Research ArticleRobust Guaranteed Cost Observer Design forSingular Markovian Jump Time-Delay Systems withGenerally Incomplete Transition Probability

Yanbo Li12 Yonggui Kao2 and Jing Xie3

1 School of Mathematical Sciences Guangxi Teachers Education University Guangxi 530001 China2Department of Mathematics Harbin Institute of Technology Weihai 264209 China3 College of Information Science and Engineering Ocean University of China Qingdao 266071 China

Correspondence should be addressed to Yonggui Kao kaoyongguisinacom

Received 29 November 2013 Revised 6 February 2014 Accepted 10 February 2014 Published 24 March 2014

Academic Editor Shen Yin

Copyright copy 2014 Yanbo Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to the investigation of the design of robust guaranteed cost observer for a class of linear singular Markovianjump time-delay systems with generally incomplete transition probability In this singular model each transition rate can becompletely unknown or only its estimate value is known Based on stability theory of stochastic differential equations and linearmatrix inequality (LMI) technique we design an observer to ensure that for all uncertainties the resulting augmented system isregular impulse free and robust stochastically stablewith the proposed guaranteed cost performance Finally a convex optimizationproblem with LMI constraints is formulated to design the suboptimal guaranteed cost filters for linear singular Markovian jumptime-delay systems with generally incomplete transition probability

1 Introduction

Descriptor systems are also referred to as singular systemsimplicit systems generalized state-space systems or semis-tate systems and provide convenient and natural representa-tions in the description of economic systems power systemsrobotics network theory and circuits systems [1] The sta-bility for singular system is more complicated than that fornonsingular systems because not only the asymptotic stabilitybut also the system regularity and impulse elimination areneeded to be addressed [2ndash5]

In practice in many physical systems such as aircraftcontrol solar receiver control power systems manufacturingsystems networked control systems air intake systems andother practical systems abrupt variations may happen intheir structure due to random failures repair of componentssudden environmental disturbances changing subsysteminterconnections or abrupt variations in the operating pointsof a nonlinear plant [6ndash19] Therefore more and moreattention has been paid to the problem of stochastic stabilityand stochastic admissibility for singular Markovian jump

systems (SMJSs) [20ndash30] Long et al [23] derived stochasticadmissibility for a class of singular Markovian jump systemswith mode-dependent time delays Wang and Zhang [27]focused on the asynchronous 119897

2minus119897infinfiltering for discrete-time

stochastic Markov jump systems with randomly occurringsensor nonlinearities However the TRs in the above men-tioned literatures are assumed to be completely known

In practice the TRs in some jumping processes aredifficult to be precisely estimated due to the cost and someother factors Therefore analysis and synthesis problemsfor normal MJSs with incomplete information on transitionprobability have attracted more and more attentions [31ndash49]Xiong and Lam [32] probed robust119867

2control of Markovian

jump systems with uncertain switching probabilities Karanet al [33] considered the stochastic stability robustness forcontinuous-time and discrete-time Markovian jump linearsystems (MJLSs) with upper bounded TRs Zhang andBoukas [34] discussed stability and stabilization for thecontinuous-time MJSs with partly unknown TRs Lin et al[38] considered delay-dependent 119867

infinfiltering for discrete-

time singular Markovian jump systems with time-varying

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 832891 11 pageshttpdxdoiorg1011552014832891

2 Abstract and Applied Analysis

delay and partially unknown transition probabilities Guoand Wang [49] proposed another description for the uncer-tain TRs which is called generally uncertain TRs (GUTRs)

On the other hand state estimation plays an importantrole in systems and control theory signal processing andinformation fusion [50 51] Certainly the most widely usedestimation method is the well-known Kalman filtering [5253] A common feature in the Kalman filtering is that anaccurate model is available In some applications howeverwhen the system is subject to parameter uncertainties theaccurate system model is hard to obtain To overcome thisdifficulty the guaranteed cost filtering approach has beenproposed to ensure the upper bound of guaranteed cost func-tion [54] Robust119867

infinfiltering for uncertainMarkovian jump

systems with mode-dependent time delays was proposed in[55] In [56] guaranteed cost and 119867

infinfiltering for time-

delay systems were presented in terms of LMIs Howeverto the best of our knowledge there are few consideringthe robust guaranteed cost observer for a class of linearsingular Markovian jump time-delay systems with generallyincomplete transition probability which is still an openproblem

In this paper based on LMI method we address thedesign problem of the robust guaranteed cost observer for aclass of uncertain descriptor time-delay systems withMarko-vian jumping parameters and generally uncertain transitionrates The design problem proposed here is to design a mem-oryless observer such that for all uncertainties includinggenerally uncertain transition rates the resulting augmentedsystem is regular impulse-free and robust stochastically sta-ble and satisfies the proposed guaranteed cost performance

2 Problem Formulation

Consider the following descriptor time-delay systems withMarkovian jumping parameters

119864 (119905) = 119860 (119903119905 119905) 119909 (119905) + 119860

119889(119903119905 119905) 119909 (119905 minus 119889)

119910 (119905) = 119862 (119903119905 119905) 119909 (119905) + 119862

119889(119903119905 119905) 119909 (119905 minus 119889)

119909 (119905) = 120593 (119905) forall119905 isin [minus119889 0]

(1)

where 119909(119905) isin 119877119899 and 119910(119905) isin 119877

119903 are the state vector andthe controlled output respectively 119889 represents the state timedelay For convenience the input terms in system (1) havebeen omitted 120593(119905) isin 119871

2[minus119889 0] is a continuous vector-valued

initial function The random parameter 120574(119905) represents acontinuous-time discrete-state Markov process taking valuesin a finite set S = 1 2 119904 and having the transitionprobability matrix Π = [120587

119894119895] 119894 119895 isin 119873 The transition

probability from mode 119894 to mode 119895 is defined by

Pr 119903119905+Δ

= 119895 | 119903119905= 119894 =

120587119894119895Δ + 119900 (Δ) 119894 = 119895

1 + 120587119894119895Δ + 119900 (Δ) 119894 = 119895

(2)

where Δ gt 0 satisfies limΔrarr0

(119900(Δ)Δ) = 0 120587119894119895ge 0 is the

transition probability from mode 119894 to mode 119895 and satisfies

120587119894119894= minus

119904

sum

119895=1119895 = 119894

120587119894119895le 0 (3)

In this paper the transition rates of the jumping processare assumed to be partly available that is some elements inmatrix Λ have been exactly known some have been merelyknown with lower and upper bounds and others may haveno information to use For instance for system (1) withfour operation modes the transition rate matrix might bedescribed by

Λ =

[[[[

[

11+ Δ11

sdot sdot sdot

23+ Δ23

sdot sdot sdot 2119904+ Δ2119904

d

1199042+ Δ1199042

sdot sdot sdot

]]]]

]

(4)

where 119894119895and Δ

119894119895isin [minus120590

119894119895 120590119894119895] (120590119894119895

ge 0) represent theestimate value and estimate error of the uncertain TR 120587

119894119895

respectively where 119894119895and 120590

119894119895are known represents the

complete unknown TR which means that its estimate value119894119895and estimate error bound are unknownFor notational clarity for all 119894 isin S the set 119880

119894

denotes 119880119894

= 119880119894

119896cup 119880119894

119906119896with 119880

119894

119896= 119895 The estimate

value of 120587119894119895is known for 119895 isin S 119880119894

119906119896= 119895 The estimate

value of 120587119894119895is unknown for 119895 isin S Moreover if 119880119894

119896= 0 it is

further described as 119880119894119896= 119896119894

1 119896119894

2 119896

119894

119898 where 119896119894

119898isin N+

represents the 119898th bound-known element with the index119896119894

119898in the 119894th row of matrix Π We assume that the known

estimate values of the TRs are well defined That is

Assumption 1 If 119880119894119896

= S then 119894119895minus 120590119894119895

ge 0 (for all 119895 isin

S 119895 = 119894) 119894119894= minussum

119873

119895=1119895 = 119894119894119895and 120590

119894119894= minussum

119904

119895=1119895 = 119894120590119894119895

Assumption 2 If 119880119894

119896=S and 119894 isin 119880

119894

119896 then

119894119895minus 120590119894119895

ge

0 (for all 119895 isin S 119895 = 119894) 119894119894+ 120590119894119894le 0 and sum

119895isin119880119894

119896

119894119895

Assumption 3 If 119880119894

119896=S and 119894 notin 119880

119894

119896 then

119894119895minus 120590119894119895

ge

0 (for all 119895 isin S)

Remark 4 The above assumption is reasonable since it isthe direct result from the properties of the TRs (eg 120587

119894119895ge

0 (for all 119894 119895 isin S 119895 = 119894) and 120587119894119894= minussum

119904

119895=1119895 = 119894120587119894119895) The above

description about uncertain TRs is more general than eitherthe MJSs model with bounded uncertain TRs or the MJSsmodel with partly uncertain TRs If 119880119894

119906119896= 0 for all 119894 isin S

then generally uncertain TR matrix (4) reduces to boundeduncertain TR matrix (5) as follows

[[[[

[

11+ Δ11

12+ Δ12

sdot sdot sdot 1119904+ Δ1119904

21+ Δ21

22+ Δ22

sdot sdot sdot 2119904+ Δ2119904

d

1199041+ Δ1199041

1199042+ Δ1199042

sdot sdot sdot 119904119904+ Δ119904119904

]]]]

]

(5)

where 119894119895minusΔ119894119895ge 0 (for all 119895 isin S 119895 = 119894)

119894119894= minussum

119904

119895=1119895 = 119894119894119895le

0 and Δ119894119894= sum119904

119895=1119895 = 119894Δ119894119895 if 120590119894119895= 0 for all 119894 isin S for all 119895 isin

Abstract and Applied Analysis 3

119880119894

119896 then generally uncertain TR matrix (4) reduces to partly

uncertain TR matrix (6) as follows

[[[[

[

12058711

sdot sdot sdot

12058723

sdot sdot sdot 1205872119904

d

1205871199042

sdot sdot sdot

]]]]

]

(6)

Our results in this paper can be applicable to the generalMarkovian jump systems with bounded uncertain or partlyuncertain TR matrix

119860(120574(119905) 119905)119860119889(120574(119905) 119905)119862(120574(119905) 119905) and119862

119889(120574(119905) 119905) arematrix

functions of the random jumping process 120574(119905) To simplify thenotion the notation 119860

119894(119905) represents 119860(120574(119905) 119905) when 120574(119905) =

119894 For example 119860119889(120574(119905) 119905) is denoted by 119860

119889119894(119905) and so on

Further for each 120574(119905) = 119894 isin 119873 it is assumed that thematrices 119860

119894(119905) 119860

119889119894(119905) 119862119894(119905) and 119862

119889119894(119905) can be described by

the following form

119860119894(119905) = 119860

119894+ Δ119860119894(119905) 119860

119889119894(119905) = 119860

119889119894+ Δ119860119889119894(119905)

119862119894(119905) = 119862

119894+ Δ119862119894(119905) 119862

119889119894(119905) = 119862

119889119894+ Δ119862119889119894(119905)

(7)

where 119860119894 119860119889119894 119862119894are 119862

119889119894known real coefficient matrices

with appropriate dimensions Time-varying matricesΔ119860119894(119905) Δ119860

119889119894(119905) Δ119862

119894(119905) and Δ119862

119889119894(119905) represent norm-

bounded uncertainties and satisfy

[Δ119860119894(119905) Δ119860

119889119894(119905)

Δ119862119894(119905) Δ119862

119889119894(119905)

] = [1198721119894

1198722119894

]119865119894(119905) [1198731119894 1198732119894] (8)

where119872111989411987221198941198721119894 and119873

2119894are known constant real matri-

ces of appropriate dimensions which represent the structureof uncertainties and 119865

119894(119905) is an unknown matrix function

with Lebesgue measurable elements and satisfies 119865119894(119905)119865119879

119894(119905) le

119868Further for convenience we assume that the system has

the same dimension at each mode and the Markov processis irreducible Consider the following nominal unforceddescriptor time-delay system

119864 (119905) = 119860119894119909 (119905) + 119860

119889119894119909 (119905 minus 119889)

119909 (119905) = 120593 (119905) forall119905 isin [minus119889 0]

(9)

Let 1199090 1199030 and 119909(119905 120593 119903

0) be the initial state initial mode

and the corresponding solution of the system (9) at time 119905respectively

Definition 5 System (9) is said to be stochastically stable iffor all 120593(119905) isin 119871

2[minus119889 0] and initial mode 119903

0isin 119873 there exists

a matrix119872 gt 0 such that

119864int

infin

0

1003817100381710038171003817119909 (119905 120593 1199030)10038171003817100381710038172

119889119905 | 1199030 119909 (119905) = 120593 (119905) 119905 isin [minus119889 0]

le 119909119879

01198721199090

(10)

The following definition can be regarded as an extensionof the definition in [2]

Definition 6 (1) System (9) is said to be regular if det(119904E minus

119860119894) 119894 = 1 2 119904 are not identically zero(2) System (9) is said to be impulse free if deg(det(119904119864 minus

119860119894)) = rank 119864

119894 119894 = 1 2 119904

(3) System (9) is said to be admissible if it is regularimpulse free and stochastically stable

The linearmemoryless observer under consideration is asfollows

119864 119909 (119905) = 1198701119894119909 (119905) + 119870

2119894119910 (119905)

1199090= 0 119903 (0) = 119903

0

(11)

where 119909(119905) isin 119877119899 is the observer state and the constant

matrices1198701119894and119870

2119894are observer parameters to be designed

Denote the error state 119890(119905) = 119909(119905) minus 119909(119905) and theaugmented state vector 119909

119891= [119909119879(119905) 119890119879(119905)]119879

Let 119909(119905) = 119871119890(119905)

represent the output of the error states where 119871 is a knownconstant matrix Define

119860119891119894= [

119860119894

0

119860119894minus 1198701119894minus 11987021198941198621198941198701119894

]

119860119891119889119894

= [119860119889119894

0

119860119889119894minus 1198702119894119862119889119894

0] 119864

119891= [

119864 0

0 119864]

119872119891119894= 1198721198911119894

= [1198721119894

1198721119894minus 11987021198941198722119894

] 119873119891119894= [1198731119894 0]

Δ119860119891119894= 119872119891119894119865119894(119905)119873119891119894 119873

1198911119894= [1198732119894 0]

Δ119860119891119889119894

= 1198721198911119894

119865119894(119905)1198731198911119894

119862119891= [0 119871]

(12)

and combine (1) and (11) then we derive the augmentedsystems as follows

119864119891119891(119905) = (119860

119891119894+ Δ119860119891119894) 119909119891(119905)

+ (119860119891119889119894

+ Δ119860119891119889119894

) 119909119891(119905 minus 119889)

119911 (119905) = 119862119891119909119891(119905)

1199091198910

(119905) = [120593119879(119905) 120593119879(119905)]119879

forall119905 isin [minus119889 0]

(13)

Similar to [5] it is also assumed in this paper that for all 120589 isin[minus119889 0] there exists a scalar ℎ gt 0 such that 119909

119891(119905 + 120589) le

ℎ119909119891(119905)Associated with system (13) is the cost function

J = Eint

infin

0

119911119879(119905) 119911 (119905) 119889119905 (14)

Definition 7 Consider the augmented system (13) if thereexist the observer parameters 119870

1119894 1198702119894and a positive scalar

Jlowast for all uncertainties such that the augmented system(13) is robust stochastically stable and the value of the costfunction (14) satisfies J le Jlowast then Jlowast is said to be arobust guaranteed cost and observer (11) is said to be a robustguaranteed cost observer for system (1) with (4)

Problem 8 (robust guaranteed cost observer problem for aclass of linear singular Markovian jump time-delay systems

4 Abstract and Applied Analysis

with generally incomplete transition probability) Given sys-tem (1) with GUTRMatrix (4) can we determine an observer(11) with parameters 119870

1119894and 119870

2119894such that the observer is a

robust guaranteed cost observer for system (1) with GUTRMatrix (4)

Lemma 9 Given any real number 120576 and any matrix Q thematrix inequality 120576(119876 + 119876

119879) le 1205762119879 + 119876119879

minus1119876119879 holds for any

matrix 119879 gt 0

3 Main Results

Theorem 10 Consider the augmented system (13)with GUTRMatrix (4) and the cost function (14) Then the robust guar-anteed cost observer (11) with parameters 119870

1119894and 119870

2119894can

be designed if there exist matrices 119875119894 1198701119894 and 119870

2119894 119894 =

1 2 119904 and symmetric positive definite matrix Q satisfyingthe following LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 there exist a set of

symmetric positive definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin

119880119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (15)

[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (16)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (17)

Case 2 If 119894 isin 119880119894

119896 119880119894119896= 119896119894

1 119896

119894

119898 and 119880119894

119906119896= 0 there exist a

set of symmetric positive definite matrices 119881119894119895119897

isin R119899times119899 (119894 119895 isin

119880119894

119896 119897 isin 119880

119894

119906119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (18)

[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (19)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 there exist a set of symmetric

positive definite matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (20)

[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (21)

where

Π119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

Ω119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119897) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897

Δ119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isinS119895 = 119894

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isinS119895 = 119894

1

41205902

119894119895119882119894119895

1= [119864119879

119891(119875119894119896119894

1

minus 119875119894) 119864119879

119891(119875119894119896119894

2

minus 119875119894) 119864

119879

119891(119875119894119896119894

119898

minus 119875119894)]

2= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1= [119864119879

119891(119875119896119894

1

minus 119875119897) 119864119879

119891(119875119896119894

2

minus 119875119897) 119864

119879

119891(119875119896119894

119898

minus 119875119897)]

2= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

2= diag minus119882

1198941 minus119882

119894119904

(22)

Proof According to Definition 2 and Theorem 1 in [2] wecan derive from (15)ndash(21) that system (13) is regular andimpulse free Let the mode at time 119905 be 119894 and consider thefollowing Lyapunov function with respect to the augmentedsystem (13)

119881(119909119891(119905) 120574 (119905) = 119894) = 119909

119879

119891(119905) 119864119879

119891119875119894119909119891(119905)

+ int

119905

119905minus119889

119909119879

119891(119904) 119876119909

119891(119904) 119889119905

(23)

where 119876 is the symmetric positive definite matrix to be cho-sen and 119875

119894is a matrix satisfying (15)ndash(21)The weak infinites-

imal operatorL of the stochastic process 120574(119905) 119909119891(119905) 119905 ge 0

is presented by

L119881(119909119891(119905) 120574 (119905) = 119894)

= limΔrarr0

1

Δ[119864119891119881 (119909 (119905 + Δ) 120574 (119905 + Δ)) 119909 (119905) 120574 (119905) = 119894

minus119881 (119909 (119905) 120574 (119905) = 119894) ]

= 119909119879

119891(119905) [

[

(119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+

119904

sum

119895=1

120587119894119895119864119879

119891119875119895+ 119876]

]

119909119891(119905)

+ 2119909119879

119891(119905) 119875119894(1198601198911119894

+ Δ1198601198911119894

) 119909119891(119905 minus 119889)

minus 119909119879

119891(119905 minus 119889)119876119909

119891(119905 minus 119889)

(24)

Abstract and Applied Analysis 5

Case 1 (119894 notin 119880119894

119896) Note that in this case sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895

=

minussum119895isinU119894119896119895 = 119894

120587119894119895minus 120587119894119894and 120587

119894119895ge 0 119895 isin 119880

119894

119906119896 119895 = 119894 then from

(24) we have

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895119864119879

119891119875119895+ 120587119894119894119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ (minus120587

119894119894minus sum

119895isin119880119894

119896

120587119894119895)119864119879

119891119875119894

+120587119894119894119864119879

119891119875119894]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(25)

On the other hand in view of Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119894)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119894) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119894)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

(26)

Case 2 (119894 isin 119880119894

119896and119880119894

119906119896= 0) Because of119880119894

119896= 119896119894

1 119896

119894

119898 and

119880119894

119906119896= 119906119894

1 119906

119894

119904minus119898 there must be 119897 isin 119880

119894

119906119896so that 119864119879

119891119875119897ge

119864119879

119891119875119895(for all 119895 isin 119880

119894

119906119896)

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

le 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895minus ( sum

119895isin119880119894

119896119895 = 119894

120587119894119895)119864119879

119891119875119897]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119897) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

(27)

By using Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119897)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119897) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119897)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

(28)

Case 3 (119894 isin 119880119894

119896and 119880

119894

119906119896= 0) Consider

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

119894119895(119875119895minus 119875119894)

+

119904

sum

119895=1119895 = 119894

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(29)

Case 1 Substituting (25) and (26) into (24) it results in

L119881 le Λ119879(119905) Φ (119894) Λ (119905) (30)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Φ119894=[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(31)

Case 2 Substituting (27) and (28) into (24) it results in

L119881 le Λ119879(119905) Ψ (119894) Λ (119905) (32)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Ψ119894=[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(33)

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

delay and partially unknown transition probabilities Guoand Wang [49] proposed another description for the uncer-tain TRs which is called generally uncertain TRs (GUTRs)

On the other hand state estimation plays an importantrole in systems and control theory signal processing andinformation fusion [50 51] Certainly the most widely usedestimation method is the well-known Kalman filtering [5253] A common feature in the Kalman filtering is that anaccurate model is available In some applications howeverwhen the system is subject to parameter uncertainties theaccurate system model is hard to obtain To overcome thisdifficulty the guaranteed cost filtering approach has beenproposed to ensure the upper bound of guaranteed cost func-tion [54] Robust119867

infinfiltering for uncertainMarkovian jump

systems with mode-dependent time delays was proposed in[55] In [56] guaranteed cost and 119867

infinfiltering for time-

delay systems were presented in terms of LMIs Howeverto the best of our knowledge there are few consideringthe robust guaranteed cost observer for a class of linearsingular Markovian jump time-delay systems with generallyincomplete transition probability which is still an openproblem

In this paper based on LMI method we address thedesign problem of the robust guaranteed cost observer for aclass of uncertain descriptor time-delay systems withMarko-vian jumping parameters and generally uncertain transitionrates The design problem proposed here is to design a mem-oryless observer such that for all uncertainties includinggenerally uncertain transition rates the resulting augmentedsystem is regular impulse-free and robust stochastically sta-ble and satisfies the proposed guaranteed cost performance

2 Problem Formulation

Consider the following descriptor time-delay systems withMarkovian jumping parameters

119864 (119905) = 119860 (119903119905 119905) 119909 (119905) + 119860

119889(119903119905 119905) 119909 (119905 minus 119889)

119910 (119905) = 119862 (119903119905 119905) 119909 (119905) + 119862

119889(119903119905 119905) 119909 (119905 minus 119889)

119909 (119905) = 120593 (119905) forall119905 isin [minus119889 0]

(1)

where 119909(119905) isin 119877119899 and 119910(119905) isin 119877

119903 are the state vector andthe controlled output respectively 119889 represents the state timedelay For convenience the input terms in system (1) havebeen omitted 120593(119905) isin 119871

2[minus119889 0] is a continuous vector-valued

initial function The random parameter 120574(119905) represents acontinuous-time discrete-state Markov process taking valuesin a finite set S = 1 2 119904 and having the transitionprobability matrix Π = [120587

119894119895] 119894 119895 isin 119873 The transition

probability from mode 119894 to mode 119895 is defined by

Pr 119903119905+Δ

= 119895 | 119903119905= 119894 =

120587119894119895Δ + 119900 (Δ) 119894 = 119895

1 + 120587119894119895Δ + 119900 (Δ) 119894 = 119895

(2)

where Δ gt 0 satisfies limΔrarr0

(119900(Δ)Δ) = 0 120587119894119895ge 0 is the

transition probability from mode 119894 to mode 119895 and satisfies

120587119894119894= minus

119904

sum

119895=1119895 = 119894

120587119894119895le 0 (3)

In this paper the transition rates of the jumping processare assumed to be partly available that is some elements inmatrix Λ have been exactly known some have been merelyknown with lower and upper bounds and others may haveno information to use For instance for system (1) withfour operation modes the transition rate matrix might bedescribed by

Λ =

[[[[

[

11+ Δ11

sdot sdot sdot

23+ Δ23

sdot sdot sdot 2119904+ Δ2119904

d

1199042+ Δ1199042

sdot sdot sdot

]]]]

]

(4)

where 119894119895and Δ

119894119895isin [minus120590

119894119895 120590119894119895] (120590119894119895

ge 0) represent theestimate value and estimate error of the uncertain TR 120587

119894119895

respectively where 119894119895and 120590

119894119895are known represents the

complete unknown TR which means that its estimate value119894119895and estimate error bound are unknownFor notational clarity for all 119894 isin S the set 119880

119894

denotes 119880119894

= 119880119894

119896cup 119880119894

119906119896with 119880

119894

119896= 119895 The estimate

value of 120587119894119895is known for 119895 isin S 119880119894

119906119896= 119895 The estimate

value of 120587119894119895is unknown for 119895 isin S Moreover if 119880119894

119896= 0 it is

further described as 119880119894119896= 119896119894

1 119896119894

2 119896

119894

119898 where 119896119894

119898isin N+

represents the 119898th bound-known element with the index119896119894

119898in the 119894th row of matrix Π We assume that the known

estimate values of the TRs are well defined That is

Assumption 1 If 119880119894119896

= S then 119894119895minus 120590119894119895

ge 0 (for all 119895 isin

S 119895 = 119894) 119894119894= minussum

119873

119895=1119895 = 119894119894119895and 120590

119894119894= minussum

119904

119895=1119895 = 119894120590119894119895

Assumption 2 If 119880119894

119896=S and 119894 isin 119880

119894

119896 then

119894119895minus 120590119894119895

ge

0 (for all 119895 isin S 119895 = 119894) 119894119894+ 120590119894119894le 0 and sum

119895isin119880119894

119896

119894119895

Assumption 3 If 119880119894

119896=S and 119894 notin 119880

119894

119896 then

119894119895minus 120590119894119895

ge

0 (for all 119895 isin S)

Remark 4 The above assumption is reasonable since it isthe direct result from the properties of the TRs (eg 120587

119894119895ge

0 (for all 119894 119895 isin S 119895 = 119894) and 120587119894119894= minussum

119904

119895=1119895 = 119894120587119894119895) The above

description about uncertain TRs is more general than eitherthe MJSs model with bounded uncertain TRs or the MJSsmodel with partly uncertain TRs If 119880119894

119906119896= 0 for all 119894 isin S

then generally uncertain TR matrix (4) reduces to boundeduncertain TR matrix (5) as follows

[[[[

[

11+ Δ11

12+ Δ12

sdot sdot sdot 1119904+ Δ1119904

21+ Δ21

22+ Δ22

sdot sdot sdot 2119904+ Δ2119904

d

1199041+ Δ1199041

1199042+ Δ1199042

sdot sdot sdot 119904119904+ Δ119904119904

]]]]

]

(5)

where 119894119895minusΔ119894119895ge 0 (for all 119895 isin S 119895 = 119894)

119894119894= minussum

119904

119895=1119895 = 119894119894119895le

0 and Δ119894119894= sum119904

119895=1119895 = 119894Δ119894119895 if 120590119894119895= 0 for all 119894 isin S for all 119895 isin

Abstract and Applied Analysis 3

119880119894

119896 then generally uncertain TR matrix (4) reduces to partly

uncertain TR matrix (6) as follows

[[[[

[

12058711

sdot sdot sdot

12058723

sdot sdot sdot 1205872119904

d

1205871199042

sdot sdot sdot

]]]]

]

(6)

Our results in this paper can be applicable to the generalMarkovian jump systems with bounded uncertain or partlyuncertain TR matrix

119860(120574(119905) 119905)119860119889(120574(119905) 119905)119862(120574(119905) 119905) and119862

119889(120574(119905) 119905) arematrix

functions of the random jumping process 120574(119905) To simplify thenotion the notation 119860

119894(119905) represents 119860(120574(119905) 119905) when 120574(119905) =

119894 For example 119860119889(120574(119905) 119905) is denoted by 119860

119889119894(119905) and so on

Further for each 120574(119905) = 119894 isin 119873 it is assumed that thematrices 119860

119894(119905) 119860

119889119894(119905) 119862119894(119905) and 119862

119889119894(119905) can be described by

the following form

119860119894(119905) = 119860

119894+ Δ119860119894(119905) 119860

119889119894(119905) = 119860

119889119894+ Δ119860119889119894(119905)

119862119894(119905) = 119862

119894+ Δ119862119894(119905) 119862

119889119894(119905) = 119862

119889119894+ Δ119862119889119894(119905)

(7)

where 119860119894 119860119889119894 119862119894are 119862

119889119894known real coefficient matrices

with appropriate dimensions Time-varying matricesΔ119860119894(119905) Δ119860

119889119894(119905) Δ119862

119894(119905) and Δ119862

119889119894(119905) represent norm-

bounded uncertainties and satisfy

[Δ119860119894(119905) Δ119860

119889119894(119905)

Δ119862119894(119905) Δ119862

119889119894(119905)

] = [1198721119894

1198722119894

]119865119894(119905) [1198731119894 1198732119894] (8)

where119872111989411987221198941198721119894 and119873

2119894are known constant real matri-

ces of appropriate dimensions which represent the structureof uncertainties and 119865

119894(119905) is an unknown matrix function

with Lebesgue measurable elements and satisfies 119865119894(119905)119865119879

119894(119905) le

119868Further for convenience we assume that the system has

the same dimension at each mode and the Markov processis irreducible Consider the following nominal unforceddescriptor time-delay system

119864 (119905) = 119860119894119909 (119905) + 119860

119889119894119909 (119905 minus 119889)

119909 (119905) = 120593 (119905) forall119905 isin [minus119889 0]

(9)

Let 1199090 1199030 and 119909(119905 120593 119903

0) be the initial state initial mode

and the corresponding solution of the system (9) at time 119905respectively

Definition 5 System (9) is said to be stochastically stable iffor all 120593(119905) isin 119871

2[minus119889 0] and initial mode 119903

0isin 119873 there exists

a matrix119872 gt 0 such that

119864int

infin

0

1003817100381710038171003817119909 (119905 120593 1199030)10038171003817100381710038172

119889119905 | 1199030 119909 (119905) = 120593 (119905) 119905 isin [minus119889 0]

le 119909119879

01198721199090

(10)

The following definition can be regarded as an extensionof the definition in [2]

Definition 6 (1) System (9) is said to be regular if det(119904E minus

119860119894) 119894 = 1 2 119904 are not identically zero(2) System (9) is said to be impulse free if deg(det(119904119864 minus

119860119894)) = rank 119864

119894 119894 = 1 2 119904

(3) System (9) is said to be admissible if it is regularimpulse free and stochastically stable

The linearmemoryless observer under consideration is asfollows

119864 119909 (119905) = 1198701119894119909 (119905) + 119870

2119894119910 (119905)

1199090= 0 119903 (0) = 119903

0

(11)

where 119909(119905) isin 119877119899 is the observer state and the constant

matrices1198701119894and119870

2119894are observer parameters to be designed

Denote the error state 119890(119905) = 119909(119905) minus 119909(119905) and theaugmented state vector 119909

119891= [119909119879(119905) 119890119879(119905)]119879

Let 119909(119905) = 119871119890(119905)

represent the output of the error states where 119871 is a knownconstant matrix Define

119860119891119894= [

119860119894

0

119860119894minus 1198701119894minus 11987021198941198621198941198701119894

]

119860119891119889119894

= [119860119889119894

0

119860119889119894minus 1198702119894119862119889119894

0] 119864

119891= [

119864 0

0 119864]

119872119891119894= 1198721198911119894

= [1198721119894

1198721119894minus 11987021198941198722119894

] 119873119891119894= [1198731119894 0]

Δ119860119891119894= 119872119891119894119865119894(119905)119873119891119894 119873

1198911119894= [1198732119894 0]

Δ119860119891119889119894

= 1198721198911119894

119865119894(119905)1198731198911119894

119862119891= [0 119871]

(12)

and combine (1) and (11) then we derive the augmentedsystems as follows

119864119891119891(119905) = (119860

119891119894+ Δ119860119891119894) 119909119891(119905)

+ (119860119891119889119894

+ Δ119860119891119889119894

) 119909119891(119905 minus 119889)

119911 (119905) = 119862119891119909119891(119905)

1199091198910

(119905) = [120593119879(119905) 120593119879(119905)]119879

forall119905 isin [minus119889 0]

(13)

Similar to [5] it is also assumed in this paper that for all 120589 isin[minus119889 0] there exists a scalar ℎ gt 0 such that 119909

119891(119905 + 120589) le

ℎ119909119891(119905)Associated with system (13) is the cost function

J = Eint

infin

0

119911119879(119905) 119911 (119905) 119889119905 (14)

Definition 7 Consider the augmented system (13) if thereexist the observer parameters 119870

1119894 1198702119894and a positive scalar

Jlowast for all uncertainties such that the augmented system(13) is robust stochastically stable and the value of the costfunction (14) satisfies J le Jlowast then Jlowast is said to be arobust guaranteed cost and observer (11) is said to be a robustguaranteed cost observer for system (1) with (4)

Problem 8 (robust guaranteed cost observer problem for aclass of linear singular Markovian jump time-delay systems

4 Abstract and Applied Analysis

with generally incomplete transition probability) Given sys-tem (1) with GUTRMatrix (4) can we determine an observer(11) with parameters 119870

1119894and 119870

2119894such that the observer is a

robust guaranteed cost observer for system (1) with GUTRMatrix (4)

Lemma 9 Given any real number 120576 and any matrix Q thematrix inequality 120576(119876 + 119876

119879) le 1205762119879 + 119876119879

minus1119876119879 holds for any

matrix 119879 gt 0

3 Main Results

Theorem 10 Consider the augmented system (13)with GUTRMatrix (4) and the cost function (14) Then the robust guar-anteed cost observer (11) with parameters 119870

1119894and 119870

2119894can

be designed if there exist matrices 119875119894 1198701119894 and 119870

2119894 119894 =

1 2 119904 and symmetric positive definite matrix Q satisfyingthe following LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 there exist a set of

symmetric positive definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin

119880119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (15)

[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (16)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (17)

Case 2 If 119894 isin 119880119894

119896 119880119894119896= 119896119894

1 119896

119894

119898 and 119880119894

119906119896= 0 there exist a

set of symmetric positive definite matrices 119881119894119895119897

isin R119899times119899 (119894 119895 isin

119880119894

119896 119897 isin 119880

119894

119906119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (18)

[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (19)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 there exist a set of symmetric

positive definite matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (20)

[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (21)

where

Π119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

Ω119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119897) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897

Δ119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isinS119895 = 119894

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isinS119895 = 119894

1

41205902

119894119895119882119894119895

1= [119864119879

119891(119875119894119896119894

1

minus 119875119894) 119864119879

119891(119875119894119896119894

2

minus 119875119894) 119864

119879

119891(119875119894119896119894

119898

minus 119875119894)]

2= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1= [119864119879

119891(119875119896119894

1

minus 119875119897) 119864119879

119891(119875119896119894

2

minus 119875119897) 119864

119879

119891(119875119896119894

119898

minus 119875119897)]

2= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

2= diag minus119882

1198941 minus119882

119894119904

(22)

Proof According to Definition 2 and Theorem 1 in [2] wecan derive from (15)ndash(21) that system (13) is regular andimpulse free Let the mode at time 119905 be 119894 and consider thefollowing Lyapunov function with respect to the augmentedsystem (13)

119881(119909119891(119905) 120574 (119905) = 119894) = 119909

119879

119891(119905) 119864119879

119891119875119894119909119891(119905)

+ int

119905

119905minus119889

119909119879

119891(119904) 119876119909

119891(119904) 119889119905

(23)

where 119876 is the symmetric positive definite matrix to be cho-sen and 119875

119894is a matrix satisfying (15)ndash(21)The weak infinites-

imal operatorL of the stochastic process 120574(119905) 119909119891(119905) 119905 ge 0

is presented by

L119881(119909119891(119905) 120574 (119905) = 119894)

= limΔrarr0

1

Δ[119864119891119881 (119909 (119905 + Δ) 120574 (119905 + Δ)) 119909 (119905) 120574 (119905) = 119894

minus119881 (119909 (119905) 120574 (119905) = 119894) ]

= 119909119879

119891(119905) [

[

(119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+

119904

sum

119895=1

120587119894119895119864119879

119891119875119895+ 119876]

]

119909119891(119905)

+ 2119909119879

119891(119905) 119875119894(1198601198911119894

+ Δ1198601198911119894

) 119909119891(119905 minus 119889)

minus 119909119879

119891(119905 minus 119889)119876119909

119891(119905 minus 119889)

(24)

Abstract and Applied Analysis 5

Case 1 (119894 notin 119880119894

119896) Note that in this case sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895

=

minussum119895isinU119894119896119895 = 119894

120587119894119895minus 120587119894119894and 120587

119894119895ge 0 119895 isin 119880

119894

119906119896 119895 = 119894 then from

(24) we have

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895119864119879

119891119875119895+ 120587119894119894119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ (minus120587

119894119894minus sum

119895isin119880119894

119896

120587119894119895)119864119879

119891119875119894

+120587119894119894119864119879

119891119875119894]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(25)

On the other hand in view of Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119894)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119894) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119894)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

(26)

Case 2 (119894 isin 119880119894

119896and119880119894

119906119896= 0) Because of119880119894

119896= 119896119894

1 119896

119894

119898 and

119880119894

119906119896= 119906119894

1 119906

119894

119904minus119898 there must be 119897 isin 119880

119894

119906119896so that 119864119879

119891119875119897ge

119864119879

119891119875119895(for all 119895 isin 119880

119894

119906119896)

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

le 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895minus ( sum

119895isin119880119894

119896119895 = 119894

120587119894119895)119864119879

119891119875119897]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119897) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

(27)

By using Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119897)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119897) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119897)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

(28)

Case 3 (119894 isin 119880119894

119896and 119880

119894

119906119896= 0) Consider

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

119894119895(119875119895minus 119875119894)

+

119904

sum

119895=1119895 = 119894

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(29)

Case 1 Substituting (25) and (26) into (24) it results in

L119881 le Λ119879(119905) Φ (119894) Λ (119905) (30)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Φ119894=[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(31)

Case 2 Substituting (27) and (28) into (24) it results in

L119881 le Λ119879(119905) Ψ (119894) Λ (119905) (32)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Ψ119894=[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(33)

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

119880119894

119896 then generally uncertain TR matrix (4) reduces to partly

uncertain TR matrix (6) as follows

[[[[

[

12058711

sdot sdot sdot

12058723

sdot sdot sdot 1205872119904

d

1205871199042

sdot sdot sdot

]]]]

]

(6)

Our results in this paper can be applicable to the generalMarkovian jump systems with bounded uncertain or partlyuncertain TR matrix

119860(120574(119905) 119905)119860119889(120574(119905) 119905)119862(120574(119905) 119905) and119862

119889(120574(119905) 119905) arematrix

functions of the random jumping process 120574(119905) To simplify thenotion the notation 119860

119894(119905) represents 119860(120574(119905) 119905) when 120574(119905) =

119894 For example 119860119889(120574(119905) 119905) is denoted by 119860

119889119894(119905) and so on

Further for each 120574(119905) = 119894 isin 119873 it is assumed that thematrices 119860

119894(119905) 119860

119889119894(119905) 119862119894(119905) and 119862

119889119894(119905) can be described by

the following form

119860119894(119905) = 119860

119894+ Δ119860119894(119905) 119860

119889119894(119905) = 119860

119889119894+ Δ119860119889119894(119905)

119862119894(119905) = 119862

119894+ Δ119862119894(119905) 119862

119889119894(119905) = 119862

119889119894+ Δ119862119889119894(119905)

(7)

where 119860119894 119860119889119894 119862119894are 119862

119889119894known real coefficient matrices

with appropriate dimensions Time-varying matricesΔ119860119894(119905) Δ119860

119889119894(119905) Δ119862

119894(119905) and Δ119862

119889119894(119905) represent norm-

bounded uncertainties and satisfy

[Δ119860119894(119905) Δ119860

119889119894(119905)

Δ119862119894(119905) Δ119862

119889119894(119905)

] = [1198721119894

1198722119894

]119865119894(119905) [1198731119894 1198732119894] (8)

where119872111989411987221198941198721119894 and119873

2119894are known constant real matri-

ces of appropriate dimensions which represent the structureof uncertainties and 119865

119894(119905) is an unknown matrix function

with Lebesgue measurable elements and satisfies 119865119894(119905)119865119879

119894(119905) le

119868Further for convenience we assume that the system has

the same dimension at each mode and the Markov processis irreducible Consider the following nominal unforceddescriptor time-delay system

119864 (119905) = 119860119894119909 (119905) + 119860

119889119894119909 (119905 minus 119889)

119909 (119905) = 120593 (119905) forall119905 isin [minus119889 0]

(9)

Let 1199090 1199030 and 119909(119905 120593 119903

0) be the initial state initial mode

and the corresponding solution of the system (9) at time 119905respectively

Definition 5 System (9) is said to be stochastically stable iffor all 120593(119905) isin 119871

2[minus119889 0] and initial mode 119903

0isin 119873 there exists

a matrix119872 gt 0 such that

119864int

infin

0

1003817100381710038171003817119909 (119905 120593 1199030)10038171003817100381710038172

119889119905 | 1199030 119909 (119905) = 120593 (119905) 119905 isin [minus119889 0]

le 119909119879

01198721199090

(10)

The following definition can be regarded as an extensionof the definition in [2]

Definition 6 (1) System (9) is said to be regular if det(119904E minus

119860119894) 119894 = 1 2 119904 are not identically zero(2) System (9) is said to be impulse free if deg(det(119904119864 minus

119860119894)) = rank 119864

119894 119894 = 1 2 119904

(3) System (9) is said to be admissible if it is regularimpulse free and stochastically stable

The linearmemoryless observer under consideration is asfollows

119864 119909 (119905) = 1198701119894119909 (119905) + 119870

2119894119910 (119905)

1199090= 0 119903 (0) = 119903

0

(11)

where 119909(119905) isin 119877119899 is the observer state and the constant

matrices1198701119894and119870

2119894are observer parameters to be designed

Denote the error state 119890(119905) = 119909(119905) minus 119909(119905) and theaugmented state vector 119909

119891= [119909119879(119905) 119890119879(119905)]119879

Let 119909(119905) = 119871119890(119905)

represent the output of the error states where 119871 is a knownconstant matrix Define

119860119891119894= [

119860119894

0

119860119894minus 1198701119894minus 11987021198941198621198941198701119894

]

119860119891119889119894

= [119860119889119894

0

119860119889119894minus 1198702119894119862119889119894

0] 119864

119891= [

119864 0

0 119864]

119872119891119894= 1198721198911119894

= [1198721119894

1198721119894minus 11987021198941198722119894

] 119873119891119894= [1198731119894 0]

Δ119860119891119894= 119872119891119894119865119894(119905)119873119891119894 119873

1198911119894= [1198732119894 0]

Δ119860119891119889119894

= 1198721198911119894

119865119894(119905)1198731198911119894

119862119891= [0 119871]

(12)

and combine (1) and (11) then we derive the augmentedsystems as follows

119864119891119891(119905) = (119860

119891119894+ Δ119860119891119894) 119909119891(119905)

+ (119860119891119889119894

+ Δ119860119891119889119894

) 119909119891(119905 minus 119889)

119911 (119905) = 119862119891119909119891(119905)

1199091198910

(119905) = [120593119879(119905) 120593119879(119905)]119879

forall119905 isin [minus119889 0]

(13)

Similar to [5] it is also assumed in this paper that for all 120589 isin[minus119889 0] there exists a scalar ℎ gt 0 such that 119909

119891(119905 + 120589) le

ℎ119909119891(119905)Associated with system (13) is the cost function

J = Eint

infin

0

119911119879(119905) 119911 (119905) 119889119905 (14)

Definition 7 Consider the augmented system (13) if thereexist the observer parameters 119870

1119894 1198702119894and a positive scalar

Jlowast for all uncertainties such that the augmented system(13) is robust stochastically stable and the value of the costfunction (14) satisfies J le Jlowast then Jlowast is said to be arobust guaranteed cost and observer (11) is said to be a robustguaranteed cost observer for system (1) with (4)

Problem 8 (robust guaranteed cost observer problem for aclass of linear singular Markovian jump time-delay systems

4 Abstract and Applied Analysis

with generally incomplete transition probability) Given sys-tem (1) with GUTRMatrix (4) can we determine an observer(11) with parameters 119870

1119894and 119870

2119894such that the observer is a

robust guaranteed cost observer for system (1) with GUTRMatrix (4)

Lemma 9 Given any real number 120576 and any matrix Q thematrix inequality 120576(119876 + 119876

119879) le 1205762119879 + 119876119879

minus1119876119879 holds for any

matrix 119879 gt 0

3 Main Results

Theorem 10 Consider the augmented system (13)with GUTRMatrix (4) and the cost function (14) Then the robust guar-anteed cost observer (11) with parameters 119870

1119894and 119870

2119894can

be designed if there exist matrices 119875119894 1198701119894 and 119870

2119894 119894 =

1 2 119904 and symmetric positive definite matrix Q satisfyingthe following LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 there exist a set of

symmetric positive definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin

119880119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (15)

[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (16)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (17)

Case 2 If 119894 isin 119880119894

119896 119880119894119896= 119896119894

1 119896

119894

119898 and 119880119894

119906119896= 0 there exist a

set of symmetric positive definite matrices 119881119894119895119897

isin R119899times119899 (119894 119895 isin

119880119894

119896 119897 isin 119880

119894

119906119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (18)

[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (19)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 there exist a set of symmetric

positive definite matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (20)

[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (21)

where

Π119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

Ω119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119897) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897

Δ119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isinS119895 = 119894

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isinS119895 = 119894

1

41205902

119894119895119882119894119895

1= [119864119879

119891(119875119894119896119894

1

minus 119875119894) 119864119879

119891(119875119894119896119894

2

minus 119875119894) 119864

119879

119891(119875119894119896119894

119898

minus 119875119894)]

2= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1= [119864119879

119891(119875119896119894

1

minus 119875119897) 119864119879

119891(119875119896119894

2

minus 119875119897) 119864

119879

119891(119875119896119894

119898

minus 119875119897)]

2= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

2= diag minus119882

1198941 minus119882

119894119904

(22)

Proof According to Definition 2 and Theorem 1 in [2] wecan derive from (15)ndash(21) that system (13) is regular andimpulse free Let the mode at time 119905 be 119894 and consider thefollowing Lyapunov function with respect to the augmentedsystem (13)

119881(119909119891(119905) 120574 (119905) = 119894) = 119909

119879

119891(119905) 119864119879

119891119875119894119909119891(119905)

+ int

119905

119905minus119889

119909119879

119891(119904) 119876119909

119891(119904) 119889119905

(23)

where 119876 is the symmetric positive definite matrix to be cho-sen and 119875

119894is a matrix satisfying (15)ndash(21)The weak infinites-

imal operatorL of the stochastic process 120574(119905) 119909119891(119905) 119905 ge 0

is presented by

L119881(119909119891(119905) 120574 (119905) = 119894)

= limΔrarr0

1

Δ[119864119891119881 (119909 (119905 + Δ) 120574 (119905 + Δ)) 119909 (119905) 120574 (119905) = 119894

minus119881 (119909 (119905) 120574 (119905) = 119894) ]

= 119909119879

119891(119905) [

[

(119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+

119904

sum

119895=1

120587119894119895119864119879

119891119875119895+ 119876]

]

119909119891(119905)

+ 2119909119879

119891(119905) 119875119894(1198601198911119894

+ Δ1198601198911119894

) 119909119891(119905 minus 119889)

minus 119909119879

119891(119905 minus 119889)119876119909

119891(119905 minus 119889)

(24)

Abstract and Applied Analysis 5

Case 1 (119894 notin 119880119894

119896) Note that in this case sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895

=

minussum119895isinU119894119896119895 = 119894

120587119894119895minus 120587119894119894and 120587

119894119895ge 0 119895 isin 119880

119894

119906119896 119895 = 119894 then from

(24) we have

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895119864119879

119891119875119895+ 120587119894119894119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ (minus120587

119894119894minus sum

119895isin119880119894

119896

120587119894119895)119864119879

119891119875119894

+120587119894119894119864119879

119891119875119894]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(25)

On the other hand in view of Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119894)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119894) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119894)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

(26)

Case 2 (119894 isin 119880119894

119896and119880119894

119906119896= 0) Because of119880119894

119896= 119896119894

1 119896

119894

119898 and

119880119894

119906119896= 119906119894

1 119906

119894

119904minus119898 there must be 119897 isin 119880

119894

119906119896so that 119864119879

119891119875119897ge

119864119879

119891119875119895(for all 119895 isin 119880

119894

119906119896)

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

le 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895minus ( sum

119895isin119880119894

119896119895 = 119894

120587119894119895)119864119879

119891119875119897]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119897) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

(27)

By using Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119897)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119897) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119897)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

(28)

Case 3 (119894 isin 119880119894

119896and 119880

119894

119906119896= 0) Consider

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

119894119895(119875119895minus 119875119894)

+

119904

sum

119895=1119895 = 119894

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(29)

Case 1 Substituting (25) and (26) into (24) it results in

L119881 le Λ119879(119905) Φ (119894) Λ (119905) (30)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Φ119894=[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(31)

Case 2 Substituting (27) and (28) into (24) it results in

L119881 le Λ119879(119905) Ψ (119894) Λ (119905) (32)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Ψ119894=[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(33)

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

with generally incomplete transition probability) Given sys-tem (1) with GUTRMatrix (4) can we determine an observer(11) with parameters 119870

1119894and 119870

2119894such that the observer is a

robust guaranteed cost observer for system (1) with GUTRMatrix (4)

Lemma 9 Given any real number 120576 and any matrix Q thematrix inequality 120576(119876 + 119876

119879) le 1205762119879 + 119876119879

minus1119876119879 holds for any

matrix 119879 gt 0

3 Main Results

Theorem 10 Consider the augmented system (13)with GUTRMatrix (4) and the cost function (14) Then the robust guar-anteed cost observer (11) with parameters 119870

1119894and 119870

2119894can

be designed if there exist matrices 119875119894 1198701119894 and 119870

2119894 119894 =

1 2 119904 and symmetric positive definite matrix Q satisfyingthe following LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 there exist a set of

symmetric positive definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin

119880119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (15)

[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (16)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (17)

Case 2 If 119894 isin 119880119894

119896 119880119894119896= 119896119894

1 119896

119894

119898 and 119880119894

119906119896= 0 there exist a

set of symmetric positive definite matrices 119881119894119895119897

isin R119899times119899 (119894 119895 isin

119880119894

119896 119897 isin 119880

119894

119906119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (18)

[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (19)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 there exist a set of symmetric

positive definite matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (20)

[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

lt 0 (21)

where

Π119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

Ω119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119897) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897

Δ119894= (119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+ 119876 + sum

119895isinS119895 = 119894

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isinS119895 = 119894

1

41205902

119894119895119882119894119895

1= [119864119879

119891(119875119894119896119894

1

minus 119875119894) 119864119879

119891(119875119894119896119894

2

minus 119875119894) 119864

119879

119891(119875119894119896119894

119898

minus 119875119894)]

2= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1= [119864119879

119891(119875119896119894

1

minus 119875119897) 119864119879

119891(119875119896119894

2

minus 119875119897) 119864

119879

119891(119875119896119894

119898

minus 119875119897)]

2= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

2= diag minus119882

1198941 minus119882

119894119904

(22)

Proof According to Definition 2 and Theorem 1 in [2] wecan derive from (15)ndash(21) that system (13) is regular andimpulse free Let the mode at time 119905 be 119894 and consider thefollowing Lyapunov function with respect to the augmentedsystem (13)

119881(119909119891(119905) 120574 (119905) = 119894) = 119909

119879

119891(119905) 119864119879

119891119875119894119909119891(119905)

+ int

119905

119905minus119889

119909119879

119891(119904) 119876119909

119891(119904) 119889119905

(23)

where 119876 is the symmetric positive definite matrix to be cho-sen and 119875

119894is a matrix satisfying (15)ndash(21)The weak infinites-

imal operatorL of the stochastic process 120574(119905) 119909119891(119905) 119905 ge 0

is presented by

L119881(119909119891(119905) 120574 (119905) = 119894)

= limΔrarr0

1

Δ[119864119891119881 (119909 (119905 + Δ) 120574 (119905 + Δ)) 119909 (119905) 120574 (119905) = 119894

minus119881 (119909 (119905) 120574 (119905) = 119894) ]

= 119909119879

119891(119905) [

[

(119860119891119894+ Δ119860119891119894)119879

119875119894+ 119875119894(119860119891119894+ Δ119860119891119894)

+

119904

sum

119895=1

120587119894119895119864119879

119891119875119895+ 119876]

]

119909119891(119905)

+ 2119909119879

119891(119905) 119875119894(1198601198911119894

+ Δ1198601198911119894

) 119909119891(119905 minus 119889)

minus 119909119879

119891(119905 minus 119889)119876119909

119891(119905 minus 119889)

(24)

Abstract and Applied Analysis 5

Case 1 (119894 notin 119880119894

119896) Note that in this case sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895

=

minussum119895isinU119894119896119895 = 119894

120587119894119895minus 120587119894119894and 120587

119894119895ge 0 119895 isin 119880

119894

119906119896 119895 = 119894 then from

(24) we have

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895119864119879

119891119875119895+ 120587119894119894119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ (minus120587

119894119894minus sum

119895isin119880119894

119896

120587119894119895)119864119879

119891119875119894

+120587119894119894119864119879

119891119875119894]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(25)

On the other hand in view of Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119894)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119894) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119894)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

(26)

Case 2 (119894 isin 119880119894

119896and119880119894

119906119896= 0) Because of119880119894

119896= 119896119894

1 119896

119894

119898 and

119880119894

119906119896= 119906119894

1 119906

119894

119904minus119898 there must be 119897 isin 119880

119894

119906119896so that 119864119879

119891119875119897ge

119864119879

119891119875119895(for all 119895 isin 119880

119894

119906119896)

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

le 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895minus ( sum

119895isin119880119894

119896119895 = 119894

120587119894119895)119864119879

119891119875119897]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119897) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

(27)

By using Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119897)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119897) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119897)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

(28)

Case 3 (119894 isin 119880119894

119896and 119880

119894

119906119896= 0) Consider

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

119894119895(119875119895minus 119875119894)

+

119904

sum

119895=1119895 = 119894

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(29)

Case 1 Substituting (25) and (26) into (24) it results in

L119881 le Λ119879(119905) Φ (119894) Λ (119905) (30)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Φ119894=[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(31)

Case 2 Substituting (27) and (28) into (24) it results in

L119881 le Λ119879(119905) Ψ (119894) Λ (119905) (32)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Ψ119894=[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(33)

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

Submit your manuscripts athttpwwwhindawicom

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Case 1 (119894 notin 119880119894

119896) Note that in this case sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895

=

minussum119895isinU119894119896119895 = 119894

120587119894119895minus 120587119894119894and 120587

119894119895ge 0 119895 isin 119880

119894

119906119896 119895 = 119894 then from

(24) we have

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ sum

119895isin119880119894

119906119896119895 = 119894

120587119894119895119864119879

119891119875119895+ 120587119894119894119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895+ (minus120587

119894119894minus sum

119895isin119880119894

119896

120587119894119895)119864119879

119891119875119894

+120587119894119894119864119879

119891119875119894]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(25)

On the other hand in view of Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119894)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119894) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119894)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119879119894119895+ 119864119879

119891(119875119895minus 119875119894) 119879minus1

119894119895(119875119895minus 119875119894) 119864119891]

(26)

Case 2 (119894 isin 119880119894

119896and119880119894

119906119896= 0) Because of119880119894

119896= 119896119894

1 119896

119894

119898 and

119880119894

119906119896= 119906119894

1 119906

119894

119904minus119898 there must be 119897 isin 119880

119894

119906119896so that 119864119879

119891119875119897ge

119864119879

119891119875119895(for all 119895 isin 119880

119894

119906119896)

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

le 119909119879

119891(119905) [

[

sum

119895isin119880119894

119896

120587119894119895119864119879

119891119875119895minus ( sum

119895isin119880119894

119896119895 = 119894

120587119894119895)119864119879

119891119875119897]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

120587119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119897) + sum

119895isin119880119894

119896

Δ119894119895(119875119895minus 119875119897)]

]

119909119891(119905)

(27)

By using Lemma 9 we have

sum

119895isin119880119894

119896

Δ119894119895119864119879

119891(119875119895minus 119875119897)

= sum

119895isin119880119894

119896

[1

2Δ119894119895119864119879

119891(119875119895minus 119875119897) +

1

2Δ119894119895119864119879

119891(119875119895minus 119875119897)]

le sum

119895isin119880119894

119896

[(1

2Δ119894119895)

2

119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

le sum

119895isin119880119894

119896

[1

41205902

119894119895119881119894119895119897+ 119864119879

119891(119875119895minus 119875119897)119881minus1

119894119895119897(119875119895minus 119875119897) 119864119879

119891]

(28)

Case 3 (119894 isin 119880119894

119896and 119880

119894

119906119896= 0) Consider

119909119879

119891(119905) [

[

119904

sum

119895=1

120587119894119895119864119879

119891119875119895]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

120587119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

= 119909119879

119891(119905) 119864119879

119891[

[

119904

sum

119895=1119895 = 119894

119894119895(119875119895minus 119875119894)

+

119904

sum

119895=1119895 = 119894

Δ119894119895(119875119895minus 119875119894)]

]

119909119891(119905)

(29)

Case 1 Substituting (25) and (26) into (24) it results in

L119881 le Λ119879(119905) Φ (119894) Λ (119905) (30)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Φ119894=[[

[

Π119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(31)

Case 2 Substituting (27) and (28) into (24) it results in

L119881 le Λ119879(119905) Ψ (119894) Λ (119905) (32)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Ψ119894=[[

[

Ω119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(33)

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Case 3 Substituting (29) into (24) we get

L119881 le Λ119879(119905) Γ (119894) Λ (119905) (34)

where Λ119879(119905) = [119909119879

119891(119905) 119909119879

119891(119905 minus 119889)] and

Γ119894=[[

[

Δ119894+ 119862119879

119891119862119891

119875119894(119860119891119889119894

+ Δ119860119891119889119894

) 1

(119860119891119889119894

+ Δ119860119891119889119894

)119879

119875119894

minus119876 0

lowast lowast 2

]]

]

(35)

Similar to [5] usingDynkinrsquos formula we drive for each 119894 isin 119873

lim119879rarrinfin

Eint

119879

0

119909119879

119891(119905) 119909119891(119905) 119889119905 | 120593

119891 1205740= 119894 le 119909

119879

11989101198721199091198910 (36)

By Definition 5 it is easy to see that the augmented system(13) is stochastically stable Furthermore from (16) (19) and(21) we have

L119881 le minus119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) lt 0 (37)

On the other hand we have

J = Eint

infin

0

119909119879

119891(119905) 119862119879

119891119862119891119909119891(119905) 119889119905 lt minusint

infin

0

L119881119889119905

= minus E lim119905rarrinfin

119881 (119909 (119905) 120574 (119905)) + 119881 (1199090 1205740)

(38)

As the augmented system (13) is stochastically stable itfollows from (38) that 119869 lt 119881(119909

1198910 1199030) From Definition 7 it

is concluded that a robust guaranteed cost for the augmentedsystem (13) can be given by 119869

lowast= 119909119879

1198910(119905)119864119879

1198911199030119875(1199030)1199091198910

+

int0

minus119889119909119879

119891(119905)119876119909

119891(119905)119889119905

In the following based on the above sufficient conditionthe design of robust guaranteed cost observers can be turnedinto the solvability of a system of LMIs

Theorem 11 Consider system (13)with GUTRMatrix (4) andthe cost function (14) If there exist matrices 119884

1119894and 119884

2119894 119894 =

1 2 119904 positive scalars 120576119894 119894 = 1 2 119904 symmetric positive

definite matrix Q and the full rank matrices 1198752119894 and matrices

119875119894= diag(119875

1119894 1198752119894) 119894 = 1 2 119904 satisfying the following

LMIs respectively

Case 1 If 119894 notin 119880119894

119896and 119880

119894

119896= 119896119894

1 119896

119894

119898 a set of positive

definite matrices 119879119894119895isin R119899times119899 (119894 notin 119880

119894

119896 119895 isin 119880

119894

119896) exist such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (39)

[[[[[[[[[

[

1206011119894

1206012119894

1198731

1206013119894

120601119879

2119894minus119876 0 0

119873119879

10 119873

20

120601119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (40)

119875119894minus 119875119895ge 0 forall119895 isin 119880

119894

119906119896 119895 = 119894 (41)

Case 2 If 119894 isin 119880119894

119896(119880119894

119896= 119896119894

1 119896

119894

119898) and 119880

119894

119906119896= 0 a set of

positive definite matrices119881119894119895119897isin R119899times119899 (119894 119895 isin 119880

119894

119896 119897 isin 119880

119894

119906119896) exist

such that

119864119879

119891119875119894= 119875119879

119894119864119891ge 0 (42)

[[[[[[[[[

[

1205931119894

1205932119894

1198721

1205933119894

120593119879

2119894minus119876 0 0

119872119879

10 119872

20

120593119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (43)

Case 3 If 119894 isin 119880119894

119896and 119880

119894

119906119896= 0 a set of positive definite

matrices119882119894119895isin R119899times119899 (119894 119895 isin 119880

119894

119896) exist such that

119864119879

119891119894119875119894= 119875119879

119894119864119891119894ge 0 (44)

[[[[[[[[[

[

1205951119894

1205952119894

1198711

1205953119894

120595119879

2119894minus119876 0 0

119871119879

10 1198712

0

120595119879

31198940 0 minus120576

119894119868

]]]]]]]]]

]

lt 0 (45)

where

1206011119894= 1205931119894= 1205951119894

= [1198751119894119860119894+ 119860119879

1198941198751119894

119860119879

1198941198752119894minus 119884119879

1119894minus 119862119879

119894119884119879

2119894

1198752119894119860119894minus 1198841119894minus 1198842119894119862119894

119884119879

1119894+ 1198841119894

]

+ 119876 + 119862119879

119891119862119891+ sum

119895isin119880119894

119896

119894119895119864119879

119891(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895

1206012119894= 1205932119894= 1205952119894= [

1198751119894119860 i 0

1198752119894119860119894minus 1198841119894minus 11988421198941198621198940]

1206013119894= 1205933119894= 1205953119894= [

11987511198941198721119894

11987521198941198721119894minus 11988411198941198721119894minus 11988421198941198722119894

]

1198731= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198732= diag minus119879

119894119896119894

1

minus119879119894119896119894

119898

1198721= [119864119879

119891(119875119896119894

1

minus 119875119894) 119864119879

119891(119875119896119894

2

minus 119875119894) 119864

119879

119891(119875119896119894

119898

minus 119875119894)]

1198722= diag minus119881

119894119896119894

1119897 minus119881

119894119896119894

119898119897

1198711= [119864119879

119891(1198751minus 119875119894) 119864

119879

119891(119875119894minus1

minus 119875119894)

119864119879

119891(119875119894+1

minus 119875119894) 119864

119879

119891(119875119904minus 119875119894)]

1198712= diag minus119882

1198941 minus119882

119894119904

(46)

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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

Abstract and Applied Analysis 7

Then a suitable robust guaranteed cost observer in theform of (11) has parameters as follows

1198701119894= 119875minus1

11198941198841119894 119870

2119894= 119875minus1

21198941198842119894

(47)

and 119869lowast is a robust guaranteed cost for system (13) with GUTRMatrix (4)

Proof Define

1198601

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119879119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198731

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198732

]]]]

]

(48)

1198602

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119881119894119895119897+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198721

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198722

]]]]

]

(49)

1198603

119894=

[[[[

[

119860119879

119891119894119875119894+ 119875119894119860119891119894+ 119876 + sum

119895isin119880119894

119896

119894119895(119875119895minus 119875119894) + sum

119895isin119880119894

119896

1

41205902

119894119895119882119894119895+ 119862119879

119891119862119891

119875119894119860119891119889119894

1198711

119860119879

119891119889119894119875119894

minus119876 0

lowast lowast 1198712

]]]]

]

lt 0 (50)

Then (16) is equivalent to

1198601

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(51)

Then (19) is equivalent to

1198602

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(52)

Then (21) is equivalent to

1198603

119894+ [

[

119875119894119872119891119894

0

0

]

]

119865119894[119873119891119894 1198731198911119894 0]

+ [119873119891119894 1198651198911119894 0]119879

119865119879

119894[

[

119875119894119872119891119894

0

0

]

]

119879

lt 0

(53)

By applying Lemma 24 in [57] (50) (51) and (52) hold forall uncertainties119865i satisfying119865

119879

119894119865119894lt 119868 if and only if there exist

positive scalars 120576119894 119894 = 1 2 119904 such that

1198601

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198602

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

1198603

119894+ 120576minus1

119894[

[

119875119894119872119891119894

0

0

]

]

[

[

119875119894119872119891119894

0

0

]

]

119879

+ 120576119894[119873119891119894 1198651198911119894 0]

119879

[119873119891119894 1198651198911119894 0] lt 0

(54)

Let 119875119894= diag(119875

1119894 1198752119894) and using (47) we can conclude from

Schur complement results that the above matrix inequalitiesare equivalent to the coupled LMIs (40) (43) and (45)It further follows from Theorem 10 that 119869

lowast is a robustguaranteed cost for system (13) with (4)

Remark 12 The solution of LMIs (39)ndash(45) parameterizesthe set of the proposed robust guaranteed cost observersThis parameterized representation can be used to design theguaranteed cost observer with some additional performanceconstraints By applying the methods in [14] the suboptimal

8 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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 Abstract and Applied Analysis

guaranteed cost observer can be determined by solving acertain optimization problem This is the following theorem

Theorem 13 Consider system (13)with GUTRMatrix (4) andthe cost function (14) and suppose that the initial conditions 119903

0

and 1199091198910

are known if the following optimization problem

min1198761198751119894 1198752119894 120576119894 1198841119894 and 1198842119894

119869lowast

st LMIs (39) ndash (45)(55)

has a solution 119876 1198751119894 1198752119894 120576119894 1198841119894 and 119884

2119894 119894 = 1 2 119904

then the observer (11) is a suboptimal guaranteed costobserver for system (1) where 119869

lowast= 119909119879

1198910119864119879

1198911199030119875(1199030)1199091198910

+

tr(int0minus119889

1199091198910(119905)1199091198910(119905)119909119879

1198910119889119905119876)

Proof It follows from Theorem 11 that the observer (11)constructed in terms of the solution 119876 119875

1119894 1198752119894 120576119894 1198841119894 and

1198842119894 119894 = 1 2 119904 is a robust guaranteed cost observer By

noting that

int

0

minus119889

119909119879

1198910(119905) 119876119909

1198910(119905) 119889119905 = int

0

minus119889

tr (1199091198791198910

(119905) 1198761199091198910

(119905)) 119889119905

= tr(int0

minus119889

119909119879

1198910(119905) 1199091198910

(119905) 119889119905119876)

(56)

it follows that the suboptimal guaranteed cost observerproblem is turned into the minimization problem (55)

Remark 14 Theorem 13 gives the suboptimal guaranteed costobserver conditions of a class of linear Markovian jump-ing time-delay systems with generally incomplete transitionprobability and LMI constraints which can be easily solvedby the LMI toolbox in MATLAB

4 Numerical Example

In this section a numerical example is presented todemonstrate the effectiveness of the method mentioned inTheorem 11 Consider a 2-dimensional system (1) with 3Markovian switching modes In this numerical example thesingular system matrix is set as 119864 = [ 1 0

0 0] and the 3-mode

transition rate matrix is Λ = [minus32

2

15 21 minus36

] where Δ11 Δ31

isin

[minus015 015] Δ23 Δ33

isin [minus012 012] and Δ32

isin [minus01 01]The other system matrices are as follows

For mode 119894 = 1 there are

1198601= [

minus32 065

1 02] 119860

1198891= [

02 05

1 minus068]

1198621= [

[

12 065

minus65 19

minus021 minus18

]

]

1198621198891

= [

[

minus36 minus105

21 096

021 minus086

]

]

11987211

= [minus02

08] 119872

21= [

[

025

0875

minus2

]

]

11987311

= [minus12 31] 11987321

= [minus069 minus42]

(57)

For mode 119894 = 2 there are

1198602= [

minus1 6

2 minus36] 119860

1198892= [

minus31 minus16

3 075]

1198622= [

[

9 minus25

035 minus2

36 minus18

]

]

1198621198892

= [

[

089 minus6

minus12 09

minus24 6

]

]

11987212

= [23

minus4] 119872

22= [

[

075

minus36

25

]

]

11987312

= [minus72 minus6] 11987322

= [1 2]

(58)

For mode 119894 = 3 there are

1198603= [

minus106 29

minus03 36] 119860

1198893= [

minus56 minus12

minus3 45]

1198623= [

[

minus3 minus036

015 minus18

09 minus5

]

]

1198621198893

= [

[

minus165 5

minus12 265

minus098 minus56

]

]

11987213

= [minus82

minus03] 119872

23= [

[

minus052

25

minus36

]

]

11987313

= [105 minus5] 11987323

= [minus72 minus126]

(59)

Then we set the error state matrix 119871 = [ minus45 062 minus6

] and thepositive scalars in Theorem 11 are 120576

1= 02 120576

2= 015 120576

3=

032 According to the definitions of augmented statematricesin (12) we can easily obtain the following parameter matricesin Theorem 11 by MATLAB

11988411

= [minus84521006 00127

00127 84509001]

11988421

= [002 01291 01435

minus23520 minus04080 minus08106]

11988412

= [minus170991 269626

269626 minus246750]

11988422

= [minus200744 minus132941 529388

216893 180120 minus506693]

11988413

= [minus6751329 224456

224456 minus8976976]

11988423

= [minus136021 minus1461726 540500

minus34324 minus195125 minus103068]

1198751=[[[

[

63029 0 0 0

0 48620 0 0

0 0 22914 0

0 0 0 03169

]]]

]

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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

Abstract and Applied Analysis 9

1198752=[[[

[

08914 0 0 0

0 12505 0 0

0 0 73629 0

0 0 0 30056

]]]

]

1198753=[[[

[

30265 0 0 0

0 02156 0 0

0 0 08965 0

0 0 0 10002

]]]

]

119876 =[[[

[

05000 0 0 0

0 05001 0 0

0 0 05001 0

0 0 0 05001

]]]

]

11987911

=[[[

[

34173214 minus8707765 0 0

minus8707765 4167216 0 0

0 0 22263598 minus3207456

0 0 minus3207456 12263101

]]]

]

11987923

=[[[

[

37753231 minus27999330 0 0

minus27999330 28107685 0 0

0 0 106907366 minus107432750

0 0 minus107432750 108555053

]]]

]

11987931

=[[[

[

9518504 minus5399245 0 0

minus5399245 8962029 0 0

0 0 14773012 minus2077540

0 0 minus2077540 14791256

]]]

]

11987932

=[[[

[

21617695 minus12094164 0 0

minus12094164 20371205 0 0

0 0 14786 minus09283

0 0 minus09283 14794

]]]

]

11987933

=[[[

[

14939313 minus8398780 0 0

minus8398780 14073689 0 0

0 0 1478123 minus1336452

0 0 minus1336452 2459347

]]]

]

11988111

=[[[

[

16650 0 0 0

0 16650 0 0

0 0 16650 0

0 0 0 16650

]]]

]

11988231

=[[[

[

15426 0 0 0

0 16650 0 0

0 0 16662 0

0 0 0 16650

]]]

]

11988232

=[[[

[

15428 0 0 0

0 16650 0 0

0 0 16622 0

0 0 0 16650

]]]

]

(60)

Therefore we can design a linear memoryless observer as(11) with the constant matrices

11987011

= 119875minus1

1111988411

= [minus13409860 00020

00026 17381530]

11987021

= 119875minus1

2111988421

= [00087 00563 00626

minus74219 minus12875 minus25579]

11987012

= 119875minus1

1211988412

= [minus191823 302475

215615 minus197321]

11987022

= 119875minus1

2211988422

= [minus27264 minus18056 71899

72163 59928 minus168583]

11987013

= 119875minus1

1311988413

= [minus2231402 74186

1041076 minus41637180]

11987023

= 119875minus1

2311988423

= [minus151724 minus1630481 602900

minus34317 minus195086 minus103047]

(61)

Finally the observer (11) with the above parameter matri-ces for this numerical example is a suboptimal guaranteedcost observer byTheorems 11 and 13

5 Conclusions

In this paper the robust guaranteed cost observer problemfor a class of uncertain descriptor time-delay systems withMarkovian jumping parameters and generally uncertaintransition rates is studied by using LMI method In thisGUTR singular model each transition rate can be com-pletely unknown or only its estimate value is known Theparameterrsquos uncertainty is time varying and is assumed tobe norm-bounded Memoryless guaranteed cost observersare designed in terms of a set of linear coupled matrixinequalities The suboptimal guaranteed cost observer isdesigned by solving a certain optimization problem Ourresults can be applicable to the general Markovian jumpsystems with bounded uncertain or partly uncertain TRmatrix

Conflict of Interests

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

Acknowledgments

This research is supported by the National NaturalScience Foundations of China (10701020 1097102260973048 61272077 60974025 60673101 and 60939003)NCET-08-0755 National 863 Plan Project (2008AA04Z401 2009AA043404) SDPW IMZQWH010016the Natural Science Foundation of Shandong Province (noZR2012FM006) the Scientific and Technological Projectof Shandong Province (no 2007GG3WZ04016) and theNatural Science Foundation of Guangxi Autonomous Region(no 2012GXNSFBA053003)

10 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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 Abstract and Applied Analysis

References

[1] L Dai Singular Control Systems Springer Berlin Germany1989

[2] S Xu P V Dooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] M Fu and G Duan ldquoRobust guaranteed cost observer foruncertain descriptor time-delay systems withMarkovian jump-ing parametersrdquo Acta Automatica Sinica vol 31 pp 479ndash4832005

[4] Y Q Xia E-K Boukas P Shi and J H Zhang ldquoStabilityand stabilization of continuous-time singular hybrid systemsrdquoAutomatica vol 45 no 6 pp 1504ndash1509 2009

[5] S Xu and J Lam Robust Control and Filtering of SingularSystems Springer Berlin Germany 2006

[6] Y-Y Cao and J Lam ldquoRobust 119867infin

control of uncertainMarkovian jump systems with time-delayrdquo IEEE Transactionson Automatic Control vol 45 no 1 pp 77ndash83 2000

[7] Y Li and Y Kao ldquoStability of stochastic reaction-diffusion sys-tems with Markovian switching and impulsive perturbationsrdquoMathematical Problems in Engineering vol 2012 Article ID429568 13 pages 2012

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[9] E-K Boukas Stochastic Switching Systems Analysis and DesignBirkhauser Basel Switzerland 2005

[10] Y Kao C Wang and L Zhang ldquoDelay-dependent exponentialstability of impulsive markovian jumping Cohen-Grossbergneural networks with reaction-diffusion and mixed delaysrdquoNeural Processing Letters vol 38 no 3 pp 321ndash346 2013

[11] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012

[12] Y Kao C Wang F Zha and H Cao ldquoStability in mean ofpartial variables for stochastic reactionmdashdiffusion systems withMarkovian switchingrdquo Journal of the Franklin Institute vol 351no 1 pp 500ndash512 2014

[13] N Kumaresan and P Balasubramaniam ldquoOptimal control forstochastic nonlinear singular system using neural networksrdquoComputers amp Mathematics with Applications vol 56 no 9 pp2145ndash2154 2008

[14] L Yu and J Chu ldquoAn LMI approach to guaranteed cost controlof linear uncertain time-delay systemsrdquoAutomatica vol 35 no6 pp 1155ndash1159 1999

[15] S Yin H Luo and S Ding ldquoReal-time implementation of fault-tolerant control systems with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 64 no 5 pp 2402ndash2411 2014

[16] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[17] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[18] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic datadriven fault diagnosis and process

monitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[19] Y G Kao and C H Wang ldquoGlobal stability analysis forstochastic coupled reaction-diffusion systems on networksrdquoNonlinear Analysis Real World Applications vol 14 no 3 pp1457ndash1465 2013

[20] E K Boukas ldquoStabilization of stochastic singular nonlinearhybrid systemsrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 64 no 2 pp 217ndash228 2006

[21] E-K Boukas Control of Singular Systems with Random AbruptChanges Springer Berlin Germany 2008

[22] Y Ding H Zhu S Zhong and Y Zeng ldquoExponential mean-square stability of time-delay singular systems with Markovianswitching and nonlinear perturbationsrdquo Applied Mathematicsand Computation vol 219 no 4 pp 2350ndash2359 2012

[23] S Long S Zhong and Z Liu ldquoStochastic admissibility fora class of singular Markovian jump systems with mode-dependent time delaysrdquoAppliedMathematics and Computationvol 219 no 8 pp 4106ndash4117 2012

[24] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198682minus 119868infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[25] Z Wu P Shi H Su and J Chu ldquoStochastic synchronizationof Markovian jump neural networks with time-varying delayusing sampled-datardquo IEEE Transactions on Cybernetics vol 43no 6 pp 1796ndash1806 2013

[26] S Ma and E-K Boukas ldquoGuaranteed cost control of uncertaindiscrete-time singular Markov jump systems with indefinitequadratic costrdquo International Journal of Robust and NonlinearControl vol 21 no 9 pp 1031ndash1045 2011

[27] G Wang and Q Zhang ldquoRobust control of uncertain singularstochastic systems with Markovian switching via proportional-derivative state feedbackrdquo IET Control Theory amp Applicationsvol 6 no 8 pp 1089ndash1096 2012

[28] J Wang H Wang A Xue and R Lu ldquoDelay-dependent 119867infin

control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2013

[29] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[30] Z-GWu J H ParkH Su and J Chu ldquoStochastic stability anal-

ysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilitiesrdquoJournal of the Franklin Institute vol 349 no 9 pp 2889ndash29022012

[31] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[32] J Xiong and J Lam ldquoRobust 1198672control of Markovian jump

systems with uncertain switching probabilitiesrdquo InternationalJournal of Systems Science vol 40 no 3 pp 255ndash265 2009

[33] 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

[34] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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

Abstract and Applied Analysis 11

[35] B Du J Lam Y Zou and Z Shu ldquoStability and stabilization forMarkovian jump time-delay systems with partially unknowntransition ratesrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 60 no 2 pp 341ndash351 2013

[36] L Zhang and J Lam ldquoNecessary and sufficient conditionsfor analysis and synthesis of Markov jump linear systemswith incomplete transition descriptionsrdquo IEEE Transactions onAutomatic Control vol 55 no 7 pp 1695ndash1701 2010

[37] Y Guo and F Zhu ldquoNew results on stability and stabilizationof Markovian jump systems with partly known transitionprobabilitiesrdquoMathematical Problems in Engineering vol 2012Article ID 869842 11 pages 2012

[38] J Lin S Fei and J Shen ldquoDelay-dependent 119867infin

filtering fordiscrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilitiesrdquoSignal Processing vol 91 no 2 pp 277ndash289 2011

[39] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[40] Y Wei J Qiu H R Karimi and M Wang ldquoA new design 119867infin

filtering for contrinuous-time Markovian jump systems withtime-varying delay and partially accessible mode informationrdquoSignal Processing vol 93 pp 2392ndash2407 2013

[41] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[42] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

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

[43] L Zhang and E-K Boukas ldquoMode-dependent 119867infin

controlfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo International Journal ofRobust and Nonlinear Control vol 19 no 8 pp 868ndash883 2009

[44] Y Zhang Y He M Wu and J Zhang ldquoStabilization forMarkovian jump systems with partial information on transi-tion probability based on free-connection weighting matricesrdquoAutomatica vol 47 no 1 pp 79ndash84 2011

[45] G Zong D Yang L Hou and Q Wang ldquoRobust finite-time119867infincontrol for Markovian jump systems with partially known

transition probabilitiesrdquo Journal of the Franklin Institute vol350 no 6 pp 1562ndash1578 2013

[46] Y Ding H Zhu S Zhong and Y Zhang ldquo1198712minus 119871infin

filteringfor Markovian jump systems with time-varying delays andpartly unknown transition probabilitiesrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 7 pp3070ndash3081 2012

[47] Q Ma S Xu and Y Zou ldquoStability and synchronizationfor Markovian jump neural networks with partly unknowntransition probabilitiesrdquo Neurocomputing vol 74 pp 3403ndash3411 2011

[48] E Tian D Yue and G Wei ldquoRobust control for Markovianjump systems with partially known transition probabilities andnonlinearitiesrdquo Journal of the Franklin Institute vol 350 no 8pp 2069ndash2083 2013

[49] Y Guo and Z Wang ldquoStability of Markovian jump systemswith generally uncertain transition ratesrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2826ndash2836 2013

[50] L Zhang ldquo119867infinestimation for discrete-time piecewise homoge-

neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009

[51] Z Wu H Su and J Chu ldquostate estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 pp 2247ndash2254 2010

[52] I R Petersen and A V Savkin Robust Kalman Filteringfor Signals and Systems with Large Uncertainties BirkhauserBoston Mass USA 1999

[53] B D O Anderson and J B Moore Optimal Filtering Prentice-Hall Upper Saddle River NJ USA 1979

[54] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEETransactions on Automatic Control vol 39 no 9 pp 1971ndash19771994

[55] S Xu T Chen and J Lam ldquoRobust 119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[56] J H Kim D Oh and H Park ldquoGuaranteed cost and 119867infin

filtering for time-delay systemsrdquo in Proceeding of the AmericanControl Conference pp 4014ndash4018 IEEE Arlington Va USA2001

[57] L Xie ldquoOutput feedback119867infincontrol of systems with parameter

uncertaintyrdquo International Journal of Control vol 63 no 4 pp741ndash750 1996

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

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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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of