Research ArticleRobust119867

infinFuzzy Control for Nonlinear

Discrete-Time Stochastic Systems with MarkovianJump and Parametric Uncertainties

Huiying Sun and Long Yan

College of Electrical Engineering and Automation Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Huiying Sun sunhyinggmailcom

Received 4 May 2014 Revised 2 August 2014 Accepted 3 August 2014 Published 18 August 2014

Academic Editor Peng Shi

Copyright copy 2014 H Sun and L Yan 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 paper mainly investigates the Hinfin

fuzzy control problem for a class of nonlinear discrete-time stochastic systems withMarkovian jump and parametric uncertainties The class of systems is modeled by a state space Takagi-Sugeno (T-S) fuzzy modelthat has linear nominal parts and norm-bounded parameter uncertainties in the state and output equations AnH

infincontrol design

method is developed by using the Lyapunov function The decoupling technique makes the Lyapunov matrices and the systemmatrices separated which makes the control design feasible Then some strict linear matrix inequalities are derived on robustH

infin

norm conditions in which both robust stability and Hinfin

performance are required to be achieved Finally a computer-simulatedtruck-trailer example is given to verify the feasibility and effectiveness of the proposed design method

1 Introduction

Over the past decade there has been a rapidly growinginterest in control and filtering of nonlinear systems andthere have been many successful applications [1ndash4] In [1]based on the sum of squares approach sufficient conditionsfor the existence of a nonlinear state feedback controllerfor polynomial discrete-time systems are given in terms ofsolvability of polynomial matrix inequalities Reference [3]presents a stochastic distribution control algorithm an opti-mal control law is then obtained using the penalty functionmethod Despite the success it has become evident thatmanybasic issues remain to be addressed In particular the controltechnique based on the so-called Takagi-Sugeno (T-S) fuzzymodel [5] has attracted a great deal of attention (see [6ndash12]) This is because it is regarded as a powerful solution tobridging the gap between the fruitful linear control and thefuzzy logic control targeting complex nonlinear systems Thecommon practice of the technique is as follows First thenonlinear plant is represented by the T-S fuzzy model Thisfuzzy model is described by a set of fuzzy IF-THEN ruleswhich correspond to local linear input-output relations of

the system respectively The overall model of the system isachieved by fuzzy ldquoblendingrdquo of these fuzzy models Thenbased on this fuzzy model a control design is carriedout based on the fuzzy models via the so-called paralleldistributed compensation (PDC) scheme The idea is thata linear feedback control is designed for each local linearmodel The resulting overall controller which is in generalnonlinear is again a fuzzy blending of each individual linearcontroller Their works introduce the model-based analysismethods into fuzzy logic control

In recent years the stability issue and robust performanceof T-S fuzzy control systems have been discussed in anextensive literature Since the pioneering work on the so-called119867

infinoptimal control theory there has been a dramatic

progress in the 119867infin

control theory Recently the problem ofnonlinear 119867

infincontrol was intensively studied (see [5 7 8

13]) The design of119867infin

controller for fuzzy dynamic systemswas presented in the paper [5] Reference [4] introduces anew class of discrete-time networked nonlinear systems withmixed random delays and packet dropouts and sufficientconditions for the existence of an admissible filter are estab-lished which ensure the asymptotical stability as well as a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 235804 11 pageshttpdxdoiorg1011552014235804

2 Mathematical Problems in Engineering

prescribed 119867infin

performance The authors in [13] analyzedthe 119867

infincontrol design problem systematically for a class of

nonlinear stochastic active fault-tolerant control systemswithMarkovian parameters119867

infincontroller designwhichwas con-

sidered in [7] noted that most of the aforementioned researchefforts focused on the use of single quadratic Lyapunovfunction which tended to givemore conservative conditionsThis is because with the use of parameter-dependent orbasis-dependent Lyapunov function less conservative controlresults can be obtained than with the use of single Lyapunovquadratic function More recently there were many resultson stability analysis and control synthesis of discrete-timeuncertain systems based on parameter-dependent Lyapunovfunctions [9] or basis-dependent Lyapunov function (see[8 14]) It is shown that with the use of Lyapunov functionwell-pleasing control results can be obtained

In practice a lot of physical systems have variablestructures subject to random changes These changes mayresult from abrupt phenomena such as random failures andrepairs of the components changes in the interconnectionsof subsystems and sudden environment changes The so-called Markovian jump systems (MJSS) are special class ofhybrid systems As one of the most basic dynamics modelsMarkov jump nonlinear system can be used to representthese random failure processes in manufacturing and someinvestment portfolio models In theMJSS the random jumpsin system parameters are governed by a Markov processwhich takes values in a finite set The class of systems mayrepresent a large variety of processes including those in theproduction systems fault-tolerant systems communicationsystems and economic systems [15] In the past two decadesmany important issues of MJSS were researched extensivelysuch as the controllability and observability [16] stability andstabilization (see [17ndash22])119867

2[23] and119867

infinperformance [24ndash

27] and robustness [28 29] To the best of our knowledgedespite these efforts very few results are available for thecontrol design for nonlinear MJSS In this study a mode-dependent control design is proposed to achieve robuststability of uncertain discrete-time nonlinear MJSS via fuzzycontrol

In the paper robust 119867infin

control is investigated fornonlinear discrete-time stochastic MJSS with parametricuncertainties Under our proposed fuzzy rules the nonlineardiscrete-time stochastic MJSS could be translated into a classof equivalent T-S fuzzymodels with parametric uncertaintiesThe uncertainty could be translated into a linear fractionalform which includes the norm-bounded uncertainty as aspecial case and can describe a kind of rational nonlinearitiesWith the PDC scheme the control law is designed tomake theclosed-loop system with 119867

infinnorm bound 120574 stable Besides

some matrix variables and the Lyapunov function for robuststabilization with 119897

2-norm bound for the fuzzy discrete-time

stochastic MJSS are given It is shown that the solution of thecontrol design problem can be obtained by solving a class ofLMIs

The paper is organized as follows Section 2 discussesthe T-S fuzzy models And definitions and preliminaryresults are given for uncertain nonlinear discrete-time MJSSSection 3 gives the analysis results of robust stability with119867

infin

performance and the results are employed in the followingto develop an 119867

infincontrol design In Section 4 a numerical

simulation example is proposed to illustrate the effectivenessof the approach Finally the paper is concluded in Section 5

For convenience the following basic notations areadopted throughout the paperR119899 denotes the 119899-dimensionalreal space andR119899times119898 denotes the set of all real 119899times119898matrices1198801015840 indicates the transpose of matrix 119880 and 119880 ge 0 (119880 gt 0)

represents a nonnegative definite (positive definite) matrixSimilarly 119880 le 0 (119880 lt 0) represents a nonpositive definitematrix (negative definite) 119897

2[0infin) refers to the space of

square summable infinite vector sequences sdot 2stands for

the usual 1198972[0infin) norm 119864(sdot) represents the mathematical

expectation

2 Problem Formulation and Preliminaries

Consider the following nonlinear MJSS

119909 (119896 + 1) = 119891 (119909 (119896) 120579119896) + 119892 (119909 (119896) 120579

119896) 119906 (119896)

+ 119896 (119909 (119896) 120579119896) V (119896) + 119897 (119909 (119896) 120579

119896) 119908 (119896)

119910 (119896) = 119898 (119909 (119896) 120579119896) + 119905 (119909 (119896) 120579

119896) 119906 (119896)

+ 119899 (119909 (119896) 120579119896) V (119896)

(1)

Assume that 119891(119909(119896) 120579119896) 119892(119909(119896) 120579

119896) 119896(119909(119896) 120579

119896) 119897(119909(119896) 120579

119896)

119898(119909(119896) 120579119896) 119905(119909(119896) 120579

119896) and 119899(119909(119896) 120579

119896) are Borel measurable

on R119899 And 119909(119896) isin R119899 is the system state 119906(119896) isin R119898

and V(119896) isin R119901 represent the system control inputs anddisturbance signal and 119910(119896) isin R119902 is system output 119908(119896) isa sequence of real random variables defined on a completeprobability space (ΩF

119896 119875) which is wide sense stationary

second-order processes with 119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] =120575119894119895 where 120575

119894119895refers to a Kronecker function that is 120575

119894119895= 1

if 119894 = 119895 and 120575119894119895= 0 if 119894 = 119895 120579

119896 119896 ge 0 is a measurable

Markov chain taking values in a finite set X = 1 2 119873with transition probability matrix P = [119901

120572120573] where

119901120572120573= 119875 (120579

119896+1= 120573 | 120579

119896= 120572) forall120572 120573 isin X 119896 ge 0 (2)

The paper considers the nonlinear discrete-time MJSS whichcan be described by the following T-S fuzzy model withuncertainties

Rule i If 1199111(119896) is 119865119894

1 1199112(119896) is 119865119894

2 and 119911

119899(119896) is 119865119894

119899 then

119909 (119896 + 1) = AA119894120579119896119909 (119896) +B

119894120579119896119906 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)

119910 (119896) =M119894120579119896119909 (119896) +T

119894120579119896119906 (119896) +N

119894120579119896V (119896)

(3)

The 119865119894119895are fuzzy sets 119903 is the number of IF-THEN rules

Moreover A119894120579119896

B119894120579119896

C119894120579119896

D119894120579119896

M119894120579119896

T119894120579119896

and N119894120579119896

are system matrices with parametric uncertainties Besides1199111(119896) 1199112(119896) 119911

119899(119896) are the premise variables of the fuzzy

modes and they are the functions of state variables

Mathematical Problems in Engineering 3

Following the PDC scheme we consider a state feedbackfuzzy controller which shares the same structure of the aboveT-S fuzzy system as follows

Rule j IF 1199111(119896) is 119865119895

1 1199112(119896) is 119865119895

2 and 119911

119899(119896) is 119865119895

119899 then

119906 (119896) = 119870119895120579119896119909 (119896) (4)

where119870119895120579119896

are the local feedback gain matricesAnd the fuzzy basis functions are given by

ℎ119894[119911 (119896)] =

prod119899

119895=1120583119894119895[119911119895(119896)]

sum119903

119897=1prod119899

119895=1120583119894119895[119911119895(119896)]

119894 = 1 2 119903 (5)

where 120583119894119895[119911119895(119896)] is the grade of membership of 119911

119895(119896) in 119865119894

119895 By

definition the fuzzy basis functions satisfy

ℎ119894[119911 (119896)] ge 0 119894 = 1 2 119903

119903

sum

119894=1

ℎ119894[119911 (119896)] = 1

(6)

A more compact presentation of the discrete-time T-S fuzzymodel is given by

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) (A

119894120579119896119909 (119896) +B

119894120579119896119906 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896))

(7)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (M119894120579119896119909 (119896) +T

119894120579119896119906 (119896) +N

119894120579119896V (119896))

(8)

where

A119894120579119896= 119860119894120579119896+ Δ119860119894120579119896

B119894120579119896= 119861119894120579119896+ Δ119861119894120579119896

C119894120579119896= 119862119894120579119896+ Δ119862119894120579119896

D119894120579119896= 119863119894120579119896

M119894120579119896= 119872119894120579119896+ Δ119872

119894120579119896

T119894120579119896= 119879119894120579119896+ Δ119879119894120579119896

N119894120579119896= 119873119894120579119896+ Δ119873119894120579119896

(9)

And 119860119894120579119896

119861119894120579119896

119862119894120579119896

119863119894120579119896

119872119894120579119896

119879119894120579119896

and119873119894120579119896

are assumedto be deterministic matrices with appropriate dimensionΔ119860119894120579119896 Δ119861119894120579119896 Δ119862119894120579119896 Δ119872119894120579119896 Δ119879119894120579119896

and Δ119873119894120579119896

are unknown

matrices which represent the time-varying parameter uncer-tainties and are assumed to be of the form

[

Δ119860119894120579119896

Δ119861119894120579119896

Δ119862119894120579119896

Δ119872119894120579119896

Δ119879119894120579119896Δ119873119894120579119896

] = [

1198641119894

1198642119894

]Δ [119867111989411986721198941198673119894]

Δ = [119868 minus 119865119894120579119896119869]

minus1

119865119894120579119896

(10)

where 1198641119894 1198671119894 1198642119894 1198672119894 1198673119894 and 119869 are known real constant

matrices with appropriate dimension and unknown nonlin-ear time-varying matrix 119865

119894120579119896isin R119898times119899 satisfying 119865

1198941205791198961198651015840

119894120579119896

le 119868To guarantee that the matrix 119868 minus 119865

119894120579119896119869 is invertible for all

admissible 119865119894120579119896

it is necessary that 119868 minus 1198691198691015840 gt 0The following definition and lemmas will be used later

Definition 1 The discrete-time unforced uncertain fuzzysystem is said to be robust stable with 119867

infinnorm bound 120574

if it is stable with 119867infin

norm bound 120574 for all uncertainlyadmissible 119865

119894120579119896 For a given control law (4) and a prescribed

level of disturbance attenuation 120574 gt 0 to be achieved thediscrete-time fuzzy system (3) is said to be stabilizable with119867infin

norm bound 120574 if for all V(119896) isin 1198972[0infin) V(119896) = 0 the

closed-loop system (7)-(8) is asymptotically stable and theresponse 119910(119896) of the system under the zero initial condition(119909(0) = 119909

0= 0) satisfies

1003817100381710038171003817119910 (119896)

10038171003817100381710038172lt 120574V(119896)

2 (11)

Lemma 2 It is supposed that 119906(119896) = 0 V(119896) = 0 and Δ = 0so the discrete-time MJSS becomes

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119860

119894120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896)) (12)

The system (12) is globally asymptotically stable if there exists asymmetric piecewise matrix 119875

119894120579119896gt 0 such that

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0 (13)

where 119895120579119896= 119875119895120579119896+1

= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Lemma 3 Given a matrix 119860119894120579119896

119863119894120579119896

suppose 119875119894120579119896

gt 0119895120579119896gt 0 119875

119897120579119896gt 0 If

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0

1198601015840

119897120579119896

119895120579119896119860119897120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896minus 119875119897120579119896lt 0

(14)

then

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(15)

Proof It is noted that

119895120579119896gt 0 997904rArr (119860

119894120579119896minus 119860119897120579119896)

1015840

119895120579119896(119860119894120579119896minus 119860119897120579119896)

+ (119863119894120579119896minus 119863119897120579119896)

1015840

119895120579119896(119863119894120579119896minus 119863119897120579119896) ge 0

(16)

4 Mathematical Problems in Engineering

which gives

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896

le 1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198601015840

119897120579119896

119895120579119896119860119897120579119896

+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896

(17)

Therefore the result of Lemma 3 can be easily proven

Lemma 4 (see [14]) Suppose Δ is given by (10) With matrices119872 = 119872

1015840 119878 and119873 with appropriate dimension the inequality

119872+ 119878Δ119873 +1198731015840

Δ1015840

1198781015840

lt 0 (18)

holds for all 119865119894120579119896

such that 1198651198941205791198961198651015840

119894120579119896

le 119868 if and only if for some120575 = 1205982

gt 0

[

[

120575119872 119878 1205751198731015840

1198781015840

minus119868 1198691015840

120575119873 119869 minus119868

]

]

lt 0 (19)

3 Robust Stability and119867infin

Performance Analysis

In this section the stability and 119867infin

performance for thenominal fuzzy systemwill be analyzed Under control law (4)the closed-loop fuzzy system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) [(A

119894120579119896+B119894120579119896119870119895120579119896) 119909 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)]

(20)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times [(M119894120579119896+T119894120579119896119870119895120579119896) 119909 (119896) +N

119894120579119896V (119896)]

(21)

When Δ = 0 the nominal closed-loop system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(22)

119910 (119896) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119872119894119895120579119896119909 (119896) + 119873

119894120579119896V (119896))

(23)

where

119860119894119895120579119896= 119860119894120579119896+ 119861119894120579119896119870119895120579119896

119872119894119895120579119896= 119872119894120579119896+ 119879119894120579119896119870119895120579119896

(24)

Theorem 5 The nominal closed-loop fuzzy system (22) isstable with119867

infinnorm bound 120574 that is 119910(119896)

2lt 120574V(119896)

2for

all nonzero V(119896) isin 1198972[0infin) under the zero initial condition if

there exists matrices 119875119894120579119896gt 0119903

119894=1for all 119894 119895 119897 isin 1 2 119903

such that

[

Φ Γ

Λ Ψ] lt 0 (25)

where

119897120579119896=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896

Φ = 1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896+1198721015840

119894119895120579119896

119872119894119895120579119896minus 119875119894120579119896

Γ = 1198601015840

119894119895120579119896

119897120579119896119862119894120579119896+1198721015840

119894119895120579119896

119873119894120579119896

Λ = 1198621015840

119894120579119896

119897120579119896119860119894119895120579119896+ 1198731015840

119894120579119896

119872119894119895120579119896

Ψ = 1198621015840

119894120579119896

119897120579119896119862119894120579119896+ 1198731015840

119894120579119896

119873119894120579119896minus 1205742

119868

(26)

Proof Obviously inequality (25) implies the followinginequality

1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896minus 119875119894120579119896lt 0 (27)

It can be checked from the result of Lemma 3 that for all119894 119895 119897 119901 119902 isin 1 2 119903

1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(28)

When 119894 = 119897 the inequality (28) becomes

1198601015840

119894119901120579119896

119895120579119896119860119894119902120579119896+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+ 21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896lt 0

(29)

Let

119881 (119909 (119896) 120579119896) = 1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896) (30)

When V(119896) = 0 the system (22) becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(31)

Mathematical Problems in Engineering 5

In what follows we will drop the argument of ℎ119894(119911(119896)) for

clarity By some algebraic manipulation with ℎ+119895= ℎ119895(119911(119896 +

1)) the difference of Lyapunov function 119881(119909(119896) 120579119896) given by

Δ119881(119909(119896) 120579119896) = 119881(119909(119896 + 1) 120579

119896+1) minus 119881(119909(119896) 120579

119896) along the

solution of system (31) is

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)]

= 119864 [119881 (119909 (119896 + 1) 120579119896+1) minus 119881 (119909 (119896) 120579

119896)]

= 119864

1199091015840

(119896 + 1)[

[

119903

sum

119895=1

ℎ119895(119911 (119896)) 119875

119895120579119896+1

]

]

119909 (119896 + 1)

minus1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896)

= 1199091015840

(119896)[

[

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119897=1

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+1198631015840

119894120579119896

119895120579119896119863119897120579119896minus 119875119894120579119896)]

]

119909 (119896)

= 1199091015840

(119896)

times

119903

sum

119895=1

ℎ+

119895[

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times (1198601015840

119894119901120579119896

119895120579119896119860119894119901120579119896

+1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896)

+

119903

sum

119894=1

119903

sum

119897gt119894

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896)

+

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times (1198601015840

119894119902120579119896

119895120579119896119860119894119902120579119896

+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896) ]

119909 (119896)

(32)

where

119895120579119896= 119875119895120579119896+1

=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896 (33)

It follows from (13) (28) and (29) that

Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)

lt 0 (34)

which proves the stability of system (31)It follows from (25) that for all 119894 119895 119897 isin 1 2 119903 exists

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(35)

Let

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)] (36)

For zero initial condition 119909(0) = 1199090= 0 one has

119873minus1

sum

119896=0

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)] = 119881 (119909 (119873) 120579

119873) minus 119881 (119909 (0) 120579

0)

= 119881 (119909 (119873) 120579119873)

(37)

Therefore

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

=

119873minus1

sum

119896=0

119864 [119910(119896)1015840

119910 (119896) minus 1205742V1015840 (119896) V (119896) + Δ119881 (119909 (119896) 120579

119896)1003816100381610038161003816(22)]

minus 119881 (119909 (119873) 120579119873)

(38)

where Δ119881(119909(119896) 120579119896)|(22)

defines the difference of 119881(119909(119896) 120579119896)

along system (22) It is noted that with 120577(119896) defined by

120577 (119896) = [

119909 (119896)

V (119896)] (39)

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

prescribed 119867infin

performance The authors in [13] analyzedthe 119867

infincontrol design problem systematically for a class of

nonlinear stochastic active fault-tolerant control systemswithMarkovian parameters119867

infincontroller designwhichwas con-

sidered in [7] noted that most of the aforementioned researchefforts focused on the use of single quadratic Lyapunovfunction which tended to givemore conservative conditionsThis is because with the use of parameter-dependent orbasis-dependent Lyapunov function less conservative controlresults can be obtained than with the use of single Lyapunovquadratic function More recently there were many resultson stability analysis and control synthesis of discrete-timeuncertain systems based on parameter-dependent Lyapunovfunctions [9] or basis-dependent Lyapunov function (see[8 14]) It is shown that with the use of Lyapunov functionwell-pleasing control results can be obtained

In practice a lot of physical systems have variablestructures subject to random changes These changes mayresult from abrupt phenomena such as random failures andrepairs of the components changes in the interconnectionsof subsystems and sudden environment changes The so-called Markovian jump systems (MJSS) are special class ofhybrid systems As one of the most basic dynamics modelsMarkov jump nonlinear system can be used to representthese random failure processes in manufacturing and someinvestment portfolio models In theMJSS the random jumpsin system parameters are governed by a Markov processwhich takes values in a finite set The class of systems mayrepresent a large variety of processes including those in theproduction systems fault-tolerant systems communicationsystems and economic systems [15] In the past two decadesmany important issues of MJSS were researched extensivelysuch as the controllability and observability [16] stability andstabilization (see [17ndash22])119867

2[23] and119867

infinperformance [24ndash

27] and robustness [28 29] To the best of our knowledgedespite these efforts very few results are available for thecontrol design for nonlinear MJSS In this study a mode-dependent control design is proposed to achieve robuststability of uncertain discrete-time nonlinear MJSS via fuzzycontrol

In the paper robust 119867infin

control is investigated fornonlinear discrete-time stochastic MJSS with parametricuncertainties Under our proposed fuzzy rules the nonlineardiscrete-time stochastic MJSS could be translated into a classof equivalent T-S fuzzymodels with parametric uncertaintiesThe uncertainty could be translated into a linear fractionalform which includes the norm-bounded uncertainty as aspecial case and can describe a kind of rational nonlinearitiesWith the PDC scheme the control law is designed tomake theclosed-loop system with 119867

infinnorm bound 120574 stable Besides

some matrix variables and the Lyapunov function for robuststabilization with 119897

2-norm bound for the fuzzy discrete-time

stochastic MJSS are given It is shown that the solution of thecontrol design problem can be obtained by solving a class ofLMIs

The paper is organized as follows Section 2 discussesthe T-S fuzzy models And definitions and preliminaryresults are given for uncertain nonlinear discrete-time MJSSSection 3 gives the analysis results of robust stability with119867

infin

performance and the results are employed in the followingto develop an 119867

infincontrol design In Section 4 a numerical

simulation example is proposed to illustrate the effectivenessof the approach Finally the paper is concluded in Section 5

For convenience the following basic notations areadopted throughout the paperR119899 denotes the 119899-dimensionalreal space andR119899times119898 denotes the set of all real 119899times119898matrices1198801015840 indicates the transpose of matrix 119880 and 119880 ge 0 (119880 gt 0)

represents a nonnegative definite (positive definite) matrixSimilarly 119880 le 0 (119880 lt 0) represents a nonpositive definitematrix (negative definite) 119897

2[0infin) refers to the space of

square summable infinite vector sequences sdot 2stands for

the usual 1198972[0infin) norm 119864(sdot) represents the mathematical

expectation

2 Problem Formulation and Preliminaries

Consider the following nonlinear MJSS

119909 (119896 + 1) = 119891 (119909 (119896) 120579119896) + 119892 (119909 (119896) 120579

119896) 119906 (119896)

+ 119896 (119909 (119896) 120579119896) V (119896) + 119897 (119909 (119896) 120579

119896) 119908 (119896)

119910 (119896) = 119898 (119909 (119896) 120579119896) + 119905 (119909 (119896) 120579

119896) 119906 (119896)

+ 119899 (119909 (119896) 120579119896) V (119896)

(1)

Assume that 119891(119909(119896) 120579119896) 119892(119909(119896) 120579

119896) 119896(119909(119896) 120579

119896) 119897(119909(119896) 120579

119896)

119898(119909(119896) 120579119896) 119905(119909(119896) 120579

119896) and 119899(119909(119896) 120579

119896) are Borel measurable

on R119899 And 119909(119896) isin R119899 is the system state 119906(119896) isin R119898

and V(119896) isin R119901 represent the system control inputs anddisturbance signal and 119910(119896) isin R119902 is system output 119908(119896) isa sequence of real random variables defined on a completeprobability space (ΩF

119896 119875) which is wide sense stationary

second-order processes with 119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] =120575119894119895 where 120575

119894119895refers to a Kronecker function that is 120575

119894119895= 1

if 119894 = 119895 and 120575119894119895= 0 if 119894 = 119895 120579

119896 119896 ge 0 is a measurable

Markov chain taking values in a finite set X = 1 2 119873with transition probability matrix P = [119901

120572120573] where

119901120572120573= 119875 (120579

119896+1= 120573 | 120579

119896= 120572) forall120572 120573 isin X 119896 ge 0 (2)

The paper considers the nonlinear discrete-time MJSS whichcan be described by the following T-S fuzzy model withuncertainties

Rule i If 1199111(119896) is 119865119894

1 1199112(119896) is 119865119894

2 and 119911

119899(119896) is 119865119894

119899 then

119909 (119896 + 1) = AA119894120579119896119909 (119896) +B

119894120579119896119906 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)

119910 (119896) =M119894120579119896119909 (119896) +T

119894120579119896119906 (119896) +N

119894120579119896V (119896)

(3)

The 119865119894119895are fuzzy sets 119903 is the number of IF-THEN rules

Moreover A119894120579119896

B119894120579119896

C119894120579119896

D119894120579119896

M119894120579119896

T119894120579119896

and N119894120579119896

are system matrices with parametric uncertainties Besides1199111(119896) 1199112(119896) 119911

119899(119896) are the premise variables of the fuzzy

modes and they are the functions of state variables

Mathematical Problems in Engineering 3

Following the PDC scheme we consider a state feedbackfuzzy controller which shares the same structure of the aboveT-S fuzzy system as follows

Rule j IF 1199111(119896) is 119865119895

1 1199112(119896) is 119865119895

2 and 119911

119899(119896) is 119865119895

119899 then

119906 (119896) = 119870119895120579119896119909 (119896) (4)

where119870119895120579119896

are the local feedback gain matricesAnd the fuzzy basis functions are given by

ℎ119894[119911 (119896)] =

prod119899

119895=1120583119894119895[119911119895(119896)]

sum119903

119897=1prod119899

119895=1120583119894119895[119911119895(119896)]

119894 = 1 2 119903 (5)

where 120583119894119895[119911119895(119896)] is the grade of membership of 119911

119895(119896) in 119865119894

119895 By

definition the fuzzy basis functions satisfy

ℎ119894[119911 (119896)] ge 0 119894 = 1 2 119903

119903

sum

119894=1

ℎ119894[119911 (119896)] = 1

(6)

A more compact presentation of the discrete-time T-S fuzzymodel is given by

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) (A

119894120579119896119909 (119896) +B

119894120579119896119906 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896))

(7)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (M119894120579119896119909 (119896) +T

119894120579119896119906 (119896) +N

119894120579119896V (119896))

(8)

where

A119894120579119896= 119860119894120579119896+ Δ119860119894120579119896

B119894120579119896= 119861119894120579119896+ Δ119861119894120579119896

C119894120579119896= 119862119894120579119896+ Δ119862119894120579119896

D119894120579119896= 119863119894120579119896

M119894120579119896= 119872119894120579119896+ Δ119872

119894120579119896

T119894120579119896= 119879119894120579119896+ Δ119879119894120579119896

N119894120579119896= 119873119894120579119896+ Δ119873119894120579119896

(9)

And 119860119894120579119896

119861119894120579119896

119862119894120579119896

119863119894120579119896

119872119894120579119896

119879119894120579119896

and119873119894120579119896

are assumedto be deterministic matrices with appropriate dimensionΔ119860119894120579119896 Δ119861119894120579119896 Δ119862119894120579119896 Δ119872119894120579119896 Δ119879119894120579119896

and Δ119873119894120579119896

are unknown

matrices which represent the time-varying parameter uncer-tainties and are assumed to be of the form

[

Δ119860119894120579119896

Δ119861119894120579119896

Δ119862119894120579119896

Δ119872119894120579119896

Δ119879119894120579119896Δ119873119894120579119896

] = [

1198641119894

1198642119894

]Δ [119867111989411986721198941198673119894]

Δ = [119868 minus 119865119894120579119896119869]

minus1

119865119894120579119896

(10)

where 1198641119894 1198671119894 1198642119894 1198672119894 1198673119894 and 119869 are known real constant

matrices with appropriate dimension and unknown nonlin-ear time-varying matrix 119865

119894120579119896isin R119898times119899 satisfying 119865

1198941205791198961198651015840

119894120579119896

le 119868To guarantee that the matrix 119868 minus 119865

119894120579119896119869 is invertible for all

admissible 119865119894120579119896

it is necessary that 119868 minus 1198691198691015840 gt 0The following definition and lemmas will be used later

Definition 1 The discrete-time unforced uncertain fuzzysystem is said to be robust stable with 119867

infinnorm bound 120574

if it is stable with 119867infin

norm bound 120574 for all uncertainlyadmissible 119865

119894120579119896 For a given control law (4) and a prescribed

level of disturbance attenuation 120574 gt 0 to be achieved thediscrete-time fuzzy system (3) is said to be stabilizable with119867infin

norm bound 120574 if for all V(119896) isin 1198972[0infin) V(119896) = 0 the

closed-loop system (7)-(8) is asymptotically stable and theresponse 119910(119896) of the system under the zero initial condition(119909(0) = 119909

0= 0) satisfies

1003817100381710038171003817119910 (119896)

10038171003817100381710038172lt 120574V(119896)

2 (11)

Lemma 2 It is supposed that 119906(119896) = 0 V(119896) = 0 and Δ = 0so the discrete-time MJSS becomes

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119860

119894120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896)) (12)

The system (12) is globally asymptotically stable if there exists asymmetric piecewise matrix 119875

119894120579119896gt 0 such that

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0 (13)

where 119895120579119896= 119875119895120579119896+1

= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Lemma 3 Given a matrix 119860119894120579119896

119863119894120579119896

suppose 119875119894120579119896

gt 0119895120579119896gt 0 119875

119897120579119896gt 0 If

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0

1198601015840

119897120579119896

119895120579119896119860119897120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896minus 119875119897120579119896lt 0

(14)

then

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(15)

Proof It is noted that

119895120579119896gt 0 997904rArr (119860

119894120579119896minus 119860119897120579119896)

1015840

119895120579119896(119860119894120579119896minus 119860119897120579119896)

+ (119863119894120579119896minus 119863119897120579119896)

1015840

119895120579119896(119863119894120579119896minus 119863119897120579119896) ge 0

(16)

4 Mathematical Problems in Engineering

which gives

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896

le 1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198601015840

119897120579119896

119895120579119896119860119897120579119896

+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896

(17)

Therefore the result of Lemma 3 can be easily proven

Lemma 4 (see [14]) Suppose Δ is given by (10) With matrices119872 = 119872

1015840 119878 and119873 with appropriate dimension the inequality

119872+ 119878Δ119873 +1198731015840

Δ1015840

1198781015840

lt 0 (18)

holds for all 119865119894120579119896

such that 1198651198941205791198961198651015840

119894120579119896

le 119868 if and only if for some120575 = 1205982

gt 0

[

[

120575119872 119878 1205751198731015840

1198781015840

minus119868 1198691015840

120575119873 119869 minus119868

]

]

lt 0 (19)

3 Robust Stability and119867infin

Performance Analysis

In this section the stability and 119867infin

performance for thenominal fuzzy systemwill be analyzed Under control law (4)the closed-loop fuzzy system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) [(A

119894120579119896+B119894120579119896119870119895120579119896) 119909 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)]

(20)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times [(M119894120579119896+T119894120579119896119870119895120579119896) 119909 (119896) +N

119894120579119896V (119896)]

(21)

When Δ = 0 the nominal closed-loop system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(22)

119910 (119896) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119872119894119895120579119896119909 (119896) + 119873

119894120579119896V (119896))

(23)

where

119860119894119895120579119896= 119860119894120579119896+ 119861119894120579119896119870119895120579119896

119872119894119895120579119896= 119872119894120579119896+ 119879119894120579119896119870119895120579119896

(24)

Theorem 5 The nominal closed-loop fuzzy system (22) isstable with119867

infinnorm bound 120574 that is 119910(119896)

2lt 120574V(119896)

2for

all nonzero V(119896) isin 1198972[0infin) under the zero initial condition if

there exists matrices 119875119894120579119896gt 0119903

119894=1for all 119894 119895 119897 isin 1 2 119903

such that

[

Φ Γ

Λ Ψ] lt 0 (25)

where

119897120579119896=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896

Φ = 1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896+1198721015840

119894119895120579119896

119872119894119895120579119896minus 119875119894120579119896

Γ = 1198601015840

119894119895120579119896

119897120579119896119862119894120579119896+1198721015840

119894119895120579119896

119873119894120579119896

Λ = 1198621015840

119894120579119896

119897120579119896119860119894119895120579119896+ 1198731015840

119894120579119896

119872119894119895120579119896

Ψ = 1198621015840

119894120579119896

119897120579119896119862119894120579119896+ 1198731015840

119894120579119896

119873119894120579119896minus 1205742

119868

(26)

Proof Obviously inequality (25) implies the followinginequality

1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896minus 119875119894120579119896lt 0 (27)

It can be checked from the result of Lemma 3 that for all119894 119895 119897 119901 119902 isin 1 2 119903

1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(28)

When 119894 = 119897 the inequality (28) becomes

1198601015840

119894119901120579119896

119895120579119896119860119894119902120579119896+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+ 21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896lt 0

(29)

Let

119881 (119909 (119896) 120579119896) = 1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896) (30)

When V(119896) = 0 the system (22) becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(31)

Mathematical Problems in Engineering 5

In what follows we will drop the argument of ℎ119894(119911(119896)) for

clarity By some algebraic manipulation with ℎ+119895= ℎ119895(119911(119896 +

1)) the difference of Lyapunov function 119881(119909(119896) 120579119896) given by

Δ119881(119909(119896) 120579119896) = 119881(119909(119896 + 1) 120579

119896+1) minus 119881(119909(119896) 120579

119896) along the

solution of system (31) is

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)]

= 119864 [119881 (119909 (119896 + 1) 120579119896+1) minus 119881 (119909 (119896) 120579

119896)]

= 119864

1199091015840

(119896 + 1)[

[

119903

sum

119895=1

ℎ119895(119911 (119896)) 119875

119895120579119896+1

]

]

119909 (119896 + 1)

minus1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896)

= 1199091015840

(119896)[

[

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119897=1

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+1198631015840

119894120579119896

119895120579119896119863119897120579119896minus 119875119894120579119896)]

]

119909 (119896)

= 1199091015840

(119896)

times

119903

sum

119895=1

ℎ+

119895[

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times (1198601015840

119894119901120579119896

119895120579119896119860119894119901120579119896

+1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896)

+

119903

sum

119894=1

119903

sum

119897gt119894

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896)

+

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times (1198601015840

119894119902120579119896

119895120579119896119860119894119902120579119896

+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896) ]

119909 (119896)

(32)

where

119895120579119896= 119875119895120579119896+1

=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896 (33)

It follows from (13) (28) and (29) that

Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)

lt 0 (34)

which proves the stability of system (31)It follows from (25) that for all 119894 119895 119897 isin 1 2 119903 exists

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(35)

Let

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)] (36)

For zero initial condition 119909(0) = 1199090= 0 one has

119873minus1

sum

119896=0

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)] = 119881 (119909 (119873) 120579

119873) minus 119881 (119909 (0) 120579

0)

= 119881 (119909 (119873) 120579119873)

(37)

Therefore

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

=

119873minus1

sum

119896=0

119864 [119910(119896)1015840

119910 (119896) minus 1205742V1015840 (119896) V (119896) + Δ119881 (119909 (119896) 120579

119896)1003816100381610038161003816(22)]

minus 119881 (119909 (119873) 120579119873)

(38)

where Δ119881(119909(119896) 120579119896)|(22)

defines the difference of 119881(119909(119896) 120579119896)

along system (22) It is noted that with 120577(119896) defined by

120577 (119896) = [

119909 (119896)

V (119896)] (39)

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Following the PDC scheme we consider a state feedbackfuzzy controller which shares the same structure of the aboveT-S fuzzy system as follows

Rule j IF 1199111(119896) is 119865119895

1 1199112(119896) is 119865119895

2 and 119911

119899(119896) is 119865119895

119899 then

119906 (119896) = 119870119895120579119896119909 (119896) (4)

where119870119895120579119896

are the local feedback gain matricesAnd the fuzzy basis functions are given by

ℎ119894[119911 (119896)] =

prod119899

119895=1120583119894119895[119911119895(119896)]

sum119903

119897=1prod119899

119895=1120583119894119895[119911119895(119896)]

119894 = 1 2 119903 (5)

where 120583119894119895[119911119895(119896)] is the grade of membership of 119911

119895(119896) in 119865119894

119895 By

definition the fuzzy basis functions satisfy

ℎ119894[119911 (119896)] ge 0 119894 = 1 2 119903

119903

sum

119894=1

ℎ119894[119911 (119896)] = 1

(6)

A more compact presentation of the discrete-time T-S fuzzymodel is given by

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) (A

119894120579119896119909 (119896) +B

119894120579119896119906 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896))

(7)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (M119894120579119896119909 (119896) +T

119894120579119896119906 (119896) +N

119894120579119896V (119896))

(8)

where

A119894120579119896= 119860119894120579119896+ Δ119860119894120579119896

B119894120579119896= 119861119894120579119896+ Δ119861119894120579119896

C119894120579119896= 119862119894120579119896+ Δ119862119894120579119896

D119894120579119896= 119863119894120579119896

M119894120579119896= 119872119894120579119896+ Δ119872

119894120579119896

T119894120579119896= 119879119894120579119896+ Δ119879119894120579119896

N119894120579119896= 119873119894120579119896+ Δ119873119894120579119896

(9)

And 119860119894120579119896

119861119894120579119896

119862119894120579119896

119863119894120579119896

119872119894120579119896

119879119894120579119896

and119873119894120579119896

are assumedto be deterministic matrices with appropriate dimensionΔ119860119894120579119896 Δ119861119894120579119896 Δ119862119894120579119896 Δ119872119894120579119896 Δ119879119894120579119896

and Δ119873119894120579119896

are unknown

matrices which represent the time-varying parameter uncer-tainties and are assumed to be of the form

[

Δ119860119894120579119896

Δ119861119894120579119896

Δ119862119894120579119896

Δ119872119894120579119896

Δ119879119894120579119896Δ119873119894120579119896

] = [

1198641119894

1198642119894

]Δ [119867111989411986721198941198673119894]

Δ = [119868 minus 119865119894120579119896119869]

minus1

119865119894120579119896

(10)

where 1198641119894 1198671119894 1198642119894 1198672119894 1198673119894 and 119869 are known real constant

matrices with appropriate dimension and unknown nonlin-ear time-varying matrix 119865

119894120579119896isin R119898times119899 satisfying 119865

1198941205791198961198651015840

119894120579119896

le 119868To guarantee that the matrix 119868 minus 119865

119894120579119896119869 is invertible for all

admissible 119865119894120579119896

it is necessary that 119868 minus 1198691198691015840 gt 0The following definition and lemmas will be used later

Definition 1 The discrete-time unforced uncertain fuzzysystem is said to be robust stable with 119867

infinnorm bound 120574

if it is stable with 119867infin

norm bound 120574 for all uncertainlyadmissible 119865

119894120579119896 For a given control law (4) and a prescribed

level of disturbance attenuation 120574 gt 0 to be achieved thediscrete-time fuzzy system (3) is said to be stabilizable with119867infin

norm bound 120574 if for all V(119896) isin 1198972[0infin) V(119896) = 0 the

closed-loop system (7)-(8) is asymptotically stable and theresponse 119910(119896) of the system under the zero initial condition(119909(0) = 119909

0= 0) satisfies

1003817100381710038171003817119910 (119896)

10038171003817100381710038172lt 120574V(119896)

2 (11)

Lemma 2 It is supposed that 119906(119896) = 0 V(119896) = 0 and Δ = 0so the discrete-time MJSS becomes

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119860

119894120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896)) (12)

The system (12) is globally asymptotically stable if there exists asymmetric piecewise matrix 119875

119894120579119896gt 0 such that

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0 (13)

where 119895120579119896= 119875119895120579119896+1

= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Lemma 3 Given a matrix 119860119894120579119896

119863119894120579119896

suppose 119875119894120579119896

gt 0119895120579119896gt 0 119875

119897120579119896gt 0 If

1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896lt 0

1198601015840

119897120579119896

119895120579119896119860119897120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896minus 119875119897120579119896lt 0

(14)

then

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(15)

Proof It is noted that

119895120579119896gt 0 997904rArr (119860

119894120579119896minus 119860119897120579119896)

1015840

119895120579119896(119860119894120579119896minus 119860119897120579119896)

+ (119863119894120579119896minus 119863119897120579119896)

1015840

119895120579119896(119863119894120579119896minus 119863119897120579119896) ge 0

(16)

4 Mathematical Problems in Engineering

which gives

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896

le 1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198601015840

119897120579119896

119895120579119896119860119897120579119896

+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896

(17)

Therefore the result of Lemma 3 can be easily proven

Lemma 4 (see [14]) Suppose Δ is given by (10) With matrices119872 = 119872

1015840 119878 and119873 with appropriate dimension the inequality

119872+ 119878Δ119873 +1198731015840

Δ1015840

1198781015840

lt 0 (18)

holds for all 119865119894120579119896

such that 1198651198941205791198961198651015840

119894120579119896

le 119868 if and only if for some120575 = 1205982

gt 0

[

[

120575119872 119878 1205751198731015840

1198781015840

minus119868 1198691015840

120575119873 119869 minus119868

]

]

lt 0 (19)

3 Robust Stability and119867infin

Performance Analysis

In this section the stability and 119867infin

performance for thenominal fuzzy systemwill be analyzed Under control law (4)the closed-loop fuzzy system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) [(A

119894120579119896+B119894120579119896119870119895120579119896) 119909 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)]

(20)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times [(M119894120579119896+T119894120579119896119870119895120579119896) 119909 (119896) +N

119894120579119896V (119896)]

(21)

When Δ = 0 the nominal closed-loop system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(22)

119910 (119896) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119872119894119895120579119896119909 (119896) + 119873

119894120579119896V (119896))

(23)

where

119860119894119895120579119896= 119860119894120579119896+ 119861119894120579119896119870119895120579119896

119872119894119895120579119896= 119872119894120579119896+ 119879119894120579119896119870119895120579119896

(24)

Theorem 5 The nominal closed-loop fuzzy system (22) isstable with119867

infinnorm bound 120574 that is 119910(119896)

2lt 120574V(119896)

2for

all nonzero V(119896) isin 1198972[0infin) under the zero initial condition if

there exists matrices 119875119894120579119896gt 0119903

119894=1for all 119894 119895 119897 isin 1 2 119903

such that

[

Φ Γ

Λ Ψ] lt 0 (25)

where

119897120579119896=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896

Φ = 1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896+1198721015840

119894119895120579119896

119872119894119895120579119896minus 119875119894120579119896

Γ = 1198601015840

119894119895120579119896

119897120579119896119862119894120579119896+1198721015840

119894119895120579119896

119873119894120579119896

Λ = 1198621015840

119894120579119896

119897120579119896119860119894119895120579119896+ 1198731015840

119894120579119896

119872119894119895120579119896

Ψ = 1198621015840

119894120579119896

119897120579119896119862119894120579119896+ 1198731015840

119894120579119896

119873119894120579119896minus 1205742

119868

(26)

Proof Obviously inequality (25) implies the followinginequality

1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896minus 119875119894120579119896lt 0 (27)

It can be checked from the result of Lemma 3 that for all119894 119895 119897 119901 119902 isin 1 2 119903

1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(28)

When 119894 = 119897 the inequality (28) becomes

1198601015840

119894119901120579119896

119895120579119896119860119894119902120579119896+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+ 21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896lt 0

(29)

Let

119881 (119909 (119896) 120579119896) = 1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896) (30)

When V(119896) = 0 the system (22) becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(31)

Mathematical Problems in Engineering 5

In what follows we will drop the argument of ℎ119894(119911(119896)) for

clarity By some algebraic manipulation with ℎ+119895= ℎ119895(119911(119896 +

1)) the difference of Lyapunov function 119881(119909(119896) 120579119896) given by

Δ119881(119909(119896) 120579119896) = 119881(119909(119896 + 1) 120579

119896+1) minus 119881(119909(119896) 120579

119896) along the

solution of system (31) is

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)]

= 119864 [119881 (119909 (119896 + 1) 120579119896+1) minus 119881 (119909 (119896) 120579

119896)]

= 119864

1199091015840

(119896 + 1)[

[

119903

sum

119895=1

ℎ119895(119911 (119896)) 119875

119895120579119896+1

]

]

119909 (119896 + 1)

minus1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896)

= 1199091015840

(119896)[

[

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119897=1

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+1198631015840

119894120579119896

119895120579119896119863119897120579119896minus 119875119894120579119896)]

]

119909 (119896)

= 1199091015840

(119896)

times

119903

sum

119895=1

ℎ+

119895[

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times (1198601015840

119894119901120579119896

119895120579119896119860119894119901120579119896

+1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896)

+

119903

sum

119894=1

119903

sum

119897gt119894

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896)

+

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times (1198601015840

119894119902120579119896

119895120579119896119860119894119902120579119896

+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896) ]

119909 (119896)

(32)

where

119895120579119896= 119875119895120579119896+1

=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896 (33)

It follows from (13) (28) and (29) that

Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)

lt 0 (34)

which proves the stability of system (31)It follows from (25) that for all 119894 119895 119897 isin 1 2 119903 exists

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(35)

Let

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)] (36)

For zero initial condition 119909(0) = 1199090= 0 one has

119873minus1

sum

119896=0

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)] = 119881 (119909 (119873) 120579

119873) minus 119881 (119909 (0) 120579

0)

= 119881 (119909 (119873) 120579119873)

(37)

Therefore

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

=

119873minus1

sum

119896=0

119864 [119910(119896)1015840

119910 (119896) minus 1205742V1015840 (119896) V (119896) + Δ119881 (119909 (119896) 120579

119896)1003816100381610038161003816(22)]

minus 119881 (119909 (119873) 120579119873)

(38)

where Δ119881(119909(119896) 120579119896)|(22)

defines the difference of 119881(119909(119896) 120579119896)

along system (22) It is noted that with 120577(119896) defined by

120577 (119896) = [

119909 (119896)

V (119896)] (39)

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

which gives

1198601015840

119894120579119896

119895120579119896119860119897120579119896+ 1198601015840

119897120579119896

119895120579119896119860119894120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896

le 1198601015840

119894120579119896

119895120579119896119860119894120579119896+ 1198601015840

119897120579119896

119895120579119896119860119897120579119896

+ 1198631015840

119894120579119896

119895120579119896119863119894120579119896+ 1198631015840

119897120579119896

119895120579119896119863119897120579119896

(17)

Therefore the result of Lemma 3 can be easily proven

Lemma 4 (see [14]) Suppose Δ is given by (10) With matrices119872 = 119872

1015840 119878 and119873 with appropriate dimension the inequality

119872+ 119878Δ119873 +1198731015840

Δ1015840

1198781015840

lt 0 (18)

holds for all 119865119894120579119896

such that 1198651198941205791198961198651015840

119894120579119896

le 119868 if and only if for some120575 = 1205982

gt 0

[

[

120575119872 119878 1205751198731015840

1198781015840

minus119868 1198691015840

120575119873 119869 minus119868

]

]

lt 0 (19)

3 Robust Stability and119867infin

Performance Analysis

In this section the stability and 119867infin

performance for thenominal fuzzy systemwill be analyzed Under control law (4)the closed-loop fuzzy system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

ℎ119894(119911 (119896)) [(A

119894120579119896+B119894120579119896119870119895120579119896) 119909 (119896) +C

119894120579119896V (119896)

+D119894120579119896119909 (119896)119908 (119896)]

(20)

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times [(M119894120579119896+T119894120579119896119870119895120579119896) 119909 (119896) +N

119894120579119896V (119896)]

(21)

When Δ = 0 the nominal closed-loop system becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(22)

119910 (119896) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119872119894119895120579119896119909 (119896) + 119873

119894120579119896V (119896))

(23)

where

119860119894119895120579119896= 119860119894120579119896+ 119861119894120579119896119870119895120579119896

119872119894119895120579119896= 119872119894120579119896+ 119879119894120579119896119870119895120579119896

(24)

Theorem 5 The nominal closed-loop fuzzy system (22) isstable with119867

infinnorm bound 120574 that is 119910(119896)

2lt 120574V(119896)

2for

all nonzero V(119896) isin 1198972[0infin) under the zero initial condition if

there exists matrices 119875119894120579119896gt 0119903

119894=1for all 119894 119895 119897 isin 1 2 119903

such that

[

Φ Γ

Λ Ψ] lt 0 (25)

where

119897120579119896=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896

Φ = 1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896+1198721015840

119894119895120579119896

119872119894119895120579119896minus 119875119894120579119896

Γ = 1198601015840

119894119895120579119896

119897120579119896119862119894120579119896+1198721015840

119894119895120579119896

119873119894120579119896

Λ = 1198621015840

119894120579119896

119897120579119896119860119894119895120579119896+ 1198731015840

119894120579119896

119872119894119895120579119896

Ψ = 1198621015840

119894120579119896

119897120579119896119862119894120579119896+ 1198731015840

119894120579119896

119873119894120579119896minus 1205742

119868

(26)

Proof Obviously inequality (25) implies the followinginequality

1198601015840

119894119895120579119896

119897120579119896119860119894119895120579119896+ 1198631015840

119894120579119896

119897120579119896119863119894120579119896minus 119875119894120579119896lt 0 (27)

It can be checked from the result of Lemma 3 that for all119894 119895 119897 119901 119902 isin 1 2 119903

1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+ 1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896lt 0

(28)

When 119894 = 119897 the inequality (28) becomes

1198601015840

119894119901120579119896

119895120579119896119860119894119902120579119896+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+ 21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896lt 0

(29)

Let

119881 (119909 (119896) 120579119896) = 1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896) (30)

When V(119896) = 0 the system (22) becomes

119909 (119896 + 1)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119911 (119896)) ℎ

119895(119911 (119896))

times (119860119894119895120579119896119909 (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

(31)

Mathematical Problems in Engineering 5

In what follows we will drop the argument of ℎ119894(119911(119896)) for

clarity By some algebraic manipulation with ℎ+119895= ℎ119895(119911(119896 +

1)) the difference of Lyapunov function 119881(119909(119896) 120579119896) given by

Δ119881(119909(119896) 120579119896) = 119881(119909(119896 + 1) 120579

119896+1) minus 119881(119909(119896) 120579

119896) along the

solution of system (31) is

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)]

= 119864 [119881 (119909 (119896 + 1) 120579119896+1) minus 119881 (119909 (119896) 120579

119896)]

= 119864

1199091015840

(119896 + 1)[

[

119903

sum

119895=1

ℎ119895(119911 (119896)) 119875

119895120579119896+1

]

]

119909 (119896 + 1)

minus1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896)

= 1199091015840

(119896)[

[

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119897=1

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+1198631015840

119894120579119896

119895120579119896119863119897120579119896minus 119875119894120579119896)]

]

119909 (119896)

= 1199091015840

(119896)

times

119903

sum

119895=1

ℎ+

119895[

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times (1198601015840

119894119901120579119896

119895120579119896119860119894119901120579119896

+1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896)

+

119903

sum

119894=1

119903

sum

119897gt119894

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896)

+

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times (1198601015840

119894119902120579119896

119895120579119896119860119894119902120579119896

+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896) ]

119909 (119896)

(32)

where

119895120579119896= 119875119895120579119896+1

=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896 (33)

It follows from (13) (28) and (29) that

Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)

lt 0 (34)

which proves the stability of system (31)It follows from (25) that for all 119894 119895 119897 isin 1 2 119903 exists

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(35)

Let

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)] (36)

For zero initial condition 119909(0) = 1199090= 0 one has

119873minus1

sum

119896=0

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)] = 119881 (119909 (119873) 120579

119873) minus 119881 (119909 (0) 120579

0)

= 119881 (119909 (119873) 120579119873)

(37)

Therefore

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

=

119873minus1

sum

119896=0

119864 [119910(119896)1015840

119910 (119896) minus 1205742V1015840 (119896) V (119896) + Δ119881 (119909 (119896) 120579

119896)1003816100381610038161003816(22)]

minus 119881 (119909 (119873) 120579119873)

(38)

where Δ119881(119909(119896) 120579119896)|(22)

defines the difference of 119881(119909(119896) 120579119896)

along system (22) It is noted that with 120577(119896) defined by

120577 (119896) = [

119909 (119896)

V (119896)] (39)

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

In what follows we will drop the argument of ℎ119894(119911(119896)) for

clarity By some algebraic manipulation with ℎ+119895= ℎ119895(119911(119896 +

1)) the difference of Lyapunov function 119881(119909(119896) 120579119896) given by

Δ119881(119909(119896) 120579119896) = 119881(119909(119896 + 1) 120579

119896+1) minus 119881(119909(119896) 120579

119896) along the

solution of system (31) is

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)]

= 119864 [119881 (119909 (119896 + 1) 120579119896+1) minus 119881 (119909 (119896) 120579

119896)]

= 119864

1199091015840

(119896 + 1)[

[

119903

sum

119895=1

ℎ119895(119911 (119896)) 119875

119895120579119896+1

]

]

119909 (119896 + 1)

minus1199091015840

(119896) [

119903

sum

119894=1

ℎ119894(119911 (119896)) 119875

119894120579119896]119909 (119896)

= 1199091015840

(119896)[

[

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119897=1

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+1198631015840

119894120579119896

119895120579119896119863119897120579119896minus 119875119894120579119896)]

]

119909 (119896)

= 1199091015840

(119896)

times

119903

sum

119895=1

ℎ+

119895[

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times (1198601015840

119894119901120579119896

119895120579119896119860119894119901120579119896

+1198631015840

119894120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896)

+

119903

sum

119894=1

119903

sum

119897gt119894

119903

sum

119901=1

119903

sum

119902=1

ℎ119894ℎ119897ℎ119901ℎ119902

times (1198601015840

119894119901120579119896

119895120579119896119860119897119902120579119896

+ 1198601015840

119897119902120579119896

119895120579119896119860119894119901120579119896+ 1198631015840

119894120579119896

119895120579119896119863119897120579119896

+1198631015840

119897120579119896

119895120579119896119863119894120579119896minus 119875119894120579119896minus 119875119897120579119896)

+

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times (1198601015840

119894119902120579119896

119895120579119896119860119894119902120579119896

+ 1198601015840

119894119902120579119896

119895120579119896119860119894119901120579119896

+21198631015840

119894120579119896

119895120579119896119863119894120579119896minus 2119875119894120579119896) ]

119909 (119896)

(32)

where

119895120579119896= 119875119895120579119896+1

=

119873

sum

120579119896=1

119901120579119896120579119896+1

119875119894120579119896 (33)

It follows from (13) (28) and (29) that

Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(31)

lt 0 (34)

which proves the stability of system (31)It follows from (25) that for all 119894 119895 119897 isin 1 2 119903 exists

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(35)

Let

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)] (36)

For zero initial condition 119909(0) = 1199090= 0 one has

119873minus1

sum

119896=0

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)] = 119881 (119909 (119873) 120579

119873) minus 119881 (119909 (0) 120579

0)

= 119881 (119909 (119873) 120579119873)

(37)

Therefore

119869119873=

119873minus1

sum

119896=0

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

=

119873minus1

sum

119896=0

119864 [119910(119896)1015840

119910 (119896) minus 1205742V1015840 (119896) V (119896) + Δ119881 (119909 (119896) 120579

119896)1003816100381610038161003816(22)]

minus 119881 (119909 (119873) 120579119873)

(38)

where Δ119881(119909(119896) 120579119896)|(22)

defines the difference of 119881(119909(119896) 120579119896)

along system (22) It is noted that with 120577(119896) defined by

120577 (119896) = [

119909 (119896)

V (119896)] (39)

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

we have

119864 [1199101015840

(119896) 119910 (119896) minus 1205742V1015840 (119896) V (119896)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896] minus [

0 0

0 1205742

119868

])

120577 (119896)

119864 [Δ119881 (119909 (119896) 120579119896)1003816100381610038161003816(22)]

= 1205771015840

(119896)

times

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119895=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ+

119895ℎ119901ℎ119897ℎ119902

times ([

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

1198951205791198961198631198971205791198960

0 0

])

120577 (119896)

(40)

Then it is obtained that

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

1198721015840

119894119901120579119896

1198731015840

119894120579119896

] [119872119897119902120579119896

119873119897120579119896]

+ [

1198601015840

119894119901120579119896

1198621015840

119894120579119896

] 119895120579119896[119860119897119902120579119896

119862119897120579119896]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

=

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897=1

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

])

120577 (119896)

(41)

which can be rewritten as

119869119873le

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

ℎ2

119894ℎ2

119901

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119897gt119894

119903

sum

119902=1

ℎ119894ℎ119901ℎ119897ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

]

times [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119897120579119896

0

0 1205742

119868

]

+ [

119860119897119902120579119896

119862119897120579119896

119872119897119902120579119896

119873119897120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus[

119875119897120579119896minus 1198631015840

119897120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

+

119873minus1

sum

119896=0

1205771015840

(119896)

times

119903

sum

119895=1

ℎ+

119895

119903

sum

119894=1

119903

sum

119901=1

119903

sum

119902gt119901

ℎ2

119894ℎ119901ℎ119902

times ([

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

+ [

119860119894119902120579119896

119862119894120579119896

119872119894119902120579119896

119873119894120579119896

]

1015840

times [

119895120579119896

0

0 119868

] [

119860119894119901120579119896

119862119894120579119896

119872119894119901120579119896

119873119894120579119896

]

minus2 [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

])

120577 (119896)

(42)

By following similar line as in the proof of (34) it can bechecked that for any119873 119869

119873lt 0 which gives for any nonzero

V(119896) isin 1198972[0infin) 119910(119896) isin 119897

2[0infin) and 119910(119896)

2lt 120574V(119896)

2

Theorem 6 For the nominal fuzzy system (20) there exists astate feedback fuzzy control law (4) such that the closed-loopsystem is stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin 1 2 119903

satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (43)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(44)

Proof By using (24) and (44) (43) becomes

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (45)

which gives 0 lt 119883119894120579119896

lt Ω119895120579119896

+ Ω1015840

119895120579119896

(119883119894120579119896

ndashΩ119895120579119896)1015840

119883minus1

119894120579119896

(119883119894120579119896minus Ω119895120579119896) ge 0 implies that Ω1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896ge

Ω119895120579119896+ Ω1015840

119895120579119896

minus 119883119894120579119896

yielding

[

[

[

[

minusΩ1015840

119895120579119896

119883minus1

119894120579119896

Ω119895120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896Ω119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (46)

Note that Ω119895120579119896

is invertible Premultiplying diag(Ω1015840119895120579119896

119868 119868119868)minus1 and postmultiplying diag(Ωminus1

119895120579119896

119868 119868 119868) to (46) give

[

[

[

[

minus119883minus1

119894120579119896

lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minus119883119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

lt 0 (47)

Set119883minus1119894120579119896

= 119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

and119883minus1119897120579119896

= 119897120579119896

there is

[

[

[

[

[

minus (119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894119895120579119896

119862119894120579119896

minusminus1

119897120579119896

lowast

119872119894119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (48)

It follows from Schur complement equivalence that

[

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

1015840

[1198971205791198960

0 119868

] [

119860119894119895120579119896

119862119894120579119896

119872119894119895120579119896

119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119897120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(49)

The result then follows fromTheorem 5

Theorem 7 For the uncertain discrete-time MJSS (7) thereexists a state feedback fuzzy control law (4) such that theclosed-loop system is stable with 119867

infinnorm bound 120574 if there

exist matrices 119883119894120579119896gt 0119903

119894=1 Ω119894120579119896119903

119894=1 and 119884

119894120579119896119903

119894=1 119894 119895 119897 isin

1 2 119903 satisfying the following LMIs

[

[

[

[

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896Ω119895120579119896+ 119861119894120579119896119884119895120579119896

120598119862119894120579119896

minus119883119897120579119896

lowast lowast lowast

119872119894120579119896Ω119895120579119896+ 119879119894120579119896119884119895120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894Ω119895120579119896+ 1198671015840

2119894119884119895120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

]

lt 0

(50)

where lowast represents the transposed matrices that are readilyinferred by symmetry for all 119894 and 119895 except the pairs (119894 119895)such that ℎ

119894(119911(119896))ℎ

119895(119911(119896)) = 0 for all 119896 A robust stabilizing

controller gain can be given by

119870119895120579119896= 119884119895120579119896Ωminus1

119895120579119896

(51)

Proof By using Lemma 4 it can be checked that the feasibil-ity of inequality (50) is equivalent to

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

A119894120579119896Ω119895120579119896+B119894120579119896119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

M119894120579119896Ω119895120579119896+T119894120579119896119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (52)

where

119883119894120579119896= 120598minus1

119883119894120579119896 Ω

119895120579119896= 120598minus1

Ω119895120579119896

119895120579119896= 120598minus1

119884119895120579119896

(53)

It follows from (51)ndash(53) that

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

(A119894120579119896+B119894120579119896119870119895120579119896) Ω119895120579119896

C119894120579119896

minus119883119897120579119896

lowast

(M119894120579119896+T119894120579119896119870119895120579119896) Ω119895120579119896

N119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (54)

which implies that the closed-loop system (20) and (21) isrobustly stable with119867

infinnorm bound 120574 byTheorem 6

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Corollary 8 Consider the nominal unforced system

119909 (119896 + 1) =

119903

sum

119894=1

ℎ119894(119911 (119896))

times (119860119894120579119896119909 (119896) + 119862

119894120579119896V (119896) + 119863

119894120579119896119909 (119896)119908 (119896))

119910 (119896) =

119903

sum

119894=1

ℎ119894(119911 (119896)) (119872

119894120579119896119909 (119896) + 119873

119894120579119896V (119896))

(55)

The nominal unforced fuzzy system is stable with 119867infin

normbound 120574 that is 119910(119896)

2lt 120574V(119896)

2for all nonzero V(119896) isin

1198972[0infin) under the zero initial condition if there exist matrices119875119894120579119896gt 0119903

119894=1 for all 119894 119895 isin 1 2 119903 such that

[

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

1015840

[

119895120579119896

0

0 119868

] [

119860119894120579119896

119862119894120579119896

119872119894120579119896119873119894120579119896

]

minus [

119875119894120579119896minus 1198631015840

119894120579119896

119895120579119896119863119894120579119896

0

0 1205742

119868

] lt 0

(56)

where 119895120579119896= sum119873

120579119896=1119901120579119896120579119896+1

119875119894120579119896

Corollary 9 The nominal unforced fuzzy system (55) is stablewith 119867

infinnorm bound 120574 if there exist matrices 119883

119894120579119896gt 0119903

119894=1

Ω119894120579119896119903

119894=1 119894 119895 isin 1 2 119903 satisfying the following LMIs

[

[

[

[

[

119883119894120579119896minus (Ω119895120579119896+ Ω1015840

119895120579119896

) lowast lowast lowast

0 minus1205742

119868 lowast lowast

119860119894120579119896Ω119895120579119896

119862119894120579119896

minus119883119895120579119896

lowast

119872119894120579119896Ω119895120579119896

119873119894120579119896

0 minus119868

]

]

]

]

]

lt 0 (57)

Corollary 10 When 119906(119896) = 0 the unforced system (7) and (8)is robustly stable with119867

infinnorm bound 120574 if there exist matrices

119883119894120579119896gt 0119903

119894=1and 120598 gt 0 119894 119895 isin 1 2 119903 satisfying the

following LMIs

[

[

[

[

[

[

[

[

minus119883119894120579119896

lowast lowast lowast lowast lowast

0 minus1205981205742

119868 lowast lowast lowast lowast

119860119894120579119896119883119894120579119896

120598119862119894120579119896

minus119883119895120579119896

lowast lowast lowast

119872119894120579119896119883119894120579119896

120598119873119894120579119896

0 minus120598119868 lowast lowast

0 0 1198641015840

11198941198641015840

2119894minus119868 lowast

1198671015840

1119894119883119894120579119896

1205981198673119894

0 0 119869 minus119868

]

]

]

]

]

]

]

]

lt 0 (58)

4 Numerical Simulation Example

In this section to illustrate the proposed new 119867infin

fuzzycontrol method the backing-up control of a computer-simulated truck-trailer is considered [30] For example thetruck-trailer model is given by

1199091(119896 + 1) = (1 minus

V119905119871

) 1199091(119896) + 120575

V119905ℓ

119906 (119896) (59)

1199092(119896 + 1) = 119909

2(119896) +

V119905119871

1199091(119896) (60)

1199093(119896 + 1) = 119909

3(119896) + V119905 sin(119909

2(119896) +

V1199052119871

1199091(119896)) (61)

where 1199091(119896) is the angle difference between truck and trailer

1199092(119896) is the angle of trailer and 119909

3(119896) is the vertical position

of rear end of trailer The parameter 120575 isin [0 1] is used todescribe the actuator failure where 120575 = 1 implies no failure120575 = 0 implies a total failure and 0 lt 120575 lt 1 implies apartial failure It is assumed that there is no actuator failureEquations (59) and (60) are linear but (61) is nonlinear Themodel parameters are given as 119871 = 55 ℓ = 28 V = minus10119905 = 20 and 120575 = 0

The transition probability matrix that relates the threeoperation modes is given as follows

P = [

[

048 029 023

06 01 03

01 065 025

]

]

(62)

As in [30] we set 120596 = 001120587 and the nonlinear termsin(119911(119896)) as

sin (119911 (119896)) = ℎ1(119911 (119896)) 119911 (119896) + ℎ

2(119911 (119896)) 120596119911 (119896) (63)

where ℎ1(119911(119896)) ℎ

2(119911(119896)) isin [0 1] and ℎ

1(119911(119896))+ℎ

2(119911(119896)) = 1

By solving the equations it can be seen that the membershipfunctions ℎ

1(119911(119896)) ℎ

2(119911(119896)) have the following relations

When 119911(119896) is about 0 rad ℎ1(119911(119896)) = 1 ℎ

2(119911(119896)) = 0 and

when 119911(119896) is about plusmn120587 rad ℎ1(119911(119896)) = 0 ℎ

2(119911(119896)) = 1 Then

the following fuzzy models can be used to design the fuzzycontroller for the uncertain nonlinear MJSS

Rule 1 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about 0 rad then

119909 (119896 + 1) = (1198601120579119896+ Δ1198601120579119896) 119909 (119896) + (119861

1120579119896+ Δ1198611120579119896) 119906 (119896)

+ (1198621120579119896+ Δ1198621120579119896) V (119896) + 119863

11205791198961199091(119896) 119908 (119896)

(64)

Rule 2 If 119911(119896) = 1199092(119896) + (V1199052119871)119909

1(119896) is about plusmn120587 rad then

119909 (119896 + 1) = (1198602120579119896+ Δ1198602120579119896) 119909 (119896) + (119861

2120579119896+ Δ1198612120579119896) 119906 (119896)

+ (1198622120579119896+ Δ1198622120579119896) V (119896) + 119863

2120579119896(119896) 119908 (119896)

(65)

where

1198601120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

V21199052

2119871

V119905 1

]

]

]

]

]

]

]

]

]

1198602120579119896=

[

[

[

[

[

[

[

[

[

1 minus

V119905119871

0 0

V119905119871

1 0

120596V21199052

2119871

120596V119905 1

]

]

]

]

]

]

]

]

]

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1198611120579119896= 1198612120579119896= [

V119905119871

0 0]

1015840

1198621120579119896= 1198622120579119896= [minus01 02 015]

1015840

1198631120579119896= 1198632120579119896=[

[

01 0 0

0 001 0

0 0 0

]

]

(66)

In the above uncertain fuzzy models the uncertainty todescribe the modeling error is assumed to be in the followingform

Δ1198601120579119896= 11986411Δ11986711 Δ119861

1120579119896= 11986411Δ11986721

Δ1198621120579119896= 11986411Δ11986731

Δ1198602120579119896= 11986412Δ11986712 Δ119861

2120579119896= 11986412Δ11986722

Δ1198622120579119896= 11986412Δ11986732

(67)

where

11986411= 11986412=[

[

006 0 0

0 006 0

0 0 006

]

]

11986421= 11986422=[

[

01 0 0

0 01 0

0 0 01

]

]

11986711= 11986712=[

[

03 0 0

0 03 0

0 0 03

]

]

11986721= 11986722= 0 119867

31= 11986732=[

[

0

0

0

]

]

(68)

We choose the following matrices for the uncertain discrete-time MJSS

1198721120579119896= 1198722120579119896=[

[

01 0 0

0 01 0

0 0 03

]

]

1198731120579119896= 1198732120579119896= [05 0 0]

1015840

1198791120579119896= 1198792120579119896= [05 0 0]

1015840

119869 =[

[

0 0 0

0 0 0

0 0 0

]

]

(69)

We set 119894 = 1 119895 = 1 119897 = 1 120579119896= 1 and 119894 = 2 119895 = 2 119897 =

2 120579119896= 2 respectively Based on Theorem 7 and using LMI

control toolbox in Matlab to solve LMIs (50) we can obtainthe feasible set of solutions as follows

11988311=[

[

02389 01410 01770

01410 01387 03577

01770 03577 19030

]

]

Ω11=[

[

03259 01769 02347

01768 01730 04467

02286 04594 23108

]

]

11988322=[

[

02725 02913 minus00027

02913 08440 00210

minus00027 00210 00402

]

]

Ω22=[

[

03559 02937 minus00047

02937 09264 00186

minus00047 00186 00370

]

]

11988411= [08566 02418 00753]

11988422= [08592 02091 minus00203]

(70)

By (51) there are the local state feedback gains given by

11987011= [50451 minus51128 05086]

11987022= [30255 minus07375 02056]

(71)

Then we can get

11987511=[

[

117492 minus157505 20651

minus157505 374327 minus47133

20651 minus47133 14101

]

]

11987522=[

[

1072937 minus351096 100427

minus351096 290490 minus85669

100427 minus85669 33892

]

]

(72)

It can be found that the LMIs ofTheorem 7 have some feasiblesolution for the uncertain discrete-time MJSS Setting 120598 = 1120574 = 26 the aforementioned simulation results are obtainedEmploying the feedback gains 119870 the fuzzy controller can beobtained by (4) By using Theorem 7 with the fuzzy controlapplied if the LMIs in (50) and (25) have a positive-definitesolution for 119870

119895120579119896and 119875

119894120579119896 respectively then the system (7)

and (8) driven by the designed fuzzy controller is stable withsatisfying the119867

infinperformance constraint

5 Conclusions

In the paper the robust 119867infin

control has been discussed fora class of nonlinear discrete-time stochastic MJSS First anew LMI characterization of stability with 119867

infinnorm bound

for uncertain discrete-time stochastic MJSS has been givenMoreover sufficient conditions on robust stabilization and119867infinperformance analysis and control have been presented on

LMIs Furthermore there are some corollaries of the stabilityand the nominal unforced system has been given Finally anumerical simulation example has been presented to showthe effectiveness

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (no 61304080 and no 61174078) aProject of Shandong Province Higher Educational Scienceand Technology Program (no J12LN14) the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina and the State Key Laboratory of Alternate Elec-trical Power System with Renewable Energy Sources (noLAPS13018)

References

[1] S Saat D Huang and S K Nguang ldquoRobust state feedbackcontrol of uncertain polynomial discrete-time systems anintegral action approachrdquo International Journal of InnovativeComputing Information andControl vol 9 no 3 pp 1233ndash12442013

[2] R N Yang P Shi G P Liu and H J Gao ldquoNetwork-basedfeedback control for systems with mixed delays based onquantization and dropout compensationrdquo Automatica vol 47no 12 pp 2805ndash2809 2011

[3] J H Zhang M F Ren Y Tian G L Hou and F Fang ldquoCon-strained stochastic distribution control for nonlinear stochasticsystems with non-Gaussian noisesrdquo International Journal ofInnovative Computing Information and Control vol 9 no 4 pp1759ndash1767 2013

[4] R Yang P Shi and G Liu ldquoFiltering for discrete-time net-worked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[5] S G Cao N W Rees and G Feng ldquo119867infin

control of uncertaindynamical fuzzy discrete-time systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 31 no 5 pp 802ndash812 2001

[6] X J Su P Shi L G Wu and S K Nguang ldquoInduced l2filtering

of fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1251ndash1264 2013

[7] Y Y Cao and P M Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[8] D J Choi and P Park ldquo119867infinstate-feedback controller design for

discrete-time fuzzy systems using fuzzy weighting-dependentlyapunov functionsrdquo IEEE Transactions on Fuzzy Systems vol11 no 2 pp 271ndash278 2003

[9] G Feng ldquoStability analysis of piecewise discrete-time linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no7 pp 1108ndash1112 2002

[10] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1343 2001

[11] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO systems using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[12] X J Su L G Wu and P Shi ldquoSensor networks with randomlink failures distributed filtering for T-S fuzzy systemsrdquo IEEETransactions on Industrial Informatics vol 9 no 3 pp 1739ndash1750 2013

[13] Z Lin Y Lin and W H Zhang ldquo119867infin

stabilisation of non-linear stochastic active fault-tolerant control systems fuzzy-interpolation approachrdquo IET Control Theory amp Applicationsvol 4 no 10 pp 2003ndash2017 2010

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] M Mariton Jump Linear Systems in Automatic Control MarcelDekker New York NY USA 1990

[16] Y D Ji and H J Chizeck ldquoControllability observability anddiscrete-time Markovian jump linear quadratic controlrdquo Inter-national Journal of Control vol 48 no 2 pp 481ndash498 1988

[17] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[18] W H Zhang and B S Chen ldquoOn stabilizability and exactobservability of stochastic systems with their applicationsrdquoAutomatica vol 40 no 1 pp 87ndash94 2004

[19] Y G Fang and K A Loparo ldquoStabilization of continuous-timejump linear systemsrdquo IEEE Transactions on Automatic Controlvol 47 no 10 pp 1590ndash1603 2002

[20] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[21] L Arnold W Kliemann and E Oelijeklaus ldquoLyapunov expo-nents of linear stochastic systemsrdquo in Lyapunov Exponents vol1186 of Lecture Notes in Mathematics pp 85ndash125 SpringerBerlin Germany 1986

[22] O L V Costa J B R do Val and J C Geromel ldquoA convex pro-gramming approach to H

2-control of discrete-time Markovian

jump linear systemsrdquo International Journal of Control vol 66no 4 pp 557ndash580 1997

[23] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012

[24] O L V Costa and R P Marques ldquoMixed 1198672119867infincontrol of

discrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 43 no 1 pp 95ndash100 1998

[25] D Hinrichsen and A J Pritchard ldquoStochastic 119867infinrdquo SIAM

Journal on Control and Optimization vol 36 no 5 pp 1504ndash1538 1998

[26] W Zhang and B Chen ldquoState feedback 119867infin

control for a classof nonlinear stochastic systemsrdquo SIAM Journal on Control andOptimization vol 44 no 6 pp 1973ndash1991 2006

[27] E Boukas and P Shi ldquoStochastic stability and guaranteed costcontrol of discrete-time uncertain systems with Markovianjumping parametersrdquo International Journal of Robust and Non-linear Control vol 8 no 13 pp 1155ndash1167 1998

[28] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[29] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[30] K Tanaka and M Sano ldquoRobust stabilization problem of fuzzycontrol systems and its application to backing up control of atruck-trailerrdquo IEEE Transactions on Fuzzy Systems vol 2 no 2pp 119ndash134 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of