Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 520503 11 pageshttpdxdoiorg1011552013520503

Research ArticleH

infinControl for Two-Dimensional Markovian Jump

Systems with State-Delays and Defective Mode Information

Yanling Wei1 Mao Wang1 Hamid Reza Karimi2 and Jianbin Qiu1

1 Research Institute of Intelligent Control and Systems Harbin Institute of Technology Harbin 150001 China2Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Yanling Wei yanlingweihiteducn

Received 12 July 2013 Revised 13 September 2013 Accepted 16 September 2013

Academic Editor Bo Shen

Copyright copy 2013 Yanling Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the problem of Hinfin

state-feedback control for a class of two-dimensional (2D) discrete-time Markovianjump linear time-delay systems with defective mode information The mathematical model of the 2D system is established basedon the well-known Fornasini-Marchesini local state-space model and the defective mode information simultaneously consists ofthe exactly known partially unknown and uncertain transition probabilities By carefully analyzing the features of the transitionprobability matrices together with the convexification of uncertain domains a new H

infinperformance analysis criterion for the

underlying system is firstly derived and then theHinfinstate-feedback controller synthesis is developed via a linearisation technique

It is shown that the controller gains can be constructed by solving a set of linear matrix inequalities Finally an illustrative exampleis provided to verify the effectiveness of the proposed design method

1 Introduction

During the past decades two-dimensional (2D) systems havedrawn considerable attention due to their extensive appli-cations of both theoretical and practical interests in manymodern engineering fields such as process control (thermalprocesses gas absorption water stream heating etc) [1]multidimensional digital filtering [2] and image processing(enhancement deblurring seismographic data processingetc) [3] In particular since the well-known Roesser state-space model and the Fornasini-Marchesini local state-space(FMLSS) model were proposed theory on 2D systems hasprogressed greatly [4ndash11]

On the other hand as a class of stochastic hybrid systemsMarkovian jump linear systems (MJLSs) have been exten-sively investigated [12ndash26] The driving force behind this isthat MJLSs can model different classes of dynamic systemssubject to random abrupt variations in their structures forexample manufacturing systems power systems and net-worked control systems where random failure repairs andsudden environment changes may occur in Markov chains[27ndash29] It is known that MJLSs are described by a set ofclassical differential (or difference) equations and a Markov

stochastic process (or Markov chain) [30] As a decisive fac-tor transition probabilities (TPs) in the jumping processdetermine the system behavior to a large extent and so farmany studies on the analysis and synthesis of MJLSs havebeen carried out in the context of perfect information on TPs[12ndash15 25] In practice however defective mode informationis often encountered especially when adequate efforts toobtain the accurate TPs are costly or time consumingThus itis more practical and interesting to study more general jumpsystems with defective mode information Recently therehave appeared some results on the analysis and synthesis ofMJLSs with uncertain TPs or partially unknown TPs [26 31ndash38] To mention a few the authors in [31] studied the H

infin

filter synthesis problem for a class of MJLSs with partiallyunknown TPs The author in [32] considered the robuststability analysis and stabilization problems for a class ofMJLSs with polytopic uncertain TPs the authors in [36]addressed the robust stability analysis problem for a class ofMJLSs with norm-bounded uncertain TPs

However the above-mentioned works were only con-cerned with one-dimensional (1D) systems Inevitably when2D systems are employed to model dynamic systems withrandom abrupt changes in their structures or parameters

2 Mathematical Problems in Engineering

such as chemical process control themathematical modelingof such physical systems would be naturally dependent onjumping parameters For example information propagationoccurs from pass to pass and along a given pass in a gasabsorption water stream heating and air drying [39] There-fore 2D MJLSs emerge as a more reasonable description toaccount for the parameter jumping phenomenon and havea great potential in engineering applications Recently thestability analysis and synthesis of discrete-time delay-free 2DMJLSs described by Roesser model were reported in [39 40]which are obtained based on the traditional assumption ofcomplete knowledge on TPs To the authorsrsquo best knowledgethe analysis and synthesis for 2DMJLSs with state-delays anddefective mode information have not drawn much attentionyet which motivates us for this study

In this paper the Hinfin

control problem for a class of 2Ddiscrete-time MJLSs with state-delays and defective modeinformation will be studied The mathematical model of the2D system is established in terms of the FMLSSmodel subjectto state-delays and the defective mode information simulta-neously includes the exactly known partially unknown anduncertain TPs By fully considering the properties of thetransition probabilitymatrices together with the convexifica-tion of uncertain domains a new H

infinperformance analysis

criterion for the closed-loop system will be firstly derived Bya linearisation procedure the corresponding H

infincontroller

synthesis will then be converted into a convex optimizationproblem in terms of a set of linear matrix inequalitiesFinally an illustrative example will be performed to show theeffectiveness of the proposed controller synthesis method

Notations The notations used throughout the paper are stan-dard R119899 and R119898times119899 denote respectively the 119899-dimensionalEuclidean space and the set of all 119898 times 119899 real matrices N+represents the sets of positive integers the notation 119875 gt 0

means that 119875 is real symmetric and positive definite I and 0represent the identity matrix and a zero matrix respectively(SFP) denotes a complete probability space in whichS is the sample space F is the 120590 algebra of subsets ofthe sample space and P is the probability measure on FE[sdot] stands for the mathematical expectation sdot refersto the Euclidean norm of a vector or its induced normof a matrix 119897

2[0infin) [0infin) denotes the space of square

summable sequences on [0infin) [0infin) Matrices if notexplicitly stated are assumed to have appropriate dimensionsfor algebra operations

2 Problem Formulation and Preliminaries

Fix a complete probability space (SFP) and consider thefollowing two-dimensional (2D) discrete-time Markovianjump linear systems (MJLSs) described by the Fornasini-Marchesini local state-space (FMLSS) model with time-delays in the states

(Σ) 119909 (119894 + 1 119895 + 1) = 1198601(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

+ 1198602(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

+ 1198601198891

(119903 (119894 119895 + 1)) 119909 (119894 minus 1198891 119895 + 1)

+ 1198601198892

(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895 minus 1198892)

+ 1198611(119903 (119894 119895 + 1)) 119906 (119894 119895 + 1)

+ 1198612(119903 (119894 + 1 119895)) 119906 (119894 + 1 119895)

+ 1198631(119903 (119894 119895 + 1))119908 (119894 119895 + 1)

+ 1198632(119903 (119894 + 1 119895)) 119908 (119894 + 1 119895)

119911 (119894 119895) = 119862 (119903 (119894 119895)) 119909 (119894 119895) + 1198613(119903 (119894 119895)) 119906 (119894 119895)

+ 1198633(119903 (119894 119895)) 119908 (119894 119895)

(1)

where 119909(119894 119895) isin R119899119909 is the state vector 119906(119894 119895) isin R119899119906 is thecontrol input 119911(119894 119895) isin R119899119911 is the controlled output 119908(119894 119895) isin

R119899119908 denotes the disturbance input vector which belongs to1198972[0infin) [0infin) and 119889

1and 119889

2are two constant positive

integers representing delays along vertical and horizon-tal directions respectively 119860

1(119903(119894 119895 + 1)) 119860

2(119903(119894 + 1 119895))

1198601198891

(119903(119894 119895 + 1)) 1198601198892

(119903(119894 + 1 119895)) 1198611(119903(119894 119895 + 1)) 119861

2(119903(119894 +

1 119895))1198631(119903(119894 119895 + 1))119863

2(119903(119894 + 1 119895)) 119862(119903(119894 119895)) 119861

3(119903(119894 119895)) and

1198633(119903(119894 119895)) are real-valued system matrices These matrices

are functions of 119903(119894 119895) which is described by a discrete-timediscrete-state homogeneous Markov chain with a finite-statespaceI = 1 119873 and a stationary transition probabilitymatrix (TPM) Π = [120587

119898119899]119873times119873

where

120587119898119899

= Pr (119903 (119894 + 1 119895 + 1) = 119899 | 119903 (119894 119895 + 1) = 119898)

= Pr (119903 (119894 + 1 119895+1) = 119899 | 119903 (119894+1 119895) = 119898) forall119898 119899 isin I

(2)

with 120587119898119899

ge 0 and sum119873

119899=1120587119898119899

= 1 For 119903(119894 + 1 119895) = 119898 isin

I or 119903(119894 119895 + 1) = 119898 isin I the system matrices of the119898th mode are denoted by (119860

1119898 1198602119898

1198601198891119898

1198601198892119898

1198611119898

1198612119898

1198631119898

1198632119898

119862119898 1198613119898

1198633119898

) which are known and with appro-priate dimensions Unless otherwise stated similar simplifi-cation is also applied to other matrices in the following

In this paper the transition probabilities (TPs) of thejumping process are assumed to be uncertain and partiallyaccessed that is the TPM Π = [120587

119898119899]119873times119873

is assumedto belong to a given polytope P

Πwith vertices Π

119904 119904 =

1 2 119872 PΠ

= Π | Π = sum119872

119904=1120572119904Π119904 120572119904ge 0 sum

119872

119904=1120572119904= 1

where Π119904= [120587119898119899

]119873times119873

119898 119899 isin I are given TPMs containingunknown elements still For instance for system (Σ)with fouroperation modes the TPMmay be as

[

[

[

[

12058711

12

13

12058714

21

12058722

23

12058724

12058731

32

12058733

34

12058741

42

43

44

]

]

]

]

(3)

where the elements labeled with ldquordquo and ldquosimrdquo represent theunknown information and polytopic uncertainties on TPsrespectively and the others are known TPs For notational

Mathematical Problems in Engineering 3

clarity for all119898 isin I we denoteI = I(119898)

KcupI(119898)

UCcupI(119898)

UKas

follows

I(119898)

K = 119899 120587119898119899

is known

I(119898)

UC = 119899 119898119899

is uncertain

I(119898)

UK = 119899 119898119899

is unknown

(4)

Moreover ifI(119898)K

= 0 andI(119898)

UC= 0 it is further described as

I(119898)

K = K1(119898)

K119905(119898)

forall1 le 119905(119898)

le 119873 minus 2

I(119898)

UC = U1(119898)

UV(119898)

forall1 le V(119898)

le 119873

(5)

where K119905(119898)

isin N+ represents the 119905(119898)

th known element withthe index K

119905(119898)

in the 119898th row of the TPM and UV(119898)

isin N+

represents the V(119898)

th uncertain element with the index UV(119898)

in the 119898th row of the TPM Obviously 1 le 119905(119898)

+ V(119898)

le 119873Also we denote

120587(119898119904)

UK = sum

119899isinI(119898)

UK

119898119899

= 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899 (6)

where (119904)

119898119899represents an uncertain TP in the 119904th polytope for

all 119904 = 1 119872The boundary conditions of system (Σ) in (1) are defined

by

119909 (119894 119895) = 120601 (119894 119895) forall119895 ge 0 minus1198891le 119894 le 0

119909 (119894 119895) = 120593 (119894 119895) forall119894 ge 0 minus1198892le 119895 le 0

120601 (0 0) = 120593 (0 0)

(7)

Throughout this paper the following assumption is made

Assumption 1 The boundary conditions are assumed to sat-isfy

lim1198791rarrinfin

E

1198791

sum

119895=0

0

sum

119894=minus1198891

(120601T(119894 119895) 120601 (119894 119895))

+ lim1198792rarrinfin

E

1198792

sum

119894=0

0

sum

119895=minus1198892

(120593T(119894 119895) 120593 (119894 119895))

lt infin

(8)

In this paper we are interested in the Hinfin

controllersynthesis forMJLSs (1)The followingmode-dependent state-feedback control law is used

119906 (119894 119895) = 119870 (119903 (119894 119895)) 119909 (119894 119895) (9)

where 119870(119903(119894 119895)) isin R119899119906times119899119909

Then the corresponding closed-loop system can be repre-sented as follows

(Σ) 119909 (119894 + 1 119895 + 1) = 1198601(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

+ 1198602(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

+ 1198601198891

(119903 (119894 119895 + 1)) 119909 (119894 minus 1198891 119895 + 1)

+ 1198601198892

(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895 minus 1198892)

+ 1198631(119903 (119894 119895 + 1))119908 (119894 119895 + 1)

+ 1198632(119903 (119894 + 1 119895)) 119908 (119894 + 1 119895)

119911 (119894 119895) = 119862 (119903 (119894 119895)) 119909 (119894 119895) + 1198633(119903 (119894 119895)) 119908 (119894 119895)

(10)

where

1198601(119903 (119894 119895 + 1))

= 1198601(119903 (119894 119895 + 1)) + 119861

1(119903 (119894 119895 + 1))119870 (119903 (119894 119895 + 1))

1198602(119903 (119894 + 1 119895))

= 1198602(119903 (119894 + 1 119895)) + 119861

2(119903 (119894 + 1 119895))119870 (119903 (119894 + 1 119895))

119862 (119903 (119894 119895)) = 119862 (119903 (119894 119895)) + 1198613(119903 (119894 119895))119870 (119903 (119894 119895))

(11)

Before proceeding further we introduce the followingdefinitions

Definition 2 System (10) is said to be stochastically stable iffor119908(119894 119895) = 0 and the boundary conditions satisfying (8) thefollowing condition holds

E

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119909 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119909 (119894 + 1 119895)

1003817100381710038171003817

2

)

lt infin (12)

Definition 3 Given a scalar 120574 gt 0 system (10) is said tobe stochastically stable with an H

infindisturbance attenuation

performance index 120574 if it is stochastically stable with119908(119894 119895) =

0 and under zero boundary conditions 120601(119894 119895) = 120593(119894 119895) = 0

in (7) for all nonzero 119908 isin 1198972[0infin) [0infin) satisfies

E2

lt 1205741199082 (13)

where

E2

= radicE

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119911 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119911 (119894 + 1 119895)

1003817100381710038171003817

2

)

1199082

= radic

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119908(119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119908(119894 + 1 119895)

1003817100381710038171003817

2

)

(14)

4 Mathematical Problems in Engineering

Therefore the purpose of this paper is to design a mode-dependent H

infinstate-feedback controller in the form of (9)

such that the resulting closed-loop system (10) with defectivemode information is stochastically stable with a prescribedHinfin

performance index 120574

3 Main Results

In this section based on a Markovian Lyapunov-Krasovskiifunctional (MLKF) a new formulation of bounded reallemma (BRL) for the two-dimensional (2D)Markovian jumplinear system (MJLS) (10) with state-delays and defectivemode information will be firstly given Then via a lineari-sation procedure theH

infincontroller synthesis will be devel-

oped

31Hinfin

Performance Analysis In this subsection by invok-ing the properties of the transition probability matrices(TPMs) together with the convexification of uncertaindomains an H

infinperformance analysis criterion for the

closed-loop system (10) with state-delays and defective modeinformation is presented which will play a significant role insolving theH

infincontroller synthesis problem

Proposition 4 The 2D MJLS in (10) with state-delays anddefective mode information is stochastically stable with a guar-anteedH

infinperformance 120574 if the matrices 119875

1119898 1198752119898

1198761 1198762 isin

R119899119909times119899119909 with 1198751119898

gt 0 1198752119898

gt 0 1198761gt 0 and 119876

2gt 0119898 isin I

such that the following matrix inequalities hold

AT119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(15)

where

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899)

+ sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899) + 120587

(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(16)

Proof Consider the following MLKF for the 2D MJLS (10)

119881 (119894 119895) =

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

+

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(17)

where

1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1) 119875

1(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

=

119894minus1

sum

119896=119894minus1198891

119909T(119896 119895 + 1)119876

1119909 (119896 119895 + 1)

1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119875

2(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

=

119895minus1

sum

119896=119895minus1198892

119909T(119894 + 1 119896) 119876

2119909 (119894 + 1 119896)

(18)

Then based on the MLKF defined in (17) it is knownthat the following condition (19) guarantees that the 2Dclosed-loop system (10) is stochastically stable with an H

infin

performance 120574 under zero boundary conditions for anynonzero 119908(119894 119895) isin 119897

2[0infin) [0infin)

Ω = Δ119881 (119894 119895) + 2

E2

minus 1205742

1199082

2lt 0 (19)

where

Δ119881 (119894 119895) = E

2

sum

119896=1

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898

+ E

4

sum

119896=3

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898

Mathematical Problems in Engineering 5

minus

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

minus

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(20)

and E2

and 1199082are defined in (14)

Taking the time difference of119881(119894 119895) along the trajectoriesof the 2D system in (10) yields

Δ1198811= E [119881

1(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198751119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198751119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198751119899

+ sum

119899isinI(119898)

UK

119898119899

1198751119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

Δ1198812= E [119881

2(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1)119876

1119909 (119894 119895 + 1)

minus 119909T(119894 minus 1198891 119895 + 1)119876

1119909 (119894 minus 119889

1 119895 + 1)

Δ1198813= E [119881

3(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198752119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198752119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198752119899

+ sum

119899isinI(119898)

UK

119898119899

1198752119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

Δ1198814= E [119881

4(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119876

2119909 (119894 + 1 119895)

minus 119909T(119894 + 1 119895 minus 119889

2) 1198762119909 (119894 + 1 119895 minus 119889

2)

(21)

Therefore based on the MLKF defined in (17) together withconsideration of (10) and (21) we have

Ω = 120589T(119894 119895) [A

T119898

(P1119899

+ P2119899)A119898

+ CT119898C119898

+ Θ119898]

times 120589 (119894 119895) 119898 119899 isin I

(22)

where

120589 (119894 119895) = [119909T(119894 119895 + 1) 119909

T(119894 + 1 119895) 119909

T(119894 minus 1198891 119895 + 1)

119909T(119894 + 1 119895 minus 119889

2) 119908

T(119894 119895 + 1) 119908

T(119894 + 1 119895) ]

T

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P119897119899

= sum

119899isinI(119898)

K

120587119898119899

119875119897119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)119875119897119899

+ sum

119899isinI(119898)

UK

119898119899

119875119897119899 119897 = 1 2

(23)

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

such as chemical process control themathematical modelingof such physical systems would be naturally dependent onjumping parameters For example information propagationoccurs from pass to pass and along a given pass in a gasabsorption water stream heating and air drying [39] There-fore 2D MJLSs emerge as a more reasonable description toaccount for the parameter jumping phenomenon and havea great potential in engineering applications Recently thestability analysis and synthesis of discrete-time delay-free 2DMJLSs described by Roesser model were reported in [39 40]which are obtained based on the traditional assumption ofcomplete knowledge on TPs To the authorsrsquo best knowledgethe analysis and synthesis for 2DMJLSs with state-delays anddefective mode information have not drawn much attentionyet which motivates us for this study

In this paper the Hinfin

control problem for a class of 2Ddiscrete-time MJLSs with state-delays and defective modeinformation will be studied The mathematical model of the2D system is established in terms of the FMLSSmodel subjectto state-delays and the defective mode information simulta-neously includes the exactly known partially unknown anduncertain TPs By fully considering the properties of thetransition probabilitymatrices together with the convexifica-tion of uncertain domains a new H

infinperformance analysis

criterion for the closed-loop system will be firstly derived Bya linearisation procedure the corresponding H

infincontroller

synthesis will then be converted into a convex optimizationproblem in terms of a set of linear matrix inequalitiesFinally an illustrative example will be performed to show theeffectiveness of the proposed controller synthesis method

Notations The notations used throughout the paper are stan-dard R119899 and R119898times119899 denote respectively the 119899-dimensionalEuclidean space and the set of all 119898 times 119899 real matrices N+represents the sets of positive integers the notation 119875 gt 0

means that 119875 is real symmetric and positive definite I and 0represent the identity matrix and a zero matrix respectively(SFP) denotes a complete probability space in whichS is the sample space F is the 120590 algebra of subsets ofthe sample space and P is the probability measure on FE[sdot] stands for the mathematical expectation sdot refersto the Euclidean norm of a vector or its induced normof a matrix 119897

2[0infin) [0infin) denotes the space of square

summable sequences on [0infin) [0infin) Matrices if notexplicitly stated are assumed to have appropriate dimensionsfor algebra operations

2 Problem Formulation and Preliminaries

Fix a complete probability space (SFP) and consider thefollowing two-dimensional (2D) discrete-time Markovianjump linear systems (MJLSs) described by the Fornasini-Marchesini local state-space (FMLSS) model with time-delays in the states

(Σ) 119909 (119894 + 1 119895 + 1) = 1198601(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

+ 1198602(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

+ 1198601198891

(119903 (119894 119895 + 1)) 119909 (119894 minus 1198891 119895 + 1)

+ 1198601198892

(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895 minus 1198892)

+ 1198611(119903 (119894 119895 + 1)) 119906 (119894 119895 + 1)

+ 1198612(119903 (119894 + 1 119895)) 119906 (119894 + 1 119895)

+ 1198631(119903 (119894 119895 + 1))119908 (119894 119895 + 1)

+ 1198632(119903 (119894 + 1 119895)) 119908 (119894 + 1 119895)

119911 (119894 119895) = 119862 (119903 (119894 119895)) 119909 (119894 119895) + 1198613(119903 (119894 119895)) 119906 (119894 119895)

+ 1198633(119903 (119894 119895)) 119908 (119894 119895)

(1)

where 119909(119894 119895) isin R119899119909 is the state vector 119906(119894 119895) isin R119899119906 is thecontrol input 119911(119894 119895) isin R119899119911 is the controlled output 119908(119894 119895) isin

R119899119908 denotes the disturbance input vector which belongs to1198972[0infin) [0infin) and 119889

1and 119889

2are two constant positive

integers representing delays along vertical and horizon-tal directions respectively 119860

1(119903(119894 119895 + 1)) 119860

2(119903(119894 + 1 119895))

1198601198891

(119903(119894 119895 + 1)) 1198601198892

(119903(119894 + 1 119895)) 1198611(119903(119894 119895 + 1)) 119861

2(119903(119894 +

1 119895))1198631(119903(119894 119895 + 1))119863

2(119903(119894 + 1 119895)) 119862(119903(119894 119895)) 119861

3(119903(119894 119895)) and

1198633(119903(119894 119895)) are real-valued system matrices These matrices

are functions of 119903(119894 119895) which is described by a discrete-timediscrete-state homogeneous Markov chain with a finite-statespaceI = 1 119873 and a stationary transition probabilitymatrix (TPM) Π = [120587

119898119899]119873times119873

where

120587119898119899

= Pr (119903 (119894 + 1 119895 + 1) = 119899 | 119903 (119894 119895 + 1) = 119898)

= Pr (119903 (119894 + 1 119895+1) = 119899 | 119903 (119894+1 119895) = 119898) forall119898 119899 isin I

(2)

with 120587119898119899

ge 0 and sum119873

119899=1120587119898119899

= 1 For 119903(119894 + 1 119895) = 119898 isin

I or 119903(119894 119895 + 1) = 119898 isin I the system matrices of the119898th mode are denoted by (119860

1119898 1198602119898

1198601198891119898

1198601198892119898

1198611119898

1198612119898

1198631119898

1198632119898

119862119898 1198613119898

1198633119898

) which are known and with appro-priate dimensions Unless otherwise stated similar simplifi-cation is also applied to other matrices in the following

In this paper the transition probabilities (TPs) of thejumping process are assumed to be uncertain and partiallyaccessed that is the TPM Π = [120587

119898119899]119873times119873

is assumedto belong to a given polytope P

Πwith vertices Π

119904 119904 =

1 2 119872 PΠ

= Π | Π = sum119872

119904=1120572119904Π119904 120572119904ge 0 sum

119872

119904=1120572119904= 1

where Π119904= [120587119898119899

]119873times119873

119898 119899 isin I are given TPMs containingunknown elements still For instance for system (Σ)with fouroperation modes the TPMmay be as

[

[

[

[

12058711

12

13

12058714

21

12058722

23

12058724

12058731

32

12058733

34

12058741

42

43

44

]

]

]

]

(3)

where the elements labeled with ldquordquo and ldquosimrdquo represent theunknown information and polytopic uncertainties on TPsrespectively and the others are known TPs For notational

Mathematical Problems in Engineering 3

clarity for all119898 isin I we denoteI = I(119898)

KcupI(119898)

UCcupI(119898)

UKas

follows

I(119898)

K = 119899 120587119898119899

is known

I(119898)

UC = 119899 119898119899

is uncertain

I(119898)

UK = 119899 119898119899

is unknown

(4)

Moreover ifI(119898)K

= 0 andI(119898)

UC= 0 it is further described as

I(119898)

K = K1(119898)

K119905(119898)

forall1 le 119905(119898)

le 119873 minus 2

I(119898)

UC = U1(119898)

UV(119898)

forall1 le V(119898)

le 119873

(5)

where K119905(119898)

isin N+ represents the 119905(119898)

th known element withthe index K

119905(119898)

in the 119898th row of the TPM and UV(119898)

isin N+

represents the V(119898)

th uncertain element with the index UV(119898)

in the 119898th row of the TPM Obviously 1 le 119905(119898)

+ V(119898)

le 119873Also we denote

120587(119898119904)

UK = sum

119899isinI(119898)

UK

119898119899

= 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899 (6)

where (119904)

119898119899represents an uncertain TP in the 119904th polytope for

all 119904 = 1 119872The boundary conditions of system (Σ) in (1) are defined

by

119909 (119894 119895) = 120601 (119894 119895) forall119895 ge 0 minus1198891le 119894 le 0

119909 (119894 119895) = 120593 (119894 119895) forall119894 ge 0 minus1198892le 119895 le 0

120601 (0 0) = 120593 (0 0)

(7)

Throughout this paper the following assumption is made

Assumption 1 The boundary conditions are assumed to sat-isfy

lim1198791rarrinfin

E

1198791

sum

119895=0

0

sum

119894=minus1198891

(120601T(119894 119895) 120601 (119894 119895))

+ lim1198792rarrinfin

E

1198792

sum

119894=0

0

sum

119895=minus1198892

(120593T(119894 119895) 120593 (119894 119895))

lt infin

(8)

In this paper we are interested in the Hinfin

controllersynthesis forMJLSs (1)The followingmode-dependent state-feedback control law is used

119906 (119894 119895) = 119870 (119903 (119894 119895)) 119909 (119894 119895) (9)

where 119870(119903(119894 119895)) isin R119899119906times119899119909

Then the corresponding closed-loop system can be repre-sented as follows

(Σ) 119909 (119894 + 1 119895 + 1) = 1198601(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

+ 1198602(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

+ 1198601198891

(119903 (119894 119895 + 1)) 119909 (119894 minus 1198891 119895 + 1)

+ 1198601198892

(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895 minus 1198892)

+ 1198631(119903 (119894 119895 + 1))119908 (119894 119895 + 1)

+ 1198632(119903 (119894 + 1 119895)) 119908 (119894 + 1 119895)

119911 (119894 119895) = 119862 (119903 (119894 119895)) 119909 (119894 119895) + 1198633(119903 (119894 119895)) 119908 (119894 119895)

(10)

where

1198601(119903 (119894 119895 + 1))

= 1198601(119903 (119894 119895 + 1)) + 119861

1(119903 (119894 119895 + 1))119870 (119903 (119894 119895 + 1))

1198602(119903 (119894 + 1 119895))

= 1198602(119903 (119894 + 1 119895)) + 119861

2(119903 (119894 + 1 119895))119870 (119903 (119894 + 1 119895))

119862 (119903 (119894 119895)) = 119862 (119903 (119894 119895)) + 1198613(119903 (119894 119895))119870 (119903 (119894 119895))

(11)

Before proceeding further we introduce the followingdefinitions

Definition 2 System (10) is said to be stochastically stable iffor119908(119894 119895) = 0 and the boundary conditions satisfying (8) thefollowing condition holds

E

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119909 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119909 (119894 + 1 119895)

1003817100381710038171003817

2

)

lt infin (12)

Definition 3 Given a scalar 120574 gt 0 system (10) is said tobe stochastically stable with an H

infindisturbance attenuation

performance index 120574 if it is stochastically stable with119908(119894 119895) =

0 and under zero boundary conditions 120601(119894 119895) = 120593(119894 119895) = 0

in (7) for all nonzero 119908 isin 1198972[0infin) [0infin) satisfies

E2

lt 1205741199082 (13)

where

E2

= radicE

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119911 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119911 (119894 + 1 119895)

1003817100381710038171003817

2

)

1199082

= radic

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119908(119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119908(119894 + 1 119895)

1003817100381710038171003817

2

)

(14)

4 Mathematical Problems in Engineering

Therefore the purpose of this paper is to design a mode-dependent H

infinstate-feedback controller in the form of (9)

such that the resulting closed-loop system (10) with defectivemode information is stochastically stable with a prescribedHinfin

performance index 120574

3 Main Results

In this section based on a Markovian Lyapunov-Krasovskiifunctional (MLKF) a new formulation of bounded reallemma (BRL) for the two-dimensional (2D)Markovian jumplinear system (MJLS) (10) with state-delays and defectivemode information will be firstly given Then via a lineari-sation procedure theH

infincontroller synthesis will be devel-

oped

31Hinfin

Performance Analysis In this subsection by invok-ing the properties of the transition probability matrices(TPMs) together with the convexification of uncertaindomains an H

infinperformance analysis criterion for the

closed-loop system (10) with state-delays and defective modeinformation is presented which will play a significant role insolving theH

infincontroller synthesis problem

Proposition 4 The 2D MJLS in (10) with state-delays anddefective mode information is stochastically stable with a guar-anteedH

infinperformance 120574 if the matrices 119875

1119898 1198752119898

1198761 1198762 isin

R119899119909times119899119909 with 1198751119898

gt 0 1198752119898

gt 0 1198761gt 0 and 119876

2gt 0119898 isin I

such that the following matrix inequalities hold

AT119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(15)

where

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899)

+ sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899) + 120587

(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(16)

Proof Consider the following MLKF for the 2D MJLS (10)

119881 (119894 119895) =

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

+

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(17)

where

1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1) 119875

1(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

=

119894minus1

sum

119896=119894minus1198891

119909T(119896 119895 + 1)119876

1119909 (119896 119895 + 1)

1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119875

2(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

=

119895minus1

sum

119896=119895minus1198892

119909T(119894 + 1 119896) 119876

2119909 (119894 + 1 119896)

(18)

Then based on the MLKF defined in (17) it is knownthat the following condition (19) guarantees that the 2Dclosed-loop system (10) is stochastically stable with an H

infin

performance 120574 under zero boundary conditions for anynonzero 119908(119894 119895) isin 119897

2[0infin) [0infin)

Ω = Δ119881 (119894 119895) + 2

E2

minus 1205742

1199082

2lt 0 (19)

where

Δ119881 (119894 119895) = E

2

sum

119896=1

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898

+ E

4

sum

119896=3

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898

Mathematical Problems in Engineering 5

minus

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

minus

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(20)

and E2

and 1199082are defined in (14)

Taking the time difference of119881(119894 119895) along the trajectoriesof the 2D system in (10) yields

Δ1198811= E [119881

1(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198751119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198751119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198751119899

+ sum

119899isinI(119898)

UK

119898119899

1198751119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

Δ1198812= E [119881

2(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1)119876

1119909 (119894 119895 + 1)

minus 119909T(119894 minus 1198891 119895 + 1)119876

1119909 (119894 minus 119889

1 119895 + 1)

Δ1198813= E [119881

3(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198752119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198752119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198752119899

+ sum

119899isinI(119898)

UK

119898119899

1198752119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

Δ1198814= E [119881

4(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119876

2119909 (119894 + 1 119895)

minus 119909T(119894 + 1 119895 minus 119889

2) 1198762119909 (119894 + 1 119895 minus 119889

2)

(21)

Therefore based on the MLKF defined in (17) together withconsideration of (10) and (21) we have

Ω = 120589T(119894 119895) [A

T119898

(P1119899

+ P2119899)A119898

+ CT119898C119898

+ Θ119898]

times 120589 (119894 119895) 119898 119899 isin I

(22)

where

120589 (119894 119895) = [119909T(119894 119895 + 1) 119909

T(119894 + 1 119895) 119909

T(119894 minus 1198891 119895 + 1)

119909T(119894 + 1 119895 minus 119889

2) 119908

T(119894 119895 + 1) 119908

T(119894 + 1 119895) ]

T

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P119897119899

= sum

119899isinI(119898)

K

120587119898119899

119875119897119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)119875119897119899

+ sum

119899isinI(119898)

UK

119898119899

119875119897119899 119897 = 1 2

(23)

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

clarity for all119898 isin I we denoteI = I(119898)

KcupI(119898)

UCcupI(119898)

UKas

follows

I(119898)

K = 119899 120587119898119899

is known

I(119898)

UC = 119899 119898119899

is uncertain

I(119898)

UK = 119899 119898119899

is unknown

(4)

Moreover ifI(119898)K

= 0 andI(119898)

UC= 0 it is further described as

I(119898)

K = K1(119898)

K119905(119898)

forall1 le 119905(119898)

le 119873 minus 2

I(119898)

UC = U1(119898)

UV(119898)

forall1 le V(119898)

le 119873

(5)

where K119905(119898)

isin N+ represents the 119905(119898)

th known element withthe index K

119905(119898)

in the 119898th row of the TPM and UV(119898)

isin N+

represents the V(119898)

th uncertain element with the index UV(119898)

in the 119898th row of the TPM Obviously 1 le 119905(119898)

+ V(119898)

le 119873Also we denote

120587(119898119904)

UK = sum

119899isinI(119898)

UK

119898119899

= 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899 (6)

where (119904)

119898119899represents an uncertain TP in the 119904th polytope for

all 119904 = 1 119872The boundary conditions of system (Σ) in (1) are defined

by

119909 (119894 119895) = 120601 (119894 119895) forall119895 ge 0 minus1198891le 119894 le 0

119909 (119894 119895) = 120593 (119894 119895) forall119894 ge 0 minus1198892le 119895 le 0

120601 (0 0) = 120593 (0 0)

(7)

Throughout this paper the following assumption is made

Assumption 1 The boundary conditions are assumed to sat-isfy

lim1198791rarrinfin

E

1198791

sum

119895=0

0

sum

119894=minus1198891

(120601T(119894 119895) 120601 (119894 119895))

+ lim1198792rarrinfin

E

1198792

sum

119894=0

0

sum

119895=minus1198892

(120593T(119894 119895) 120593 (119894 119895))

lt infin

(8)

In this paper we are interested in the Hinfin

controllersynthesis forMJLSs (1)The followingmode-dependent state-feedback control law is used

119906 (119894 119895) = 119870 (119903 (119894 119895)) 119909 (119894 119895) (9)

where 119870(119903(119894 119895)) isin R119899119906times119899119909

Then the corresponding closed-loop system can be repre-sented as follows

(Σ) 119909 (119894 + 1 119895 + 1) = 1198601(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

+ 1198602(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

+ 1198601198891

(119903 (119894 119895 + 1)) 119909 (119894 minus 1198891 119895 + 1)

+ 1198601198892

(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895 minus 1198892)

+ 1198631(119903 (119894 119895 + 1))119908 (119894 119895 + 1)

+ 1198632(119903 (119894 + 1 119895)) 119908 (119894 + 1 119895)

119911 (119894 119895) = 119862 (119903 (119894 119895)) 119909 (119894 119895) + 1198633(119903 (119894 119895)) 119908 (119894 119895)

(10)

where

1198601(119903 (119894 119895 + 1))

= 1198601(119903 (119894 119895 + 1)) + 119861

1(119903 (119894 119895 + 1))119870 (119903 (119894 119895 + 1))

1198602(119903 (119894 + 1 119895))

= 1198602(119903 (119894 + 1 119895)) + 119861

2(119903 (119894 + 1 119895))119870 (119903 (119894 + 1 119895))

119862 (119903 (119894 119895)) = 119862 (119903 (119894 119895)) + 1198613(119903 (119894 119895))119870 (119903 (119894 119895))

(11)

Before proceeding further we introduce the followingdefinitions

Definition 2 System (10) is said to be stochastically stable iffor119908(119894 119895) = 0 and the boundary conditions satisfying (8) thefollowing condition holds

E

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119909 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119909 (119894 + 1 119895)

1003817100381710038171003817

2

)

lt infin (12)

Definition 3 Given a scalar 120574 gt 0 system (10) is said tobe stochastically stable with an H

infindisturbance attenuation

performance index 120574 if it is stochastically stable with119908(119894 119895) =

0 and under zero boundary conditions 120601(119894 119895) = 120593(119894 119895) = 0

in (7) for all nonzero 119908 isin 1198972[0infin) [0infin) satisfies

E2

lt 1205741199082 (13)

where

E2

= radicE

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119911 (119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119911 (119894 + 1 119895)

1003817100381710038171003817

2

)

1199082

= radic

infin

sum

119894=0

infin

sum

119895=0

(1003817100381710038171003817119908(119894 119895 + 1)

1003817100381710038171003817

2

+1003817100381710038171003817119908(119894 + 1 119895)

1003817100381710038171003817

2

)

(14)

4 Mathematical Problems in Engineering

Therefore the purpose of this paper is to design a mode-dependent H

infinstate-feedback controller in the form of (9)

such that the resulting closed-loop system (10) with defectivemode information is stochastically stable with a prescribedHinfin

performance index 120574

3 Main Results

In this section based on a Markovian Lyapunov-Krasovskiifunctional (MLKF) a new formulation of bounded reallemma (BRL) for the two-dimensional (2D)Markovian jumplinear system (MJLS) (10) with state-delays and defectivemode information will be firstly given Then via a lineari-sation procedure theH

infincontroller synthesis will be devel-

oped

31Hinfin

Performance Analysis In this subsection by invok-ing the properties of the transition probability matrices(TPMs) together with the convexification of uncertaindomains an H

infinperformance analysis criterion for the

closed-loop system (10) with state-delays and defective modeinformation is presented which will play a significant role insolving theH

infincontroller synthesis problem

Proposition 4 The 2D MJLS in (10) with state-delays anddefective mode information is stochastically stable with a guar-anteedH

infinperformance 120574 if the matrices 119875

1119898 1198752119898

1198761 1198762 isin

R119899119909times119899119909 with 1198751119898

gt 0 1198752119898

gt 0 1198761gt 0 and 119876

2gt 0119898 isin I

such that the following matrix inequalities hold

AT119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(15)

where

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899)

+ sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899) + 120587

(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(16)

Proof Consider the following MLKF for the 2D MJLS (10)

119881 (119894 119895) =

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

+

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(17)

where

1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1) 119875

1(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

=

119894minus1

sum

119896=119894minus1198891

119909T(119896 119895 + 1)119876

1119909 (119896 119895 + 1)

1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119875

2(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

=

119895minus1

sum

119896=119895minus1198892

119909T(119894 + 1 119896) 119876

2119909 (119894 + 1 119896)

(18)

Then based on the MLKF defined in (17) it is knownthat the following condition (19) guarantees that the 2Dclosed-loop system (10) is stochastically stable with an H

infin

performance 120574 under zero boundary conditions for anynonzero 119908(119894 119895) isin 119897

2[0infin) [0infin)

Ω = Δ119881 (119894 119895) + 2

E2

minus 1205742

1199082

2lt 0 (19)

where

Δ119881 (119894 119895) = E

2

sum

119896=1

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898

+ E

4

sum

119896=3

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898

Mathematical Problems in Engineering 5

minus

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

minus

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(20)

and E2

and 1199082are defined in (14)

Taking the time difference of119881(119894 119895) along the trajectoriesof the 2D system in (10) yields

Δ1198811= E [119881

1(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198751119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198751119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198751119899

+ sum

119899isinI(119898)

UK

119898119899

1198751119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

Δ1198812= E [119881

2(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1)119876

1119909 (119894 119895 + 1)

minus 119909T(119894 minus 1198891 119895 + 1)119876

1119909 (119894 minus 119889

1 119895 + 1)

Δ1198813= E [119881

3(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198752119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198752119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198752119899

+ sum

119899isinI(119898)

UK

119898119899

1198752119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

Δ1198814= E [119881

4(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119876

2119909 (119894 + 1 119895)

minus 119909T(119894 + 1 119895 minus 119889

2) 1198762119909 (119894 + 1 119895 minus 119889

2)

(21)

Therefore based on the MLKF defined in (17) together withconsideration of (10) and (21) we have

Ω = 120589T(119894 119895) [A

T119898

(P1119899

+ P2119899)A119898

+ CT119898C119898

+ Θ119898]

times 120589 (119894 119895) 119898 119899 isin I

(22)

where

120589 (119894 119895) = [119909T(119894 119895 + 1) 119909

T(119894 + 1 119895) 119909

T(119894 minus 1198891 119895 + 1)

119909T(119894 + 1 119895 minus 119889

2) 119908

T(119894 119895 + 1) 119908

T(119894 + 1 119895) ]

T

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P119897119899

= sum

119899isinI(119898)

K

120587119898119899

119875119897119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)119875119897119899

+ sum

119899isinI(119898)

UK

119898119899

119875119897119899 119897 = 1 2

(23)

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Therefore the purpose of this paper is to design a mode-dependent H

infinstate-feedback controller in the form of (9)

such that the resulting closed-loop system (10) with defectivemode information is stochastically stable with a prescribedHinfin

performance index 120574

3 Main Results

In this section based on a Markovian Lyapunov-Krasovskiifunctional (MLKF) a new formulation of bounded reallemma (BRL) for the two-dimensional (2D)Markovian jumplinear system (MJLS) (10) with state-delays and defectivemode information will be firstly given Then via a lineari-sation procedure theH

infincontroller synthesis will be devel-

oped

31Hinfin

Performance Analysis In this subsection by invok-ing the properties of the transition probability matrices(TPMs) together with the convexification of uncertaindomains an H

infinperformance analysis criterion for the

closed-loop system (10) with state-delays and defective modeinformation is presented which will play a significant role insolving theH

infincontroller synthesis problem

Proposition 4 The 2D MJLS in (10) with state-delays anddefective mode information is stochastically stable with a guar-anteedH

infinperformance 120574 if the matrices 119875

1119898 1198752119898

1198761 1198762 isin

R119899119909times119899119909 with 1198751119898

gt 0 1198752119898

gt 0 1198761gt 0 and 119876

2gt 0119898 isin I

such that the following matrix inequalities hold

AT119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(15)

where

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899)

+ sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899) + 120587

(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(16)

Proof Consider the following MLKF for the 2D MJLS (10)

119881 (119894 119895) =

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

+

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(17)

where

1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1) 119875

1(119903 (119894 119895 + 1)) 119909 (119894 119895 + 1)

1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

=

119894minus1

sum

119896=119894minus1198891

119909T(119896 119895 + 1)119876

1119909 (119896 119895 + 1)

1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119875

2(119903 (119894 + 1 119895)) 119909 (119894 + 1 119895)

1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

=

119895minus1

sum

119896=119895minus1198892

119909T(119894 + 1 119896) 119876

2119909 (119894 + 1 119896)

(18)

Then based on the MLKF defined in (17) it is knownthat the following condition (19) guarantees that the 2Dclosed-loop system (10) is stochastically stable with an H

infin

performance 120574 under zero boundary conditions for anynonzero 119908(119894 119895) isin 119897

2[0infin) [0infin)

Ω = Δ119881 (119894 119895) + 2

E2

minus 1205742

1199082

2lt 0 (19)

where

Δ119881 (119894 119895) = E

2

sum

119896=1

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898

+ E

4

sum

119896=3

119881119896(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898

Mathematical Problems in Engineering 5

minus

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

minus

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(20)

and E2

and 1199082are defined in (14)

Taking the time difference of119881(119894 119895) along the trajectoriesof the 2D system in (10) yields

Δ1198811= E [119881

1(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198751119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198751119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198751119899

+ sum

119899isinI(119898)

UK

119898119899

1198751119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

Δ1198812= E [119881

2(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1)119876

1119909 (119894 119895 + 1)

minus 119909T(119894 minus 1198891 119895 + 1)119876

1119909 (119894 minus 119889

1 119895 + 1)

Δ1198813= E [119881

3(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198752119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198752119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198752119899

+ sum

119899isinI(119898)

UK

119898119899

1198752119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

Δ1198814= E [119881

4(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119876

2119909 (119894 + 1 119895)

minus 119909T(119894 + 1 119895 minus 119889

2) 1198762119909 (119894 + 1 119895 minus 119889

2)

(21)

Therefore based on the MLKF defined in (17) together withconsideration of (10) and (21) we have

Ω = 120589T(119894 119895) [A

T119898

(P1119899

+ P2119899)A119898

+ CT119898C119898

+ Θ119898]

times 120589 (119894 119895) 119898 119899 isin I

(22)

where

120589 (119894 119895) = [119909T(119894 119895 + 1) 119909

T(119894 + 1 119895) 119909

T(119894 minus 1198891 119895 + 1)

119909T(119894 + 1 119895 minus 119889

2) 119908

T(119894 119895 + 1) 119908

T(119894 + 1 119895) ]

T

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P119897119899

= sum

119899isinI(119898)

K

120587119898119899

119875119897119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)119875119897119899

+ sum

119899isinI(119898)

UK

119898119899

119875119897119899 119897 = 1 2

(23)

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

minus

2

sum

119896=1

119881119896(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

minus

4

sum

119896=3

119881119896(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

(20)

and E2

and 1199082are defined in (14)

Taking the time difference of119881(119894 119895) along the trajectoriesof the 2D system in (10) yields

Δ1198811= E [119881

1(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198811(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198751119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198751119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198751119899

+ sum

119899isinI(119898)

UK

119898119899

1198751119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 119895 + 1) 119875

1119898119909 (119894 119895 + 1)

Δ1198812= E [119881

2(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 119895 + 1) 119903 (119894 119895 + 1) = 119898]

minus 1198812(119909 (119894 119895 + 1) 119903 (119894 119895 + 1))

= 119909T(119894 119895 + 1)119876

1119909 (119894 119895 + 1)

minus 119909T(119894 minus 1198891 119895 + 1)119876

1119909 (119894 minus 119889

1 119895 + 1)

Δ1198813= E [119881

3(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198813(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895 + 1)( sum

119899isinI

120587119898119899

1198752119899)119909 (119894 + 1 119895 + 1)

minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

= 119909T(119894 + 1 119895 + 1)

times ( sum

119899isinI(119898)

K

120587119898119899

1198752119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)1198752119899

+ sum

119899isinI(119898)

UK

119898119899

1198752119899)

times 119909 (119894 + 1 119895 + 1) minus 119909T(119894 + 1 119895) 119875

2119898119909 (119894 + 1 119895)

Δ1198814= E [119881

4(119909 (119894 + 1 119895 + 1) 119903 (119894 + 1 119895 + 1)) |

119909 (119894 + 1 119895) 119903 (119894 + 1 119895) = 119898]

minus 1198814(119909 (119894 + 1 119895) 119903 (119894 + 1 119895))

= 119909T(119894 + 1 119895) 119876

2119909 (119894 + 1 119895)

minus 119909T(119894 + 1 119895 minus 119889

2) 1198762119909 (119894 + 1 119895 minus 119889

2)

(21)

Therefore based on the MLKF defined in (17) together withconsideration of (10) and (21) we have

Ω = 120589T(119894 119895) [A

T119898

(P1119899

+ P2119899)A119898

+ CT119898C119898

+ Θ119898]

times 120589 (119894 119895) 119898 119899 isin I

(22)

where

120589 (119894 119895) = [119909T(119894 119895 + 1) 119909

T(119894 + 1 119895) 119909

T(119894 minus 1198891 119895 + 1)

119909T(119894 + 1 119895 minus 119889

2) 119908

T(119894 119895 + 1) 119908

T(119894 + 1 119895) ]

T

Θ119898

= diag minus1198751119898

+ 1198761 minus1198752119898

+ 1198762 minus1198761 minus1198762 minus1205742I minus120574

2I

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

P119897119899

= sum

119899isinI(119898)

K

120587119898119899

119875119897119899

+ sum

119899isinI(119898)

UC

(

119872

sum

119904=1

120572119904(119904)

119898119899)119875119897119899

+ sum

119899isinI(119898)

UK

119898119899

119875119897119899 119897 = 1 2

(23)

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Considering the fact that 0 le 120572119904le 1 sum119872

119904=1120572119904= 1 and 0 le

119898119899

120587(119898119904)

UKle 1sum

119899isinI(119898)

UK

(119898119899

120587(119898119904)

UK) = 1 (22) can be rewritten

as

Ω =

119872

sum

119904=1

120572119904

sum

119899isinI(119898)

UK

119898119899

120587(119898119904)

UK

times [120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895)]

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(24)

where

P(119904)

119899= sum

119899isinI(119898)

K

120587119898119899

(1198751119899

+ 1198752119899) + sum

119899isinI(119898)

UC

(119904)

119898119899(1198751119899

+ 1198752119899)

+ 120587(119898119904)

UK (1198751119899

+ 1198752119899)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899isinI(119898)

UK

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(25)According to (24) it is easy to see that (19) holds if and

only if for all 119904 = 1 119872

120589T(119894 119895) [A

T119898P(119904)

119899A119898

+ CT119898C119898

+ Θ119898] 120589 (119894 119895) lt 0

119898 isin I 119899 isin I(119898)

UK

(26)

which is implied by condition (15) This completes the proof

Remark 5 By fully considering the properties of TPMstogether with the convexification of uncertain domains anew version of BRL has been derived for the 2D MJLS(10) with state-delays and defective mode information inProposition 4 It is worth mentioning that the results forFornasini-Marchesini local state-space (FMLSS) model (10)could be readily applied to Roesser model after the similartransformation as performed in [8] It is also noted that thecondition given in (15) is nonconvex due to the presence ofproduct terms between the Lyapunov matrices and systemmatrices For the matrix inequality linearisation purposein the following we shall make a decoupling between theLyapunov matrices and system matrices which will be con-venient for controller synthesis purpose

In the sequel we focus on theHinfincontroller design based

on the analysis condition given in Proposition 4

32 Hinfin

Controller Synthesis In this subsection based on alinearisation procedure a unified framework for the solvabil-ity of theH

infincontroller synthesis problem will be proposed

It will be shown that the parametrised representations of

the controller gains can be constructed in terms of the feasiblesolutions to a set of strict linear matrix inequalities (LMIs)

Theorem 6 Consider 2D MJLS (1) with state-delays anddefective mode information and the state-feedback controllerin the form of (9) The closed-loop system (10) is stochasticallystable with an H

infinperformance 120574 if there exist matrices

1198831119898

1198832119898

1198771 1198772 isin R119899119909times119899119909 with 119883

1119898gt 0 119883

2119898gt 0 119877

1gt 0

and 1198772

gt 0 and matrices 119866119898

isin R119899119909times119899119909 and 119870119898

isin R119899119906times119899119909 119898 isin I such that the following LMIs hold

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(27)

where

X(119898)

119897119899

= diag 119883119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899

119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119866119898

+ 119866T119898

minus 1198831119898

119866119898

+ 119866T119898

minus 1198832119898

R 1205742I 1205742I

R = diag 1198771 1198772

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898

C119898

= 119862119898

+ 1198613119898

119870119898

120587(119898119904)

UK = 1 minus sum

119899isinI(119898)

K

120587119898119899

minus sum

119899isinI(119898)

UC

(119904)

119898119899

(28)

Moreover if the above conditions have a set of feasiblesolutions (119883

1119898 1198832119898

1198771 1198772 119866119898 119870119898) then the state-feedback

controller in the form of (9) can be constructed as

119870119898

= 119870119898119866minus1

119898 (29)

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Proof It follows from Proposition 4 that if we can show (15)then the claimed results follow By Schur complement andwith I

(119898)

K= K

1(119898)

K119905(119898)

and I(119898)

UC= U

1(119898)

UV(119898)

(15) is equivalent to

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

Λ

lowast lowast lowast lowast minusQminus1

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(30)

where

X(119898)

119897119899= diag 119883

119897K1(119898)

119883119897K119905(119898)

119883119897U1(119898)

119883119897UV(119898)

119883119897119899 119883

119897119898= 119875minus1

119897119898 119897 = 1 2

ϝ(119904)

119898= [

radic120587119898K1(119898)

I radic

120587119898K119905(119898)

I radic(119904)

119898U1(119898)

I

radic(119904)

119898UV(119898)

I radic120587(119898119904)

UKI]

A119898

= [1198601119898

1198602119898

1198601198891119898

1198601198892119898

1198631119898

1198632119898

]

C119898

= [

119862119898

0 0119899119911times2119899119909

1198633119898

00 119862119898

0119899119911times2119899119909

0 1198633119898

]

Θ119898

= diag 119883minus1

1119898 119883minus1

2119898Q 1205742I 1205742I Q = diag 119876

1 1198762

Λ = [

I119899119909

0 0119899119909times2(119899119909+119899119908)

0 I119899119909

0119899119909times2(119899119909+119899119908)

]

T

(31)

Now by introducing a nonsingular matrix 119866119898

isin R119899119909times119899119909and pre- and postmultiplying (30) by diagI

2((119905(119898)+V(119898)+1)119899119909+119899119911)

119866T119898 119866

T119898 119876minus1

1 119876minus1

2 I2(119899119908+119899119909) and its transpose the following

yields

[

[

[

[

[

[

[

minusX(119898)

11198990 0 ϝ

(119904)

119898A119898

0lowast minusX

(119898)

21198990 ϝ(119904)

119898A119898

0lowast lowast minusI C

1198980

lowast lowast lowast minusΘ119898

G119898

lowast lowast lowast lowast minusR

]

]

]

]

]

]

]

lt 0

119898 isin I 119899 isin I(119898)

UK 119904 = 1 119872

(32)

where

A119898

= [A1119898

A2119898

1198601198891119898

1198771

1198601198892119898

1198772

1198631119898

1198632119898

]

C119898

= [

C119898

0 0119899119911times2119899119909

1198633119898

00 C

1198980119899119911times2119899119909

0 1198633119898

]

G119898

= [

119866119898

0 0119899119909times2(119899119909+119899119908)

0 119866119898

0119899119909times2(119899119909+119899119908)

]

T

Θ119898

= diag 119866T119898119883minus1

1119898119866119898 119866

T119898119883minus1

2119898119866119898R 1205742I 1205742I

R = diag 1198771 1198772

A119897119898

= 119860119897119898

+ 119861119897119898

119870119898 119870119898

= 119870119898119866119898

119877119897

= 119876minus1

119897 119897 = 1 2

C119898

= 119862119898

+ 1198613119898

119870119898

(33)

andX(119898)

1119899X(119898)2119899

and ϝ(119904)

119898are defined in (31)

It follows from

(119883119897119898

minus 119866119898)T119883minus1

119897119898(119883119897119898

minus 119866119898) ge 0 119897 = 1 2 (34)

that

minus119866T119898119883minus1

119897119898119866119898

le minus119866119898

minus 119866T119898

+ 119883119897119898

119897 = 1 2 (35)

Then it is easy to see that (27) implies (32)On the other hand the condition in (27) implies that

minus119866119898

minus 119866T119898

lt 0 which means that 119866119898is nonsingular Thus

the controller gain can be constructed by (29) The proof isthus completed

Remark 7 Theorem 6 provides a sufficient condition on thefeasibility ofH

infinstate-feedback controller synthesis problem

for the 2D MJLSs with state-delays and defective modeinformation It is noted that theH

infinstate-feedback controller

synthesis problem for 2D discrete-time MJLSs has also beenconsidered in [40] However there still are some remarkabledifferences between our results and those in [40] Firstly inthis paper the state-delays were introduced in the system(1) whereas the 2D delay-free MJLSs are considered in [40]In addition in Theorem 6 the exactly known partiallyunknown and uncertain transition probabilities (TPs) havebeen simultaneously incorporated into the TPM for 2DMJLSs while in [40] the TPs were assumed to be completelyknown It has been recognized that the scenario containingtime-delays and such defective TPs is more general and theunderlying MJLSs are thereby more practicable for engineer-ing applications

4 An Illustrative Example

In this section we use a simulation example to demonstratethe effectiveness of the proposed state-feedback controllerdesign method to two-dimensional (2D) Markovian jumplinear systems (MJLSs)

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Consider a 2D MJLS with state-delays in the form of (1)with parameters as follows

[

[

11986011

11986011988911

11986111

11986311

11986021

11986011988921

11986121

11986321

1198621

11986131

11986331

]

]

=

[

[

[

[

[

[

[

[

minus05 0 0 005 minus05 02

0 05 minus002 0 05 06

05 1 004 0 05 05

0 1 0 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986012

11986011988912

11986112

11986312

11986022

11986011988922

11986122

11986322

1198622

11986132

11986332

]

]

=

[

[

[

[

[

[

[

[

06 0 0 005 05 02

03 05 minus002 004 05 06

05 05 004 0 05 05

0 08 004 002 1 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986013

11986011988913

11986113

11986313

11986023

11986011988923

11986123

11986323

1198623

11986133

11986333

]

]

=

[

[

[

[

[

[

[

[

05 0 0 005 05 02

03 03 minus002 01 05 06

02 05 004 0 05 05

0 04 01 002 02 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

[

[

11986014

11986011988914

11986114

11986314

11986024

11986011988924

11986124

11986324

1198624

11986134

11986334

]

]

=

[

[

[

[

[

[

[

[

04 0 0 005 03 02

05 06 minus002 01 05 06

06 05 004 0 05 05

0 05 01 002 04 03

02 05 05 03

05 06 02 minus05

]

]

]

]

]

]

]

]

(36)

Four different cases for the transition probability matrix(TPM) are given in Table 1 where the transition probabilities(TPs) labeled with ldquordquo and ldquosimrdquo represent the unknown anduncertain elements respectively Specifically Case 1 Case 2Case 3 andCase 4 stand for the completely knownTPs defec-tivemode information (including known partially unknownand uncertain TPs) partially unknown TPs and completelyunknown TPs respectively

Table 1 Four different TPMs

Case 1 completely known TPM Case 2 defective TPM1

[

[

[

[

[

03 02 01 04

03 02 03 02

01 05 03 01

02 02 01 05

]

]

]

]

]

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

Case 3 defective TPM2 Case 4 completely unknown TPM

[

[

[

[

[

03 02 01 04

21

22

03 02

31

32

33

34

02 42

43

44

]

]

]

]

]

[

[

[

[

[

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

]

]

]

]

]

Table 2 Comparison of minimum Hinfin

performance for differentTPMs

TPMs Case 1 Case 2 Case 3 Case 4

120574min 09884 10415 10906 22398

For Case 2 it is assumed that the uncertain TPs comprisefour vertices Π

119904 119904 = 1 2 3 4 where the third rows Π

119904(3) 119904 =

1 2 3 4 are given by

Π1(3)

= [31

02 33

04]

Π2(3)

= [31

05 33

03]

Π3(3)

= [31

03 33

01]

Π4(3)

= [31

01 33

045]

(37)

and the other rows in the four vertices are givenwith the sameelements that is

Π119904(1)

= [03 02 01 04]

Π119904(2)

= [21

22

03 02]

Π119904(4)

= [02 42

43

44] 119904 = 1 2 3 4

(38)

The objective is to design a state-feedback controller ofthe form (9) for the above system such that the 2D closed-loop system is stochastically stable with anH

infinperformance

120574 By applying Theorem 6 a detailed comparison of theobtained minimum H

infinperformance indices 120574min by the

state-feedback controller (9) with four TPM cases beingshown in Table 2 It is shown in Tables 1 and 2 that the lowerthe level of defectiveness of the TPM is the better the H

infin

performance can be obtained which is effective to reducethe design conservatism Therefore the introduction of theuncertain TPs is meaningful

Specifically in the following considering the four TPMcases shown inTable 1 and by applyingTheorem6 the feasible

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

05

1015

20

05

1015

20

0

1

2

3

4

i =1 2

j=1 2

Mod

es tr

ansit

ions

r(ij)

Figure 1 One possible system mode evolution

solution of 120574min = 10415 for the state-feedback controller isobtained under Case 2 with controller gains given by

1198701= [minus02467 minus07888]

1198702= [minus02125 minus07064]

1198703= [minus04563 minus05998]

1198704= [minus02015 minus07041]

(39)

The feasible solutions for the other three TPM cases in Table 1are omitted for brevity

In order to further illustrate the effectiveness of thedesigned H

infinstate-feedback controllers we present some

simulation results Let the boundary conditions be

119909 (119905 119894) = 119909 (119894 119905) =

[minus1 14]

T 0 le 119894 le 10

[0 0]

T 119894 gt 10

(40)

where minus4 le 119905 le 0 and choose the delays 1198891

= 4 (verticaldirection) 119889

2= 4 (horizontal direction) and disturbance

input 119908(119894 119895) as

119908 (119894 119895) =

02 0 le 119894 119895 le 10

0 otherwise(41)

With the previous obtained controllers under Case 2 inTable 1 one possible realization of the Markovian jumpingmode is plotted in Figure 1 The state responses for the open-loop and closed-loop systems are shown in Figures 2 and 3and Figures 4 and 5 respectively and Figures 6 and 7 are thecontrolled output trajectories of the closed-loop system It canbe clearly observed from the simulation curves that despitethe defective TPs the performance of the designed controlleris satisfactory

5 Conclusions

This paper has addressed the problem of Hinfin

control fora class of two-dimensional (2D) Markovian jump linear

05

1015

20

05

1015

20

minus05

0

05

1

15

2

25

Stat

e

times105

i = 1 2

j = 1 2

Figure 2 State responses of the open-loop system the 1st compo-nent

05

1015

20

05

1015

20

0

05

1

15

2

25

Stat

etimes10

5

i = 1 2

j = 1 2

Figure 3 State responses of the open-loop system the 2nd compo-nent

05

1015

20

05

1015

20

minus1

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 4 State responses of the closed-loop system the 1st compo-nent

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

05

1015

20

05

1015

20

minus05

0

05

1

15

Stat

e

i = 1 2

j =1 2

Figure 5 State responses of the closed-loop system the 2nd com-ponent

05

1015

20

05

1015

20

minus01

0

01

02

03

04

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 6 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 1st component

05

1015

20

05

1015

20

minus04

minus020

02

04

06

08

1

Con

trolle

d ou

tput

i = 1 2

j =1 2

Figure 7 Responses of the controlled output 119911(119894 119895) for the closed-loop system the 2nd component

systems (MJLSs) with state-delays and defective mode infor-mation Such defective mode information simultaneouslyincludes the exactly known partially unknown and uncer-tain transition probabilities which contributes to the prac-ticability of 2D MJLSs By fully considering the propertiesof the transition probability matrices together with theconvexification of uncertain domains an H

infinperformance

analysis criterion for the 2DFornasini-Marchesini local state-space model has been firstly developed and then via a lin-earisation procedure a unified framework has been proposedfor the state-feedback controller synthesis that guarantees thestochastic stability of the closed-loop system with an H

infin

disturbance attenuation level A simulation example has beengiven to illustrate the effectiveness of the proposed method

Acknowledgments

The authors are grateful to the Guest Editors and anonymousreviewers for their constructive comments based on whichthe presentation of this paper has been greatly improvedThiswork was supported in part by the National Natural ScienceFoundation of China (61004038 61374031) in part by theProgram for New Century Excellent Talents in University(NCET-12-0147) in part by the Fundamental Research Fundsfor the Central Universities (HITBRETIII201214) and inpart by the Alexander von Humboldt Foundation of Ger-many

References

[1] R P Roesser ldquoA discrete state-space model for linear imageprocessingrdquo IEEE Transactions on Automatic Control vol 20pp 1ndash10 1975

[2] T Kaczorek Two-Dimensional Linear Systems vol 68 SpringerBerlin Germany 1985

[3] W S Lu and A Antoniou Two-Dimensional Digital FitersMarcel Dekker New York NY USA 1992

[4] S Xu J Lam Z Lin and K Galkowski ldquoPositive real controlfor uncertain two-dimensional systemsrdquo IEEE Transactions onCircuits and Systems I vol 49 no 11 pp 1659ndash1666 2002

[5] S Xu J Lam Z Lin et al ldquoPositive real control of two-dimensional systems Roesser models and linear repetitiveprocessesrdquo International Journal of Control vol 76 no 11 pp1047ndash1058 2003

[6] L Xie C Du Y C Soh and C Zhang ldquoHinfinand robust control

of 2-D systems in FM secondmodelrdquoMultidimensional Systemsand Signal Processing vol 13 no 3 pp 265ndash287 2002

[7] J Lam S Xu Y Zou Z Lin and K Galkowski ldquoRobust outputfeedback stabilization for two-dimensional continuous systemsin Roesser formrdquoAppliedMathematics Letters vol 17 no 12 pp1331ndash1341 2004

[8] W Paszke J Lam K Gałkowski S Xu and Z Lin ldquoRobuststability and stabilisation of 2D discrete state-delayed systemsrdquoSystems amp Control Letters vol 51 no 3-4 pp 277ndash291 2004

[9] N T Hoang H D Tuan T Q Nguyen and S HosoeldquoRobust mixed generalized H

2Hinfin

filtering of 2-D nonlinearfractional transformation systemsrdquo IEEE Transactions on SignalProcessing vol 53 no 12 pp 4697ndash4706 2005

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[10] D Peng and X Guan ldquoHinfin

filtering of 2-D discrete state-delayed systemsrdquoMultidimensional Systems and Signal Process-ing vol 20 no 3 pp 265ndash284 2009

[11] X Li and H Gao ldquoRobust finite frequency Hinfin

filtering foruncertain 2-D Roesser systemsrdquo Automatica vol 48 no 6 pp1163ndash1170 2012

[12] S Xu J Lam and X Mao ldquoDelay-dependent Hinfin

controland filtering for uncertain Markovian jump systems with time-varying delaysrdquo IEEETransactions onCircuits and Systems I vol54 no 9 pp 2070ndash2077 2007

[13] H Liu DWCHo and F Sun ldquoDesign ofHinfinfilter forMarkov

jumping linear systemswith non-accessiblemode informationrdquoAutomatica vol 44 no 10 pp 2655ndash2660 2008

[14] M Liu D W C Ho and Y Niu ldquoStabilization of Markovianjump linear systemover networkswith randomcommunicationdelayrdquo Automatica vol 45 no 2 pp 416ndash421 2009

[15] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[16] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems I vol 51 no5 pp 262ndash268 2004

[17] Z Wang J Lam and X Liu ldquoRobust filtering for discrete-timeMarkovian jump delay systemsrdquo IEEE Signal Processing Lettersvol 11 no 8 pp 659ndash662 2004

[18] Z Wang Y Liu and X Liu ldquoExponential stabilization of aclass of stochastic system with Markovian jump parametersandmode-dependentmixed time-delaysrdquo IEEETransactions onAutomatic Control vol 55 no 7 pp 1656ndash1662 2010

[19] H Dong ZWang and H Gao ldquoDistributed filtering for a classof time-varying systems over sensor networkswith quantizationerrors and successive packet dropoutsrdquo IEEE Transactions onSignal Processing vol 60 no 6 pp 3164ndash3173 2012

[20] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time-delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[21] Z Wang B Shen H Shu and G Wei ldquoQuantizedHinfincontrol

for nonlinear stochastic time-delay systems with missing mea-surementsrdquo IEEE Transactions on Automatic Control vol 57 no6 pp 1431ndash1444 2012

[22] B Shen Z Wang and Y S Hung ldquoDistributedHinfin-consensus

filtering in sensor networks with multiple missing measure-ments the finite-horizon caserdquo Automatica vol 46 no 10 pp1682ndash1688 2010

[23] J Qiu G Feng and J Yang ldquoImproved delay-dependent Hinfin

filtering design for discrete-time polytopic linear delay systemsrdquoIEEE Transactions on Circuits and Systems II vol 55 no 2 pp178ndash182 2008

[24] B Shen Z Wang Y S Hung and G Chesi ldquoDistributed Hinfin

filtering for polynomial nonlinear stochastic systems in sensornetworksrdquo IEEE Transactions on Industrial Electronics vol 58no 5 pp 1971ndash1979 2011

[25] G He J-A Fang and X Wu ldquoRobust stability of Markovianjumping genetic regulatory networks with mode-dependentdelaysrdquoMathematical Problems in Engineering vol 2012 ArticleID 504378 18 pages 2012

[26] Y Wei M Wang H R Karimi N Wang and J Qiu ldquoHinfin

model reduction for discrete-time Markovian jump systems

with deficient mode informationrdquo Mathematical Problems inEngineerIng vol 2013 Article ID 537174 11 pages 2013

[27] J Qiu G Feng and J Yang ldquoA new design of delay-dependentrobust H

infinfiltering for discretetime T-S fuzzy systems with

time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol17 no 5 pp 1044ndash1058 2009

[28] J Qiu G Feng and J Yang ldquoDelay-dependent non-synchronized robust H

infinstate estimation for discrete-time

piecewise linear delay systemsrdquo International Journal of Adapt-ive Control and Signal Processing vol 23 no 2 pp 1082ndash10962009

[29] J Qiu G Feng and J Yang ldquoNew results on robust Hinfin

fil-tering design for discrete-time piecewise linear delay systemsrdquoInternational Journal of Control vol 82 no 1 pp 183ndash194 2009

[30] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Boston Mass USA 2006

[31] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[32] C E de Souza ldquoRobust stability and stabilization of uncertaindiscrete-time Markovian jump linear systemsrdquo IEEE Transac-tions on Automatic Control vol 51 no 5 pp 836ndash841 2006

[33] Y Zhao L Zhang S Shen and H Gao ldquoRobust stabilitycriterion for discrete-time uncertainMarkovian jumping neuralnetworks with defective statistics of modes transitionsrdquo IEEETransactions on Neural Network vol 22 no 1 pp 164ndash170 2011

[34] H Dong Z Wang D W C Ho and H Gao ldquoRobust Hinfin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[35] H Dong Z Wang and H Gao ldquoFault detection for Markovianjump systems with sensor saturations and randomly varyingnonlinearitiesrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2354ndash2362 2012

[36] M Karan P Shi and C Y Kaya ldquoTransition probabilitybounds for the stochastic stability robustness of continuous-and discrete-timeMarkovian jump linear systemsrdquoAutomaticavol 42 no 12 pp 2159ndash2168 2006

[37] Y Wei M Wang and J Qiu ldquoA new approach to delay-dependent H

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory and Applications vol 7 no5 pp 684ndash696 2013

[38] YWei J Qiu H R Karimi andMWang ldquoA new design ofHinfin

filtering for continuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 no 9 pp 2392ndash2407 2013

[39] L Wu P Shi H Gao and C Wang ldquoHinfin

filtering for 2DMarkovian jump systemsrdquo Automatica vol 44 no 7 pp 1849ndash1858 2008

[40] H Gao J Lam S Xu and C Wang ldquoStabilization and Hinfin

control of two-dimensional Markovian jump systemsrdquo IMAJournal of Mathematical Control and Information vol 21 no 4pp 377ndash392 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of