909(1198962)10038171003817100381710038171003817

=100381710038171003817100381710038171003817[119890119909(1198961) + 119890119909(1198962)]119879

1198751[119890119909(1198961) minus 119890119909(1198962)]100381710038171003817100381710038171003817

le 120582max (1198751)1003817100381710038171003817119890119909 (1198961) + 119890119909 (1198962)

1003817100381710038171003817

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817

(48)



1003817100381710038171003817 le 119872 (49)

Thus1003817100381710038171003817119881 (1198961) minus 119881 (119896

2)1003817100381710038171003817 le 119872

1003817100381710038171003817119890119909 (1198961) minus 119890119909 (1198962)1003817100381710038171003817 (50)

Δ119881 (119896)

le 119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896) minus 119890

119879

119909(119896) 1198751119890119909(119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

+ 2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

+ 1205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896) + 2120591

2

ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

+ 1205912

119890119879

119891(119896) (119879119865)

119879

1198751(119879119865) 119890

119891(119896)

(51)


2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879ΔΦ (119896)

le 1205981119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205981

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

(52)

where 1205981gt 0

2120591119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

1198751119879119865119890119891(119896)

le 1205982119890119879

119909(119896) (119879 (119868

119899+ 120591119860) minus 119871119862)

119879

times 1198751(119879 (119868119899+ 120591119860) minus 119871119862) 119890

119909(119896)

+1205912

1205982

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(53)


2120591ΔΦ119879

(119896) 119879119879

1198751119879119865119890119891(119896)

le 1205983ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+1205912

1205983

119890119879

119891(119879119865)119879

1198751119879119865119890119891(119896)

(54)

where 1205983gt 0


Δ119881 (119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

ΔΦ119879

(119896) 119879119879

1198751119879ΔΦ (119896)

+ 12059031205912

119890119879

119891(119896) (119879119865)

119879

1198751119879119865119890119891(119896)

le 119890119879

119909(119896)

times (1205901(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198751(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

1)

times 119890119909(119896) + 120590

21205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205912

1205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(55)

where 1205901= 1 + 120598

1+ 1205982 1205902= 1 + (1120598

1) + 1205983 and 120590

3= 1 +

(11205982) + (1120598

3) with 120598

119894gt 0 119894 = 1 2 3



Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119881 (119896)

120591le minus

1003817100381710038171003817119890119909 (119896)1003817100381710038171003817

2

+ 12059021205911198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

+ 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(56)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582min (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

1205880(120591) ≜ 120591

]0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) ≜ 12059021205912

1198712

1198921205902

(119879) 120582max (1198751)1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817

2

]1(119909 (119896)) ≜ 0 ]

2(119906 (119896)) ≜ 0

]3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817) = 12059031205911205902

(119879119865) 120582max (1198751)10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817

2

(57)


1205721(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) le 119881 (119896) le 1205722(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

119881 (119896 + 1) minus 119881 (119896)

120591

le minus1205723(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817)

+ 1205880(120591) []0(1003817100381710038171003817119890119909 (119896)

1003817100381710038171003817) + ]1(119909 (119896))

+]2(119906 (119896)) + ]

3(10038171003817100381710038171003817119890119891(119896)

10038171003817100381710038171003817)]

(58)



infinand









2isin


(119879 (119868119899+ 120591119860) minus 119871119862)

119879

1198752(119879 (119868119899+ 120591119860) minus 119871119862) minus 119875

2= minus1198762

(59)

120591 le 120591lowast

=120582max (1198762) minus 1205984 [120582max (1198752) + 120582max (1198762)]

2119871119892120590 (119879) 120582max (1198752)radic1 + (120582max (1198762) 120582max (1198752))

(60)





119860 =

[[[

[

0 1 0 0


0 0 0 1


]]]

]

Φ (119909 (119905) 119905) =

[[[

[

0

0

0

minus0333 sin (1199093(119905))

]]]

]

119861 =

[[[

[

0

2160

0

0

]]]

]

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

1198631=

[[[

[

01 0

0 01

0 0

0 0

]]]

]

1198632= 01119868

3

(61)






119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879 =

[[[

[

0 0 0 0

0 05 0 0

0 0 05 0

0 0 0 10

]]]

]

119867 =

[[[

[

10 0 0

0 05 0

0 0 05

0 0 0

]]]

]

(62)





2= 20230 120574

3= 20230



119871 =

[[[

[

00174 00014 01951

00014 16401 00359

00358 minus00030 03299

13353 minus02860 88951

]]]

]

119870 = [

[

minus00067

106257

minus04187

]

]

119879

(63)


Choosing 1198761= 001119868

4and 120590

1= 101 matrix 119875

1can be


1198751=

[[[

[

00961 minus00119 10957 minus00976

minus00119 00020 minus01611 00164

10957 minus01611 150770 minus13221

minus00976 00164 minus13221 01317

]]]

]

(64)


1198752= 102

times

[[[

[

35324 minus00081 02805 minus00369

minus00081 00006 minus00374 00047

02805 minus00374 66747 minus03458

minus00369 00047 minus03458 00376

]]]

]

1198762= 102

times

[[[

[


minus00044 00001 00069 00006

minus00268 00069 23426 00201

minus00077 00006 00201 00042

]]]

]

(65)

and 120591lowast



1= 03 120574

2=


to

119871 =

[[[

[


minus01134 03838 00023

002766 01753 05899

04578 15168 144412

]]]

]

119870 = [

[

00121

06371

minus01017

]

]

119879

(66)

0

005

01

0 1 2 3 4 5


t (s)

minus005

Mot

or p

ositi

onx1

(rad

)






2(119896) are



119891 (119896) = 02 2 s le 119896120591 le 6 s0 others

(67)


119891 (119896) =


(68)



0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10

0

005

01

015

02

t (s)

minus01

minus005

Mot

or v

eloc

ityx2

(rad

s)



0

002

004

006

008

0 1 2 3 4 5


t (s)

Link

pos

ition

x3

(rad

)


5 Conclusions


infintechniques


0 1 2 3 4 5

0

02

04

minus02


t (s)

Link

vel

ocity

x4

(rad

s)


0 2 4 6 8

0

02

04

The c

hang

e of b

us v

olta

ge (V

)


t (s)

minus02






0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


0 2 4 6 8

The c

hang

e of b

us v

olta

ge (V

)


0

02

04

minus02

t (s)


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Robust Fault Reconstruction in Discrete...

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