Research ArticleState Estimation for Time-Delay Systems with Markov JumpParameters and Missing Measurements

Yushun Tan Qiong Qi and Jinliang Liu

Department of Applied Mathematics Nanjing University of Finance and Economics Nanjing Jiangsu 210023 China

Correspondence should be addressed to Jinliang Liu liujinliang2009163com

Received 22 January 2014 Accepted 18 February 2014 Published 3 April 2014

Academic Editor Shuping He

Copyright copy 2014 Yushun Tan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the state estimation problem for a class of time-delay systems with Markovian jump parameters andmissing measurements considering the fact that data missing may occur in the process of transmission and its failure rates aregoverned by random variables satisfying certain probabilistic distribution By employing a new Lyapunov function and using theconvexity property of thematrix inequality a sufficient condition for the existence of the desired state estimator forMarkovian jumpsystems with missing measurements can be achieved by solving some linear matrix inequalities which can be easily facilitated byusing the standard numerical software Furthermore the gain of state estimator can also be derived based on the known conditionsFinally a numerical example is exploited to demonstrate the effectiveness of the proposed method

1 Introduction

As a class of multimodal systems Markovian jump systems(MJSs) have received considerable attention in the past twodecades [1ndash6] The system parameters usually jump in afinite mode set in which the transitions among differentmodes are governed by a Markov chain Due to the fact thatmany dynamical systems subject to random abrupt variationscan be modeled by MJSs many applications of MJSs canbe showed such as power systems failure prone man-ufacturing systems communication systems biochemicalsystems with diverse changes of environmental conditionsand economy system Quite a number of useful results havebeen extensively studied such as stability and stabilizationrobust control optimal control 119867

infincontrol synchroniza-

tion 119867infin

filtering and sliding mode control [7ndash19] Forexample the author in [7] studied the problem of unbiasedestimation of Markov jump systems with distributed delaysand sufficient conditions are obtained for the unbiased 119867

infin

filtering scheme to MJSs by stochastic Lyapunov-Krasovskiifunctional framework The author in [8] considered robust119867infin

control problems for stochastic fuzzy neutral MJSs with

parameters uncertainties and multiple time delays and asufficient condition and119867

infincontrol criteria are formulated in

the form of linearmatrix inequalities by selecting appropriateLyapunov functions In term of the peak-to-peak filteringproblem for a class of MJSs with uncertain parametersthe author in [9] investigated it Sufficient conditions thatthe solution of the peak-to-peak filter existed are given byusing the constructed Lyapunov functional and linear matrixinequalities More details on this topic can be found in [20]and the references therein

In recent years due to the fact that for many practicalstate estimation applications the problem of state estimationwith linear or nonlinear time-delay systems has receivedmuch attention it is of great significance to estimate systemsstates and then utilize the estimated systems states to achievecertain design objectives At the same time in the procedureof state estimator design time delays cannot be neglectedand their existence often results in a poor performance Somenice results on state estimation for time-delay systems havebeen showed in the literature [21ndash23] Meanwhile some stateestimation problem for JMSs has been hot topics so thatmany important results have been reported in the literature

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 379565 11 pageshttpdxdoiorg1011552014379565

2 Abstract and Applied Analysis

[16 24 25] The author in [16] studied the state estimationand sliding-mode control problems for continuous-timeMarkovian jump singular systems with unmeasured statesThe author in [24] concerned the problem of119867

infinestimation

for a class of Markov jump linear systems (MJLSs) with time-varying transition probabilities in discrete-time domain In[25] efficient simulation-based algorithms called particlefilters were used to solve the optimal state estimates for a classof jumpMarkov linear systemsThe author in [26] consideredstate estimation for Markovian jumping delayed continuous-time recurrent neural networks where only matrix parame-tersweremode-dependentDifferent from the studies [26 27]studied state estimation problem for a class of discrete-timeneural networks with Markovian jumping parameters andmode-dependent mixed time delay where the discrete anddistributed delays were mode-dependent

Recently Liu et al [28] studied the 119867infin

filter design forMarkovian jump systems with time-varying delays Howeverthese papers do not consider the data missing of sensor inthe process of transmission Motivated by the idea of abovepapers we will investigate the problem of state estimation forMarkovian jump systems with both time delays and missingmeasurements This work is not a simple extension of [28]to MJSs Our main difficulties come from the state estimatordesign and missing measurements analysis for the MJSsThus how to design an appropriate state estimator and how toestablish a sufficient condition for the existence of the desiredstate estimator derived are the key problems to be solvedBased on the above analysis in this paper we studied stateestimator design for MJSs with both missing measurementsand time delays via employing a new Lyapunov function andusing the convexity property of the matrix inequality Withthe proposed method we established a sufficient conditionfor the existence of the desired state estimator Furthermorethe problem of state estimator design is studied that is anobserver is designed for theMJSswithmissingmeasurementsto estimate the states

In this paper the problem of state estimator design forMJSs with interval time-varying delay is narrated A newLyapunov function is established to obtain less conservativeresults in which the lower and upper delay bound of intervaltime-varying delay is included Based on above analysis theitem int

119905

119905minus120591(119905)

119890119879

(119904)1198762(120579119905)119890(119904)119889119904 can depart into two parts to deal

with respectively and the convexity of the matrix functionsis used to avoid the conservative caused by enlarging 120591(119905) to120591119872in the deriving resultsThe rest of this paper is organized as follows Section 2

presents the problem statement and preliminaries An LMI-based sufficient condition for the existence of the desiredstate estimator derived is proposed in Section 3 A numericalexample is provided in Section 4 and we conclude this paperin Section 5

R119899 and R119899times119898 denote the 119899-dimensional Euclidean spaceand the set of 119899 times 119898 real matrices the superscript ldquo119879rdquorepresents matrix transposition sdot represents the Euclideanvector normor the inducedmatrix 2-normas appropriate 119868 isthe identitymatrix of appropriate dimensionE119909 representsthe expectation of 119909 when 119909 is a stochastic variable [ 119860 lowast

119861 119862]

denote a symmetric matrix where lowast denotes the entriesimplied by symmetry for a matrix 119861 and two symmetricmatrices 119860 and 119862 The notation 119883 gt 0 (resp 119883 ge 0) for119883 isin R119899times119899 means that the matrix119883 is real symmetric positivedefinite (resp positive semidefinite)

2 Problem Statement and Preliminaries

Fix a probability space (Ω F and P) and consider thefollowing class of uncertain linear stochastic systems withMarkovian jump parameters and time-varying delays

(119905) = 119860 (120579119905) 119909 (119905) + 119860

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119860

120596(120579119905) 120596 (119905)

119910 (119905) = 119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119862

120596(120579119905) 120596 (119905)

119911 (119905) = 119871 (120579119905) 119909 (119905) + 119871

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119871

120596(120579119905) 120596 (119905)

119909 (119905) = 120601 (119905) forall119905 isin [minus120591119872 minus120591119898]

(1)

119909(119905) isin R119899 is the state vector 119910(119905) isin R119903 is the measu-rement vector 119911(119905) isin R119901 is the signal to be estimated120596(119905) isin 119871

2[0infin) is the exogenous disturbance signal and

120579119905 is a continuous-time Markovian process which has right

continuous trajectories and takes values in a finite set S =

1 2 N with stationary transition probabilities

Prob 120579119905+ℎ

= 119895 | 120579119905= 119894 =

120587119894119895ℎ + 119900 (ℎ) 119894 = 119895

1 + 120587119894119894ℎ + 119900 (ℎ) 119894 = 119895

(2)

where ℎ gt 0 limℎrarr0

(119900(ℎ)ℎ) = 0 and 120587119894119895ge 0 for 119895 = 119894 is the

transition rate from mode 119894 at time 119905 to the mode 119895 at time119905 + ℎ and

120587119894119894= minus

119873

sum

119895=1 119895 = 119894

120587119894119895 (3)

In the system (1) the time delay 120591(119905) is a time-varying contin-uous function satisfying the following assumption

0 le 120591119898le 120591 (119905) le 120591

119872lt infin 120591 (119905) le 120583 forall119905 gt 0 (4)

where 120591119872

is the upper bound and 120591119898is the lower bound of

the communication delay and 120583 is the upper bound of changerate of communication delay

When considering the datamissing in the sensor channelthe actual output of sensormeasurements in system (1) can bedescribed as

119910 (119905) = Ξ119910 (119905)

= Ξ [119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119862

120596(120579119905) 120596 (119905)]

(5)

where Ξ = diag1205851 1205852 120585

119898 = sum

119898

119894=1120585119894119870119894 119870119894

=

diag0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119894minus1

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898minus119894

and 120585119894(119894 = 1 2 119898) are unrelated

stochastic variables taking values in [0 1] The mathematicalexpectation and variance of 120585

119894are 120585119894and 1205902

119894 respectively

Abstract and Applied Analysis 3

Remark 1 It can be seen from (5) that stochastic Ξ isintroduced to reflect the unreliable sensors which describesthe status of the whole sensor and has been extensivelystudied in the literature such as [29ndash33] Generally speakingdifferent sensor has different failure rate So it is reasonable toassume that the failure rate for each individual sensor satisfiesindividual probabilistic distribution and the elements 120585

119894(119894 =

1 2 119898) of the random matrix Ξ correspond to the statusof the 119894th sensor At one moment if 120585

119894= 1 it indicates that

the 119894th sensor is well working if 120585119894= 0 it indicates that 119894th

sensor fails completely or data missing in the sensor channelif 120585119894isin (0 1) itmeans that the 119894th sensor fails partlyTherefore

while Ξ = diag1 1 1 it means the status of the wholesensor is in good working condition Thus the model whichwe will establish in this paper is more general

In this paper considering the data missing of sensor inthe process of information communication and based on themeasurement 119910(119905) we consider the following state estimatorfor system (1)

119909 (119905) = 119860 (120579119905) 119909 (119905) + 119860

119889(120579119905) 119909 (119905 minus 120591 (119905))

+ 119866 (120579119905) (1199101(119905) minus 119910 (119905))

119910 (119905) = 119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905))

(119905) = 119871 (120579119905) 119909 (119905) + 119871

119889(120579119905) 119909 (119905 minus 120591 (119905))

(6)

where 1199101(119905) = Ξ119910(119905) = Ξ[119862(120579

119905)119909(119905) + 119862

119889(120579119905)119909(119905 minus 120591(119905))]

Remark 2 Similar to (5) we also consider the data missing ofsensor in the process of information communication for thesystem (6) of state estimation

The setS contains the various operationmodes of system(1) and for each possible value of 120579

119905= 119894 119894 isin S the matrices

connected with ldquo119894th moderdquo will be denoted by

119860119894= 119860 (120579

119905= 119894) 119860

119889119894= 119860119889(120579119905= 119894)

119860120596119894= 119860120596(120579119905= 119894)

119862119894= 119862 (120579

119905= 119894) 119862

119889119894= 119862119889(120579119905= 119894)

119862120596119894= 119862120596(120579119905= 119894)

119871119894= 119871 (120579

119905= 119894) 119871

119889119894= 119871119889(120579119905= 119894)

119871120596119894= 119871120596(120579119905= 119894)

(7)

where 119860119894 119860119889119894 119860120596119894 119862119894 119862119889119894 119862120596119894 119871119894 119871119889119894 and 119871

120596119894are constant

matrices for any 119894 isin S In this paper we assume that thejumping process 120579

119905 is accessible that is the operationmode

of system (1) is known for every 119905 ge 0

Set the estimation error 119890(119905) = 119909(119905) minus 119909(119905) and (119905) =

(119905) minus 119911(119905) Then the following error dynamics of the stateestimation system will be showed as follows

119890 (119905) = 119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119894119890 (119905) + 119866

119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905))

minus 119866119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = 119871119894119890 (119905) + 119871

119889119894119890 (119905 minus 120591 (119905)) minus 119871

120596119894120596 (119905)

(8)

where

119894= 119860119894+ 119866119894Ξ119862119894

119889119894= 119860119889119894+ 119866119894Ξ119862119889119894

120596119894= minus119860

120596119894minus 119866119894Ξ119862120596119894

(9)

The state estimation problem which is addressed in thispaper is to design a state estimator in form of (8) such that

(i) the estimation error system (8) with 120596(119905) = 0 isexponentially stable

(ii) the 119867infin

performance (119905)2lt 120574120596

2is sure for all

nonzero 120596(119905) isin 1198712[0infin) and a prescribed 120574 gt 0

under the condition 119890(119905) = 0 for all 119905 isin [minus120591119872 minus120591119898]

Before giving the main results the following lemmas anddefinitions are needed in the proof of our main results

Lemma 3 (see [34]) For any constant matrix 119877 isin R 119877 =

119877119879

gt 0 vector function 119909 [minus120591119872 0] rarr R119899 and constant

120591119872gt 0 such that the following integration is well defined then

the following inequality holds

minus 120591119872int

119905

119905minus120591119872

119879

(119904) 119877 (119904) 119889119904

le [119909 (119905)

119909 (119905 minus 120591119872)]

119879

[minus119877 119877

119877 minus119877] [

119909 (119905)

119909 (119905 minus 120591119872)]

(10)

Lemma 4 (see [35]) Suppose Ξ1 Ξ2 and Ω are constant

matrices of appropriate dimensions 0 le 120591119898le 120591(119905) le 120591

119872 then

(120591 (119905) minus 120591119898) Ξ1+ (120591119872minus 120591 (119905)) Ξ

2+ Ω lt 0 (11)

if and only if the following two inequalities hold

(120591119872minus 120591119898) Ξ1+ Ω lt 0

(120591119872minus 120591119898) Ξ2+ Ω lt 0

(12)

Definition 5 The system (8) is considered to be exponentiallystable in the mean-square sense (EMSS) if there existconstants 120582 gt 0 120572 gt 0 such that 119905 gt 0

119864 119909 (119905)2

le 120572119890minus120582119905 supminus120591119872lt119904lt0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(13)

4 Abstract and Applied Analysis

Definition 6 For a given function119881 1198621198871198650([minus120591119872 0] 119877

119899

)times119878 rarr

119877 its infinitesimal operatorL [36] is defined as

L119881 (119909119905) = limΔrarr0

+

1

Δ[119864 (119881 (119909

119905+Δ| 119909119905) minus 119881 (119909

119905))] (14)

3 Main Results

Theorem 7 For some given constants 0 le 120591119898le 120591119872and 120574 the

system (8) is exponentially mean-square stable (EMSS) with aprescribed 119867

infinperformance 120574 if there exist 119875

119894gt 0 119876

0gt 0

1198761gt 0119876

2119894gt 0 119877

0gt 0 119877

1gt 0119885

1gt 0 119885

2gt 0119872

119894119896gt 0 and

119873119894119896gt 0 (119894 isin S 119896 = 1 2 5) with appropriate dimensions

so that the following matrix inequalities hold

Ψ =

[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]

]

lt 0

119904 = 1 2

(15)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (16)

where

Ψ11= 119875119894119894+ 119879

119894119875119894+ 1198760+ 1198761+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941 minus1198731198941 119875119879

119894120596119894]119879

Ψ22=

[[[

[

minus (1 minus 120583)1198762119894minus1198721198942minus119872119879

1198942+ 1198731198942+ 119873119879

1198942lowast lowast lowast

minus1198721198943+119872119879

1198942+ 1198731198943

minus1198760minus 1198770+1198721198943+119872119879

1198943lowast lowast

minus1198721198944+ 1198731198944minus 119873119879

11989421198721198944minus 119873119879

1198943minus1198761minus 1198731198944minus 119873119879

1198944lowast

minus1198721198945+ 1198731198945

1198721198945

minus1198731198945

minus1205742

119868

]]]

]

Ψ31= [

[

1205911198981198770119894

radic1205751198771119894

119871119894

]

]

Ψ32= [

[

1205911198981198770119889119894

0 0 1205911198981198770120596119894

radic1205751198771119889119894

0 0 radic1205751198771120596119894

119871119889119894

0 0 minus119871120596119894

]

]

Ψ33= diag minus119877

0 minus1198771 minus119868

Ψ41(1) = radic120575119872

119879

1198941 Ψ

41(2) = radic120575119873

119879

1198941 120575 = 120591

119872minus 120591119898

Ψ42(1) = [radic120575119872

119879

1198942

radic120575119872119879

1198943

radic120575119872119879

1198944

radic120575119872119879

1198945] Ψ

42(2) = [radic120575119873

119879

1198942

radic120575119873119879

1198943

radic120575119873119879

1198944

radic120575119873119879

1198945]

Ψ51= [1205911198981205901119862119879

119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596] Ψ

55= diag minus119877

0 minus1198770 minus119877

0

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596] Ψ

66= diag minus119877

1 minus1198771 minus119877

1

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119889119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119866119879

119894119877119879

0 minus1205911198981205902119862119879

120596119894119870119879

2119866119879

119894119877119879

0 minus120591

119898120590119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

0]119879

Θ6119889= [radic120575120590

1119862119879

119889119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119889119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119866119879

119894119877119879

1 minusradic120575120590

2119862119879

120596119894119870119879

2119866119879

119894119877119879

1 minusradic120575120590

119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

1]119879

(17)

Proof Introduce a new vector

120577119879

(119905)=[ 119890119879

(119905) 119890119879

(119905minus120591 (119905)) 119890119879

(119905minus120591119898) 119890119879

(119905minus120591119872) 120596119879

(119905) ]

(18)

and two matricesΓ1= [119894119889119894

0 0 120596119894]

Γ2= [119871119894119871119889119894

0 0 minus119871120596119894]

(19)

Abstract and Applied Analysis 5

The system (8) can be rewritten as

119890 (119896) = Γ1120577 (119905) + 119866

119894(Ξ minus Ξ) 119862

119894119890 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905)) minus 119866

119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = Γ2120577 (119905)

(20)

Let119909119905(119904) = 119909(119905+119904) (minus120591(119905) le 119904 le 0)Then the same as [37]

(119909119905 120579119905) 119905 ge 0 is a Markov process Choose the following

Lyapunov functional candidate

119881 (119909119905 120579119905) =

4

sum

119894=1

119881119894(119909119905 120579119905) (21)

where

1198811(119909119905 120579119905) = 119890119879

(119905) 119875 (120579119905) 119890 (119905)

1198812(119909119905 120579119905) = int

119905

119905minus120591119898

119890119879

(119904) 1198760119890 (119904) 119889119904 + int

119905

119905minus120591119872

119890119879

(119904) 1198761119890 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119890119879

(119904) 1198762(120579119905) 119890 (119904) 119889119904

1198813(119909119905 120579119905) = 120591119898int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198770119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198771119890 (V) 119889V 119889119904

1198814(119909119905 120579119905) = int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198851119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198852119890 (V) 119889V 119889119904

(22)

LetL be the weak infinite generator of the random pro-cess 119909

119905 120579119905 Then for each 120579

119905= 119894 119894 isin S taking expectation

on it we obtain

E L119881 (119909119905 120579119905)

le 119890119879

(119905)(2119875119894119894

+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120591

119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

(23)

Note that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

= int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ int

119905

119905minus120591119898

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

(24)

From (16) and (24) we can derive that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904 minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

= int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

le int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198852

]

]

119890 (119904) 119889119904 lt 0

(25)

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

[16 24 25] The author in [16] studied the state estimationand sliding-mode control problems for continuous-timeMarkovian jump singular systems with unmeasured statesThe author in [24] concerned the problem of119867

infinestimation

for a class of Markov jump linear systems (MJLSs) with time-varying transition probabilities in discrete-time domain In[25] efficient simulation-based algorithms called particlefilters were used to solve the optimal state estimates for a classof jumpMarkov linear systemsThe author in [26] consideredstate estimation for Markovian jumping delayed continuous-time recurrent neural networks where only matrix parame-tersweremode-dependentDifferent from the studies [26 27]studied state estimation problem for a class of discrete-timeneural networks with Markovian jumping parameters andmode-dependent mixed time delay where the discrete anddistributed delays were mode-dependent

Recently Liu et al [28] studied the 119867infin

filter design forMarkovian jump systems with time-varying delays Howeverthese papers do not consider the data missing of sensor inthe process of transmission Motivated by the idea of abovepapers we will investigate the problem of state estimation forMarkovian jump systems with both time delays and missingmeasurements This work is not a simple extension of [28]to MJSs Our main difficulties come from the state estimatordesign and missing measurements analysis for the MJSsThus how to design an appropriate state estimator and how toestablish a sufficient condition for the existence of the desiredstate estimator derived are the key problems to be solvedBased on the above analysis in this paper we studied stateestimator design for MJSs with both missing measurementsand time delays via employing a new Lyapunov function andusing the convexity property of the matrix inequality Withthe proposed method we established a sufficient conditionfor the existence of the desired state estimator Furthermorethe problem of state estimator design is studied that is anobserver is designed for theMJSswithmissingmeasurementsto estimate the states

In this paper the problem of state estimator design forMJSs with interval time-varying delay is narrated A newLyapunov function is established to obtain less conservativeresults in which the lower and upper delay bound of intervaltime-varying delay is included Based on above analysis theitem int

119905

119905minus120591(119905)

119890119879

(119904)1198762(120579119905)119890(119904)119889119904 can depart into two parts to deal

with respectively and the convexity of the matrix functionsis used to avoid the conservative caused by enlarging 120591(119905) to120591119872in the deriving resultsThe rest of this paper is organized as follows Section 2

presents the problem statement and preliminaries An LMI-based sufficient condition for the existence of the desiredstate estimator derived is proposed in Section 3 A numericalexample is provided in Section 4 and we conclude this paperin Section 5

R119899 and R119899times119898 denote the 119899-dimensional Euclidean spaceand the set of 119899 times 119898 real matrices the superscript ldquo119879rdquorepresents matrix transposition sdot represents the Euclideanvector normor the inducedmatrix 2-normas appropriate 119868 isthe identitymatrix of appropriate dimensionE119909 representsthe expectation of 119909 when 119909 is a stochastic variable [ 119860 lowast

119861 119862]

denote a symmetric matrix where lowast denotes the entriesimplied by symmetry for a matrix 119861 and two symmetricmatrices 119860 and 119862 The notation 119883 gt 0 (resp 119883 ge 0) for119883 isin R119899times119899 means that the matrix119883 is real symmetric positivedefinite (resp positive semidefinite)

2 Problem Statement and Preliminaries

Fix a probability space (Ω F and P) and consider thefollowing class of uncertain linear stochastic systems withMarkovian jump parameters and time-varying delays

(119905) = 119860 (120579119905) 119909 (119905) + 119860

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119860

120596(120579119905) 120596 (119905)

119910 (119905) = 119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119862

120596(120579119905) 120596 (119905)

119911 (119905) = 119871 (120579119905) 119909 (119905) + 119871

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119871

120596(120579119905) 120596 (119905)

119909 (119905) = 120601 (119905) forall119905 isin [minus120591119872 minus120591119898]

(1)

119909(119905) isin R119899 is the state vector 119910(119905) isin R119903 is the measu-rement vector 119911(119905) isin R119901 is the signal to be estimated120596(119905) isin 119871

2[0infin) is the exogenous disturbance signal and

120579119905 is a continuous-time Markovian process which has right

continuous trajectories and takes values in a finite set S =

1 2 N with stationary transition probabilities

Prob 120579119905+ℎ

= 119895 | 120579119905= 119894 =

120587119894119895ℎ + 119900 (ℎ) 119894 = 119895

1 + 120587119894119894ℎ + 119900 (ℎ) 119894 = 119895

(2)

where ℎ gt 0 limℎrarr0

(119900(ℎ)ℎ) = 0 and 120587119894119895ge 0 for 119895 = 119894 is the

transition rate from mode 119894 at time 119905 to the mode 119895 at time119905 + ℎ and

120587119894119894= minus

119873

sum

119895=1 119895 = 119894

120587119894119895 (3)

In the system (1) the time delay 120591(119905) is a time-varying contin-uous function satisfying the following assumption

0 le 120591119898le 120591 (119905) le 120591

119872lt infin 120591 (119905) le 120583 forall119905 gt 0 (4)

where 120591119872

is the upper bound and 120591119898is the lower bound of

the communication delay and 120583 is the upper bound of changerate of communication delay

When considering the datamissing in the sensor channelthe actual output of sensormeasurements in system (1) can bedescribed as

119910 (119905) = Ξ119910 (119905)

= Ξ [119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905)) + 119862

120596(120579119905) 120596 (119905)]

(5)

where Ξ = diag1205851 1205852 120585

119898 = sum

119898

119894=1120585119894119870119894 119870119894

=

diag0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119894minus1

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898minus119894

and 120585119894(119894 = 1 2 119898) are unrelated

stochastic variables taking values in [0 1] The mathematicalexpectation and variance of 120585

119894are 120585119894and 1205902

119894 respectively

Abstract and Applied Analysis 3

Remark 1 It can be seen from (5) that stochastic Ξ isintroduced to reflect the unreliable sensors which describesthe status of the whole sensor and has been extensivelystudied in the literature such as [29ndash33] Generally speakingdifferent sensor has different failure rate So it is reasonable toassume that the failure rate for each individual sensor satisfiesindividual probabilistic distribution and the elements 120585

119894(119894 =

1 2 119898) of the random matrix Ξ correspond to the statusof the 119894th sensor At one moment if 120585

119894= 1 it indicates that

the 119894th sensor is well working if 120585119894= 0 it indicates that 119894th

sensor fails completely or data missing in the sensor channelif 120585119894isin (0 1) itmeans that the 119894th sensor fails partlyTherefore

while Ξ = diag1 1 1 it means the status of the wholesensor is in good working condition Thus the model whichwe will establish in this paper is more general

In this paper considering the data missing of sensor inthe process of information communication and based on themeasurement 119910(119905) we consider the following state estimatorfor system (1)

119909 (119905) = 119860 (120579119905) 119909 (119905) + 119860

119889(120579119905) 119909 (119905 minus 120591 (119905))

+ 119866 (120579119905) (1199101(119905) minus 119910 (119905))

119910 (119905) = 119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905))

(119905) = 119871 (120579119905) 119909 (119905) + 119871

119889(120579119905) 119909 (119905 minus 120591 (119905))

(6)

where 1199101(119905) = Ξ119910(119905) = Ξ[119862(120579

119905)119909(119905) + 119862

119889(120579119905)119909(119905 minus 120591(119905))]

Remark 2 Similar to (5) we also consider the data missing ofsensor in the process of information communication for thesystem (6) of state estimation

The setS contains the various operationmodes of system(1) and for each possible value of 120579

119905= 119894 119894 isin S the matrices

connected with ldquo119894th moderdquo will be denoted by

119860119894= 119860 (120579

119905= 119894) 119860

119889119894= 119860119889(120579119905= 119894)

119860120596119894= 119860120596(120579119905= 119894)

119862119894= 119862 (120579

119905= 119894) 119862

119889119894= 119862119889(120579119905= 119894)

119862120596119894= 119862120596(120579119905= 119894)

119871119894= 119871 (120579

119905= 119894) 119871

119889119894= 119871119889(120579119905= 119894)

119871120596119894= 119871120596(120579119905= 119894)

(7)

where 119860119894 119860119889119894 119860120596119894 119862119894 119862119889119894 119862120596119894 119871119894 119871119889119894 and 119871

120596119894are constant

matrices for any 119894 isin S In this paper we assume that thejumping process 120579

119905 is accessible that is the operationmode

of system (1) is known for every 119905 ge 0

Set the estimation error 119890(119905) = 119909(119905) minus 119909(119905) and (119905) =

(119905) minus 119911(119905) Then the following error dynamics of the stateestimation system will be showed as follows

119890 (119905) = 119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119894119890 (119905) + 119866

119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905))

minus 119866119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = 119871119894119890 (119905) + 119871

119889119894119890 (119905 minus 120591 (119905)) minus 119871

120596119894120596 (119905)

(8)

where

119894= 119860119894+ 119866119894Ξ119862119894

119889119894= 119860119889119894+ 119866119894Ξ119862119889119894

120596119894= minus119860

120596119894minus 119866119894Ξ119862120596119894

(9)

The state estimation problem which is addressed in thispaper is to design a state estimator in form of (8) such that

(i) the estimation error system (8) with 120596(119905) = 0 isexponentially stable

(ii) the 119867infin

performance (119905)2lt 120574120596

2is sure for all

nonzero 120596(119905) isin 1198712[0infin) and a prescribed 120574 gt 0

under the condition 119890(119905) = 0 for all 119905 isin [minus120591119872 minus120591119898]

Before giving the main results the following lemmas anddefinitions are needed in the proof of our main results

Lemma 3 (see [34]) For any constant matrix 119877 isin R 119877 =

119877119879

gt 0 vector function 119909 [minus120591119872 0] rarr R119899 and constant

120591119872gt 0 such that the following integration is well defined then

the following inequality holds

minus 120591119872int

119905

119905minus120591119872

119879

(119904) 119877 (119904) 119889119904

le [119909 (119905)

119909 (119905 minus 120591119872)]

119879

[minus119877 119877

119877 minus119877] [

119909 (119905)

119909 (119905 minus 120591119872)]

(10)

Lemma 4 (see [35]) Suppose Ξ1 Ξ2 and Ω are constant

matrices of appropriate dimensions 0 le 120591119898le 120591(119905) le 120591

119872 then

(120591 (119905) minus 120591119898) Ξ1+ (120591119872minus 120591 (119905)) Ξ

2+ Ω lt 0 (11)

if and only if the following two inequalities hold

(120591119872minus 120591119898) Ξ1+ Ω lt 0

(120591119872minus 120591119898) Ξ2+ Ω lt 0

(12)

Definition 5 The system (8) is considered to be exponentiallystable in the mean-square sense (EMSS) if there existconstants 120582 gt 0 120572 gt 0 such that 119905 gt 0

119864 119909 (119905)2

le 120572119890minus120582119905 supminus120591119872lt119904lt0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(13)

4 Abstract and Applied Analysis

Definition 6 For a given function119881 1198621198871198650([minus120591119872 0] 119877

119899

)times119878 rarr

119877 its infinitesimal operatorL [36] is defined as

L119881 (119909119905) = limΔrarr0

+

1

Δ[119864 (119881 (119909

119905+Δ| 119909119905) minus 119881 (119909

119905))] (14)

3 Main Results

Theorem 7 For some given constants 0 le 120591119898le 120591119872and 120574 the

system (8) is exponentially mean-square stable (EMSS) with aprescribed 119867

infinperformance 120574 if there exist 119875

119894gt 0 119876

0gt 0

1198761gt 0119876

2119894gt 0 119877

0gt 0 119877

1gt 0119885

1gt 0 119885

2gt 0119872

119894119896gt 0 and

119873119894119896gt 0 (119894 isin S 119896 = 1 2 5) with appropriate dimensions

so that the following matrix inequalities hold

Ψ =

[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]

]

lt 0

119904 = 1 2

(15)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (16)

where

Ψ11= 119875119894119894+ 119879

119894119875119894+ 1198760+ 1198761+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941 minus1198731198941 119875119879

119894120596119894]119879

Ψ22=

[[[

[

minus (1 minus 120583)1198762119894minus1198721198942minus119872119879

1198942+ 1198731198942+ 119873119879

1198942lowast lowast lowast

minus1198721198943+119872119879

1198942+ 1198731198943

minus1198760minus 1198770+1198721198943+119872119879

1198943lowast lowast

minus1198721198944+ 1198731198944minus 119873119879

11989421198721198944minus 119873119879

1198943minus1198761minus 1198731198944minus 119873119879

1198944lowast

minus1198721198945+ 1198731198945

1198721198945

minus1198731198945

minus1205742

119868

]]]

]

Ψ31= [

[

1205911198981198770119894

radic1205751198771119894

119871119894

]

]

Ψ32= [

[

1205911198981198770119889119894

0 0 1205911198981198770120596119894

radic1205751198771119889119894

0 0 radic1205751198771120596119894

119871119889119894

0 0 minus119871120596119894

]

]

Ψ33= diag minus119877

0 minus1198771 minus119868

Ψ41(1) = radic120575119872

119879

1198941 Ψ

41(2) = radic120575119873

119879

1198941 120575 = 120591

119872minus 120591119898

Ψ42(1) = [radic120575119872

119879

1198942

radic120575119872119879

1198943

radic120575119872119879

1198944

radic120575119872119879

1198945] Ψ

42(2) = [radic120575119873

119879

1198942

radic120575119873119879

1198943

radic120575119873119879

1198944

radic120575119873119879

1198945]

Ψ51= [1205911198981205901119862119879

119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596] Ψ

55= diag minus119877

0 minus1198770 minus119877

0

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596] Ψ

66= diag minus119877

1 minus1198771 minus119877

1

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119889119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119866119879

119894119877119879

0 minus1205911198981205902119862119879

120596119894119870119879

2119866119879

119894119877119879

0 minus120591

119898120590119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

0]119879

Θ6119889= [radic120575120590

1119862119879

119889119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119889119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119866119879

119894119877119879

1 minusradic120575120590

2119862119879

120596119894119870119879

2119866119879

119894119877119879

1 minusradic120575120590

119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

1]119879

(17)

Proof Introduce a new vector

120577119879

(119905)=[ 119890119879

(119905) 119890119879

(119905minus120591 (119905)) 119890119879

(119905minus120591119898) 119890119879

(119905minus120591119872) 120596119879

(119905) ]

(18)

and two matricesΓ1= [119894119889119894

0 0 120596119894]

Γ2= [119871119894119871119889119894

0 0 minus119871120596119894]

(19)

Abstract and Applied Analysis 5

The system (8) can be rewritten as

119890 (119896) = Γ1120577 (119905) + 119866

119894(Ξ minus Ξ) 119862

119894119890 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905)) minus 119866

119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = Γ2120577 (119905)

(20)

Let119909119905(119904) = 119909(119905+119904) (minus120591(119905) le 119904 le 0)Then the same as [37]

(119909119905 120579119905) 119905 ge 0 is a Markov process Choose the following

Lyapunov functional candidate

119881 (119909119905 120579119905) =

4

sum

119894=1

119881119894(119909119905 120579119905) (21)

where

1198811(119909119905 120579119905) = 119890119879

(119905) 119875 (120579119905) 119890 (119905)

1198812(119909119905 120579119905) = int

119905

119905minus120591119898

119890119879

(119904) 1198760119890 (119904) 119889119904 + int

119905

119905minus120591119872

119890119879

(119904) 1198761119890 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119890119879

(119904) 1198762(120579119905) 119890 (119904) 119889119904

1198813(119909119905 120579119905) = 120591119898int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198770119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198771119890 (V) 119889V 119889119904

1198814(119909119905 120579119905) = int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198851119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198852119890 (V) 119889V 119889119904

(22)

LetL be the weak infinite generator of the random pro-cess 119909

119905 120579119905 Then for each 120579

119905= 119894 119894 isin S taking expectation

on it we obtain

E L119881 (119909119905 120579119905)

le 119890119879

(119905)(2119875119894119894

+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120591

119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

(23)

Note that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

= int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ int

119905

119905minus120591119898

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

(24)

From (16) and (24) we can derive that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904 minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

= int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

le int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198852

]

]

119890 (119904) 119889119904 lt 0

(25)

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Remark 1 It can be seen from (5) that stochastic Ξ isintroduced to reflect the unreliable sensors which describesthe status of the whole sensor and has been extensivelystudied in the literature such as [29ndash33] Generally speakingdifferent sensor has different failure rate So it is reasonable toassume that the failure rate for each individual sensor satisfiesindividual probabilistic distribution and the elements 120585

119894(119894 =

1 2 119898) of the random matrix Ξ correspond to the statusof the 119894th sensor At one moment if 120585

119894= 1 it indicates that

the 119894th sensor is well working if 120585119894= 0 it indicates that 119894th

sensor fails completely or data missing in the sensor channelif 120585119894isin (0 1) itmeans that the 119894th sensor fails partlyTherefore

while Ξ = diag1 1 1 it means the status of the wholesensor is in good working condition Thus the model whichwe will establish in this paper is more general

In this paper considering the data missing of sensor inthe process of information communication and based on themeasurement 119910(119905) we consider the following state estimatorfor system (1)

119909 (119905) = 119860 (120579119905) 119909 (119905) + 119860

119889(120579119905) 119909 (119905 minus 120591 (119905))

+ 119866 (120579119905) (1199101(119905) minus 119910 (119905))

119910 (119905) = 119862 (120579119905) 119909 (119905) + 119862

119889(120579119905) 119909 (119905 minus 120591 (119905))

(119905) = 119871 (120579119905) 119909 (119905) + 119871

119889(120579119905) 119909 (119905 minus 120591 (119905))

(6)

where 1199101(119905) = Ξ119910(119905) = Ξ[119862(120579

119905)119909(119905) + 119862

119889(120579119905)119909(119905 minus 120591(119905))]

Remark 2 Similar to (5) we also consider the data missing ofsensor in the process of information communication for thesystem (6) of state estimation

The setS contains the various operationmodes of system(1) and for each possible value of 120579

119905= 119894 119894 isin S the matrices

connected with ldquo119894th moderdquo will be denoted by

119860119894= 119860 (120579

119905= 119894) 119860

119889119894= 119860119889(120579119905= 119894)

119860120596119894= 119860120596(120579119905= 119894)

119862119894= 119862 (120579

119905= 119894) 119862

119889119894= 119862119889(120579119905= 119894)

119862120596119894= 119862120596(120579119905= 119894)

119871119894= 119871 (120579

119905= 119894) 119871

119889119894= 119871119889(120579119905= 119894)

119871120596119894= 119871120596(120579119905= 119894)

(7)

where 119860119894 119860119889119894 119860120596119894 119862119894 119862119889119894 119862120596119894 119871119894 119871119889119894 and 119871

120596119894are constant

matrices for any 119894 isin S In this paper we assume that thejumping process 120579

119905 is accessible that is the operationmode

of system (1) is known for every 119905 ge 0

Set the estimation error 119890(119905) = 119909(119905) minus 119909(119905) and (119905) =

(119905) minus 119911(119905) Then the following error dynamics of the stateestimation system will be showed as follows

119890 (119905) = 119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119894119890 (119905) + 119866

119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905))

minus 119866119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = 119871119894119890 (119905) + 119871

119889119894119890 (119905 minus 120591 (119905)) minus 119871

120596119894120596 (119905)

(8)

where

119894= 119860119894+ 119866119894Ξ119862119894

119889119894= 119860119889119894+ 119866119894Ξ119862119889119894

120596119894= minus119860

120596119894minus 119866119894Ξ119862120596119894

(9)

The state estimation problem which is addressed in thispaper is to design a state estimator in form of (8) such that

(i) the estimation error system (8) with 120596(119905) = 0 isexponentially stable

(ii) the 119867infin

performance (119905)2lt 120574120596

2is sure for all

nonzero 120596(119905) isin 1198712[0infin) and a prescribed 120574 gt 0

under the condition 119890(119905) = 0 for all 119905 isin [minus120591119872 minus120591119898]

Before giving the main results the following lemmas anddefinitions are needed in the proof of our main results

Lemma 3 (see [34]) For any constant matrix 119877 isin R 119877 =

119877119879

gt 0 vector function 119909 [minus120591119872 0] rarr R119899 and constant

120591119872gt 0 such that the following integration is well defined then

the following inequality holds

minus 120591119872int

119905

119905minus120591119872

119879

(119904) 119877 (119904) 119889119904

le [119909 (119905)

119909 (119905 minus 120591119872)]

119879

[minus119877 119877

119877 minus119877] [

119909 (119905)

119909 (119905 minus 120591119872)]

(10)

Lemma 4 (see [35]) Suppose Ξ1 Ξ2 and Ω are constant

matrices of appropriate dimensions 0 le 120591119898le 120591(119905) le 120591

119872 then

(120591 (119905) minus 120591119898) Ξ1+ (120591119872minus 120591 (119905)) Ξ

2+ Ω lt 0 (11)

if and only if the following two inequalities hold

(120591119872minus 120591119898) Ξ1+ Ω lt 0

(120591119872minus 120591119898) Ξ2+ Ω lt 0

(12)

Definition 5 The system (8) is considered to be exponentiallystable in the mean-square sense (EMSS) if there existconstants 120582 gt 0 120572 gt 0 such that 119905 gt 0

119864 119909 (119905)2

le 120572119890minus120582119905 supminus120591119872lt119904lt0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(13)

4 Abstract and Applied Analysis

Definition 6 For a given function119881 1198621198871198650([minus120591119872 0] 119877

119899

)times119878 rarr

119877 its infinitesimal operatorL [36] is defined as

L119881 (119909119905) = limΔrarr0

+

1

Δ[119864 (119881 (119909

119905+Δ| 119909119905) minus 119881 (119909

119905))] (14)

3 Main Results

Theorem 7 For some given constants 0 le 120591119898le 120591119872and 120574 the

system (8) is exponentially mean-square stable (EMSS) with aprescribed 119867

infinperformance 120574 if there exist 119875

119894gt 0 119876

0gt 0

1198761gt 0119876

2119894gt 0 119877

0gt 0 119877

1gt 0119885

1gt 0 119885

2gt 0119872

119894119896gt 0 and

119873119894119896gt 0 (119894 isin S 119896 = 1 2 5) with appropriate dimensions

so that the following matrix inequalities hold

Ψ =

[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]

]

lt 0

119904 = 1 2

(15)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (16)

where

Ψ11= 119875119894119894+ 119879

119894119875119894+ 1198760+ 1198761+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941 minus1198731198941 119875119879

119894120596119894]119879

Ψ22=

[[[

[

minus (1 minus 120583)1198762119894minus1198721198942minus119872119879

1198942+ 1198731198942+ 119873119879

1198942lowast lowast lowast

minus1198721198943+119872119879

1198942+ 1198731198943

minus1198760minus 1198770+1198721198943+119872119879

1198943lowast lowast

minus1198721198944+ 1198731198944minus 119873119879

11989421198721198944minus 119873119879

1198943minus1198761minus 1198731198944minus 119873119879

1198944lowast

minus1198721198945+ 1198731198945

1198721198945

minus1198731198945

minus1205742

119868

]]]

]

Ψ31= [

[

1205911198981198770119894

radic1205751198771119894

119871119894

]

]

Ψ32= [

[

1205911198981198770119889119894

0 0 1205911198981198770120596119894

radic1205751198771119889119894

0 0 radic1205751198771120596119894

119871119889119894

0 0 minus119871120596119894

]

]

Ψ33= diag minus119877

0 minus1198771 minus119868

Ψ41(1) = radic120575119872

119879

1198941 Ψ

41(2) = radic120575119873

119879

1198941 120575 = 120591

119872minus 120591119898

Ψ42(1) = [radic120575119872

119879

1198942

radic120575119872119879

1198943

radic120575119872119879

1198944

radic120575119872119879

1198945] Ψ

42(2) = [radic120575119873

119879

1198942

radic120575119873119879

1198943

radic120575119873119879

1198944

radic120575119873119879

1198945]

Ψ51= [1205911198981205901119862119879

119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596] Ψ

55= diag minus119877

0 minus1198770 minus119877

0

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596] Ψ

66= diag minus119877

1 minus1198771 minus119877

1

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119889119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119866119879

119894119877119879

0 minus1205911198981205902119862119879

120596119894119870119879

2119866119879

119894119877119879

0 minus120591

119898120590119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

0]119879

Θ6119889= [radic120575120590

1119862119879

119889119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119889119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119866119879

119894119877119879

1 minusradic120575120590

2119862119879

120596119894119870119879

2119866119879

119894119877119879

1 minusradic120575120590

119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

1]119879

(17)

Proof Introduce a new vector

120577119879

(119905)=[ 119890119879

(119905) 119890119879

(119905minus120591 (119905)) 119890119879

(119905minus120591119898) 119890119879

(119905minus120591119872) 120596119879

(119905) ]

(18)

and two matricesΓ1= [119894119889119894

0 0 120596119894]

Γ2= [119871119894119871119889119894

0 0 minus119871120596119894]

(19)

Abstract and Applied Analysis 5

The system (8) can be rewritten as

119890 (119896) = Γ1120577 (119905) + 119866

119894(Ξ minus Ξ) 119862

119894119890 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905)) minus 119866

119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = Γ2120577 (119905)

(20)

Let119909119905(119904) = 119909(119905+119904) (minus120591(119905) le 119904 le 0)Then the same as [37]

(119909119905 120579119905) 119905 ge 0 is a Markov process Choose the following

Lyapunov functional candidate

119881 (119909119905 120579119905) =

4

sum

119894=1

119881119894(119909119905 120579119905) (21)

where

1198811(119909119905 120579119905) = 119890119879

(119905) 119875 (120579119905) 119890 (119905)

1198812(119909119905 120579119905) = int

119905

119905minus120591119898

119890119879

(119904) 1198760119890 (119904) 119889119904 + int

119905

119905minus120591119872

119890119879

(119904) 1198761119890 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119890119879

(119904) 1198762(120579119905) 119890 (119904) 119889119904

1198813(119909119905 120579119905) = 120591119898int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198770119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198771119890 (V) 119889V 119889119904

1198814(119909119905 120579119905) = int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198851119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198852119890 (V) 119889V 119889119904

(22)

LetL be the weak infinite generator of the random pro-cess 119909

119905 120579119905 Then for each 120579

119905= 119894 119894 isin S taking expectation

on it we obtain

E L119881 (119909119905 120579119905)

le 119890119879

(119905)(2119875119894119894

+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120591

119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

(23)

Note that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

= int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ int

119905

119905minus120591119898

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

(24)

From (16) and (24) we can derive that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904 minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

= int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

le int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198852

]

]

119890 (119904) 119889119904 lt 0

(25)

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Definition 6 For a given function119881 1198621198871198650([minus120591119872 0] 119877

119899

)times119878 rarr

119877 its infinitesimal operatorL [36] is defined as

L119881 (119909119905) = limΔrarr0

+

1

Δ[119864 (119881 (119909

119905+Δ| 119909119905) minus 119881 (119909

119905))] (14)

3 Main Results

Theorem 7 For some given constants 0 le 120591119898le 120591119872and 120574 the

system (8) is exponentially mean-square stable (EMSS) with aprescribed 119867

infinperformance 120574 if there exist 119875

119894gt 0 119876

0gt 0

1198761gt 0119876

2119894gt 0 119877

0gt 0 119877

1gt 0119885

1gt 0 119885

2gt 0119872

119894119896gt 0 and

119873119894119896gt 0 (119894 isin S 119896 = 1 2 5) with appropriate dimensions

so that the following matrix inequalities hold

Ψ =

[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]

]

lt 0

119904 = 1 2

(15)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (16)

where

Ψ11= 119875119894119894+ 119879

119894119875119894+ 1198760+ 1198761+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941 minus1198731198941 119875119879

119894120596119894]119879

Ψ22=

[[[

[

minus (1 minus 120583)1198762119894minus1198721198942minus119872119879

1198942+ 1198731198942+ 119873119879

1198942lowast lowast lowast

minus1198721198943+119872119879

1198942+ 1198731198943

minus1198760minus 1198770+1198721198943+119872119879

1198943lowast lowast

minus1198721198944+ 1198731198944minus 119873119879

11989421198721198944minus 119873119879

1198943minus1198761minus 1198731198944minus 119873119879

1198944lowast

minus1198721198945+ 1198731198945

1198721198945

minus1198731198945

minus1205742

119868

]]]

]

Ψ31= [

[

1205911198981198770119894

radic1205751198771119894

119871119894

]

]

Ψ32= [

[

1205911198981198770119889119894

0 0 1205911198981198770120596119894

radic1205751198771119889119894

0 0 radic1205751198771120596119894

119871119889119894

0 0 minus119871120596119894

]

]

Ψ33= diag minus119877

0 minus1198771 minus119868

Ψ41(1) = radic120575119872

119879

1198941 Ψ

41(2) = radic120575119873

119879

1198941 120575 = 120591

119872minus 120591119898

Ψ42(1) = [radic120575119872

119879

1198942

radic120575119872119879

1198943

radic120575119872119879

1198944

radic120575119872119879

1198945] Ψ

42(2) = [radic120575119873

119879

1198942

radic120575119873119879

1198943

radic120575119873119879

1198944

radic120575119873119879

1198945]

Ψ51= [1205911198981205901119862119879

119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596] Ψ

55= diag minus119877

0 minus1198770 minus119877

0

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596] Ψ

66= diag minus119877

1 minus1198771 minus119877

1

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119866119879

119894119877119879

0 1205911198981205902119862119879

119889119894119870119879

2119866119879

119894119877119879

0 120591

119898120590119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

0 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119866119879

119894119877119879

0 minus1205911198981205902119862119879

120596119894119870119879

2119866119879

119894119877119879

0 minus120591

119898120590119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

0]119879

Θ6119889= [radic120575120590

1119862119879

119889119894119870119879

1119866119879

119894119877119879

1 radic1205751205902119862119879

119889119894119870119879

2119866119879

119894119877119879

1 radic120575120590

119898119862119879

119889119894119870119879

119898119866119879

119894119877119879

1 0 0]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119866119879

119894119877119879

1 minusradic120575120590

2119862119879

120596119894119870119879

2119866119879

119894119877119879

1 minusradic120575120590

119898119862119879

120596119894119870119879

119898119866119879

119894119877119879

1]119879

(17)

Proof Introduce a new vector

120577119879

(119905)=[ 119890119879

(119905) 119890119879

(119905minus120591 (119905)) 119890119879

(119905minus120591119898) 119890119879

(119905minus120591119872) 120596119879

(119905) ]

(18)

and two matricesΓ1= [119894119889119894

0 0 120596119894]

Γ2= [119871119894119871119889119894

0 0 minus119871120596119894]

(19)

Abstract and Applied Analysis 5

The system (8) can be rewritten as

119890 (119896) = Γ1120577 (119905) + 119866

119894(Ξ minus Ξ) 119862

119894119890 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905)) minus 119866

119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = Γ2120577 (119905)

(20)

Let119909119905(119904) = 119909(119905+119904) (minus120591(119905) le 119904 le 0)Then the same as [37]

(119909119905 120579119905) 119905 ge 0 is a Markov process Choose the following

Lyapunov functional candidate

119881 (119909119905 120579119905) =

4

sum

119894=1

119881119894(119909119905 120579119905) (21)

where

1198811(119909119905 120579119905) = 119890119879

(119905) 119875 (120579119905) 119890 (119905)

1198812(119909119905 120579119905) = int

119905

119905minus120591119898

119890119879

(119904) 1198760119890 (119904) 119889119904 + int

119905

119905minus120591119872

119890119879

(119904) 1198761119890 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119890119879

(119904) 1198762(120579119905) 119890 (119904) 119889119904

1198813(119909119905 120579119905) = 120591119898int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198770119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198771119890 (V) 119889V 119889119904

1198814(119909119905 120579119905) = int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198851119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198852119890 (V) 119889V 119889119904

(22)

LetL be the weak infinite generator of the random pro-cess 119909

119905 120579119905 Then for each 120579

119905= 119894 119894 isin S taking expectation

on it we obtain

E L119881 (119909119905 120579119905)

le 119890119879

(119905)(2119875119894119894

+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120591

119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

(23)

Note that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

= int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ int

119905

119905minus120591119898

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

(24)

From (16) and (24) we can derive that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904 minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

= int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

le int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198852

]

]

119890 (119904) 119889119904 lt 0

(25)

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

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 5

The system (8) can be rewritten as

119890 (119896) = Γ1120577 (119905) + 119866

119894(Ξ minus Ξ) 119862

119894119890 (119905)

+ 119866119894(Ξ minus Ξ) 119862

119889119894119890 (119905 minus 120591 (119905)) minus 119866

119894(Ξ minus Ξ) 119862

120596119894120596 (119905)

(119905) = Γ2120577 (119905)

(20)

Let119909119905(119904) = 119909(119905+119904) (minus120591(119905) le 119904 le 0)Then the same as [37]

(119909119905 120579119905) 119905 ge 0 is a Markov process Choose the following

Lyapunov functional candidate

119881 (119909119905 120579119905) =

4

sum

119894=1

119881119894(119909119905 120579119905) (21)

where

1198811(119909119905 120579119905) = 119890119879

(119905) 119875 (120579119905) 119890 (119905)

1198812(119909119905 120579119905) = int

119905

119905minus120591119898

119890119879

(119904) 1198760119890 (119904) 119889119904 + int

119905

119905minus120591119872

119890119879

(119904) 1198761119890 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119890119879

(119904) 1198762(120579119905) 119890 (119904) 119889119904

1198813(119909119905 120579119905) = 120591119898int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198770119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198771119890 (V) 119889V 119889119904

1198814(119909119905 120579119905) = int

119905

119905minus120591119898

int

119905

119904

119890119879

(V) 1198851119890 (V) 119889V 119889119904

+ int

119905minus120591119898

119905minus120591119872

int

119905

119904

119890119879

(V) 1198852119890 (V) 119889V 119889119904

(22)

LetL be the weak infinite generator of the random pro-cess 119909

119905 120579119905 Then for each 120579

119905= 119894 119894 isin S taking expectation

on it we obtain

E L119881 (119909119905 120579119905)

le 119890119879

(119905)(2119875119894119894

+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120591

119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

(23)

Note that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

= int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

+ int

119905

119905minus120591119898

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

(24)

From (16) and (24) we can derive that

int

119905

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904 minus int

119905

119905minus120591119898

119890119879

(119904) 1198851119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

= int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904)(

119873

sum

119895=1

1205871198941198951198762119895)119890 (119904) 119889119904

minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198852119890 (119904) 119889119904

le int

119905

119905minus120591119898

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198851

]

]

119890 (119904) 119889119904

+ int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) [

[

119873

sum

119895=1

1205871198941198951198762119895minus 1198852

]

]

119890 (119904) 119889119904 lt 0

(25)

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

It follows from Lemma 3 that

minus 120591119898int

119905

119905minus120591119898

119890119879

(119904) 1198770119890 (119904) 119889119904

le [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

(26)

Combining ((23) (25) and (26)) and introducing slackmatrices119872

119894 119873119894 119894 = 1 2 5 we obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894

+1205911198981198851+ 1205751198852)119890 (119905)

+ 2119890119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905)) + 2119890

119879

(119905) 119875119894120596119894120596 (119905)

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ E 1205912

119898119890119879

(119905) 1198770119890 (119905) minus int

119905minus120591119898

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ E 120575 119890119879

(119905) 1198771119890 (119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119890 (119905 minus 120591

119898) minus 119890 (119905 minus 120591 (119905)) minus int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904]

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898)

+ 2120577119879

(119905)119873119894[119890 (119905 minus 120591 (119905)) minus 119890 (119905 minus 120591

119872)

minusint

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904]

(27)

where 119872119879

119894= [119872

119879

1198941119872119879

1198942119872119879

1198943119872119879

1198944119872119879

1198945] 119873

119879

119894=

[119873119879

1198941119873119879

1198942119873119879

1198943119873119879

1198944119873119879

1198945]

Note that

minus 2120577119879

(119905)119872119894int

119905minus120591119898

119905minus120591(119905)

119890 (119904) 119889119904

le int

119905minus120591119898

119905minus120591(119905)

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

minus 2120577119879

(119905)119873119894int

119905minus120591(119905)

119905minus120591119872

119890 (119904) 119889119904

le int

119905minus120591(119905)

119905minus120591119872

119890119879

(119904) 1198771119890 (119904) 119889119904

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

E 120575 119890119879

(119905) 1198771119890 (119905)

= 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

(28)

E 1205912

119898119890119879

(119905) 1198770119890 (119905)

= 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

(29)

Combining (27)ndash(29) we can obtain

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905)

le 119890119879

(119905)(2119875119894119894+

119873

sum

119895=1

120587119894119895119875119895+ 1198760+ 1198761+ 1198762119894+ 1205911198981198851

Abstract and Applied Analysis 7

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

+1205751198852)119890 (119905) + 2119890

119879

(119905) 119875119894119889119894119890 (119905 minus 120591 (119905))

+ 2119890119879

(119905) 119875119894120596119894120596 (119905)

+ 120575119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 119890119879

(119905 minus 120591119898) 1198760119890 (119905 minus 120591

119898) minus 119890119879

(119905 minus 120591119872) 1198761119890 (119905 minus 120591

119872)

minus (1 minus 120583) 119890119879

(119905 minus 120591 (119905)) 1198762119894119890 (119905 minus 120591 (119905))

+ 120575119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862119894119890 (119905)

+ 120575[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198771[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

minus 120575120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198771119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898119890119879

(119905) 119862119879

119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119894119890 (119905)

+ 1205912

119898119890119879

(119905 minus 120591 (119905)) 119862119879

119889119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862119889119894119890 (119905 minus 120591 (119905))

minus 1205912

119898120596119879

(119905) 119862119879

120596119894

119898

sum

119894=1

1205902

119894119870119879

119894119866119879

1198941198770119866119894119870119894119862120596119894120596 (119905)

+ 1205912

119898[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

119879

times 1198770[119894119890 (119905) +

119889119894119890 (119905 minus 120591 (119905)) +

120596119894120596 (119905)]

+ [119890 (119905)

119890 (119905 minus 120591119898)]

119879

[minus1198770

1198770

1198770

minus1198770

] [119890 (119905)

119890 (119905 minus 120591119898)]

minus 1205742

120596119879

(119905) 120596 (119905) + 120577(119905)119879

Γ119879

2Γ2120577 (119905)

+ 2120577119879

(119905)119872119894[119909 (119905 minus 120591

119898) minus 119909 (119905 minus 120591 (119905))]

+ 2120577119879

(119905)119873119894[119909 (119905 minus 120591 (119905)) minus 119909 (119905 minus 120591

119872)]

+ (120591 (119905) minus 120591119898) 120577119879

(119905)119872119894119877minus1

1119872119879

119894120577 (119905)

+ (120591119872minus 120591 (119905)) 120577

119879

(119905)119873119894119877minus1

1119873119879

119894120577 (119905)

(30)

By using Lemma 4 and Schur complement it is easy to seethat (15) and 119904 = 1 2 are sufficient conditions to guarantee

E L119881 (119909119905 120579119905) minus 120574

2

120596119879

(119905) 120596 (119905) + 119879

(119905) (119905) lt 0

(31)

Then the following inequality can be concluded

E L119881 (119909119905 119894 119905) lt minus120582min (Ψ)E 120577

119879

(119905) 120577 (119905) (32)

Define a new function as

119882(119909119905 119894 119905) = 119890

120598119905

119881 (119909119905 119894 119905) (33)

Its infinitesimal operatorL is given by

W (119909119905 119894 119905) = 120598119890

120598119905

119881 (119909119905 119894 119905) + 119890

120598119905

L119881 (119909119905 119894 119905) (34)

By the generalized Ito formula [36] we can obtain from(34) that

E 119882 (119909119905 119894 119905) minus E 119882 (119909

0 119894)

= int

119905

0

120598119890120598119904

E 119881 (119909119904 119894) 119889119904 + int

119905

0

119890120598119904

E L119881 (119909119904 119894) 119889119904

(35)

Then similar to the method of [1] we can see that thereexists a positive number 120572 such that for 119905 gt 0

E 119881 (119909119905 119894 119905) le 120572 sup

minus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

119890minus120598119905

(36)

Since 119881(119909119905 119894 119905) ge 120582min(119875119894)119909

119879

(119905)119909(119905) it can be shownfrom (36) that for 119905 ge 0

E 119909119879

(119905) 119909 (119905) le minus120598119905 supminus120591119872le119904le0

1003817100381710038171003817120601 (119904)

1003817100381710038171003817

2

(37)

where = 120572(120582min119875119894) Recalling Definition 5 the proof canbe completed

Remark 8 In the above proof a new Lyapunov function isconstructed and the term int

119905

119905minus120591(119905)

119909119879

(119904)(sum119873

119895=11205871198941198951198762119895)119909(119904)119889119904 in

(25) is separated into two parts It is easy to see that thismethod is less conservative than the ones in the literature[5 38]

Remark 9 A delay-dependent stochastic stability conditionfor MJSs with interval time-varying delays is provided inTheorem 7 In the proof ofTheorem 7 the convexity propertyof the matrix inequality is treated in terms of Lemma 4which need not enlarge 120591(119905) to 120591

119872 so the common existed

conservatism caused by this kind of enlargement in [39ndash42] can be avoided and thus the conservative result will bedecreased

Theorem 7 established some analysis results In the fol-lowing the problem of state estimator design is to beconsidered and the following results can be readily obtainedfromTheorem 7

Theorem 10 For some given constants 120574 and 0 le 120591119898le 120591119872 the

augmented system (8) is stochastically stable with a prescribed119867infin

performance 120574 if there exist 119875119894gt 0 119876

0gt 0 119876

1gt 0

1198762119894gt 0 119877

0gt 0 119877

1gt 0 119885

1gt 0 119885

2gt 0

119894 119872119894119896 and

8 Abstract and Applied Analysis

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

119873119894119896(119894 isin S 119896 = 1 2 5)with appropriate dimensions so that

the following LMIs hold for a given 120576 gt 0

Ψ =

[[[[[[[[[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]]]]]]]]]

]

lt 0

119904 = 1 2

(38)

119873

sum

119895=1

1205871198941198951198762119895le 119885119896 119896 = 1 2 (39)

where

Ψ11= 119875119894119860119894+ 119860119879

119894119875119894+ 119894119862119894+ 119862119879

119894119879

119894+ 1198760+ 1198761

+ 1198762119894minus 1198770+ 1205911198981198851+ 1205751198852+

119873

sum

119895=1

120587119894119895119875119895

Ψ21= [119875119879

119894119860119889119894+ 119894119862119889119894minus1198721198941+ 1198731198941 119877119879

0+1198721198941

minus1198731198941 minus119875119879

119894119860120596119894minus 119894119862120596119894]119879

Ψ31=

[[[

[

120591119898119875119894119860119894+ 120591119898119894119862120596119894

radic120575119875119894119860119894+ radic120575

119894119862120596119894

119871119894

]]]

]

Ψ32=

[[[[

[

120591119898119875119894119860119889119894+ 120591119898119894119862119889119894

0 0 minus120591119898119875119894119860120596119894minus 120591119898119894119862120596119894

radic120575119875119894119860119889119894+ radic120575

119894119862119889119894

0 0 minusradic120575119875119894119860120596119894minus radic120575

119894119862120596119894

119871119889119894

0 0 minus119871120596119894

]]]]

]

Ψ33= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198771 minus119868

Ψ51= [1205911198981205901119862119879

119894119870119879

1119879

119894 1205911198981205902119862119879

119894119870119879

2119879

119894

120591119898120590119898119862119879

119894119870119879

119898119879

119894 0 0]

119879

Ψ52= [Θ5119889 0 0 Θ

5120596]

Ψ55= diag minus2120576119875

119894+ 1205762

1198770 minus2120576119875

119894+ 1205762

1198770

minus2120576119875119894+ 1205762

1198770

Ψ61= [radic120575120590

1119862119879

119894119870119879

1119879

119894 radic1205751205902119862119879

119894119870119879

2119879

119894

radic120575120590119898119862119879

119894119870119879

119898119879

119894 0 0 ]

119879

Ψ62= [Θ6119889 0 0 Θ

6120596]

Ψ66= diag minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771 minus2120576119875

119894+ 1205762

1198771

Θ5119889= [0 120591

1198981205901119862119879

119889119894119870119879

1119879

119894 1205911198981205902119862119879

119889119894119870119879

2119879

119894

120591119898120590119898119862119879

119889119894119870119879

119898119879

119894 0 0]

119879

Θ5120596= [0 0 minus120591

1198981205901119862119879

120596119894119870119879

1119879

119894 minus1205911198981205902119862119879

120596119894119870119879

2119879

119894

minus120591119898120590119898119862119879

120596119894119870119879

119898119879

119894]119879

Θ6119889= [0radic120575120590

1119862119879

119889119894119870119879

1119879

119894 radic1205751205902119862119879

119889119894119870119879

2119879

119894

radic120575120590119898119862119879

119889119894119870119879

119898119879

119894 0 0 ]

119879

Θ6120596= [0 0 minusradic120575120590

1119862119879

120596119894119870119879

1119879

119894 minusradic120575120590

2119862119879

120596119894119870119879

2119879

119894

minusradic120575120590119898119862119879

120596119894119870119879

119898119879

119894]119879

(40)

and Ψ22 Ψ41(119904) Ψ

42(119904) and 120575 are as defined in Theorem 7

Moreover the state estimator gain in the form of (6) is asfollows

119866119894= 119875minus1

119894119894 (41)

Proof Defining 119894= 119875119894119866119894 from (15) and using Schur com-

plement the matrix inequality (15) holds if and only if

Ψ =

[[[[[[[[

[

Ψ11

lowast lowast lowast lowast lowast

Ψ21

Ψ22

lowast lowast lowast lowast

Ψ31

Ψ32

Ψ33

lowast lowast lowast

Ψ41(119904) Ψ

42(119904) 0 minus119877

1lowast lowast

Ψ51

Ψ52

0 0 Ψ55

lowast

Ψ61

Ψ62

0 0 0 Ψ66

]]]]]]]]

]

lt 0

119904 = 1 2

(42)

where

Ψ33= diag minus119875

119894119877minus1

0119875119894 minus119875119894119877minus1

1119875119894 minus119868 (43)

Due to

(119877119896minus 120576minus1

119875119894) 119877minus1

119896(119877119896minus 120576minus1

119875119894) ge 0 119894 isin 119878 119896 = 0 1

(44)

we can have

minus119875119894119877minus1

119896119875119894le minus2120576119875

119894+ 1205762

119877119896 119894 isin 119878 119896 = 0 1 (45)

Abstract and Applied Analysis 9

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

Substituting minus119875119894119877minus1

119896119875119894with minus2120576119875

119894+ 1205762

119877119896(119896 = 0 1) in (42)

we obtain (38) so if (38) holds we have (15) holds and fromabove proof we know that the desired state estimator gainmatrix is 119866

119894= 119875minus1

119894119894 This completes the proof

Remark 11 Inequality (45) is used to bound the termminus119875119894119877minus1

119896119875119894 This step can be improved by adopting the cone

complementary algorithm [43] which is popular in recentcontrol designs The scaling parameter 120576 gt 0 here can beused to improve conservatism in Theorem 10 In additionTheorem 10 shows that for given 120576 we can obtain the stateestimator gain by solving a set of LMIs in (38) and (39)

4 Numerical Example

In this section well-studied example is considered to illus-trate the effectiveness of above approaches proposed and alsoto explain the proposed method on state estimator design

Consider linear Markovian jump systems in the form of(1) with two modes For modes 1 and 2 the dynamics ofsystem with following parameters [28] are described as

1198601= [

[

minus3 1 0

03 minus25 1

minus01 03 minus38

]

]

1198601198891= [

[

minus02 01 06

05 minus1 minus08

0 1 minus25

]

]

1198601205961= [

[

1

0

1

]

]

1198621= [08 03 0] 119862

1198891= [02 minus03 minus06]

1198621205961= 02

1198711= [05 minus01 1] 119871

1198891= [0 0 0] 119871

1205961= 0

1198602= [

[

minus25 05 minus01

01 minus35 03

minus01 1 minus2

]

]

1198601198892= [

[

0 minus03 06

01 05 0

minus06 1 minus08

]

]

1198601205962= [

[

minus06

05

0

]

]

1198622= [05 02 03] 119862

1198892= [0 minus06 02]

1198621205962= 05

1198712= [0 1 06] 119871

1198892= [0 0 0]

1198711205962= 0

(46)

Suppose the initial conditions are given by 119909(0) =

[08 02 minus09]119879 119909(0) = [0 02 0]

119879 and the transition pro-bability matrix

120587 = [05 05

03 07] (47)

By Theorem 10 we get the maximum time delay 120591119872

=

59250 for 120591119898= 1 120583 = 05 120576 = 10 and 120574 = 12 Meanwhile

0 5 10 15 20 25 3005

1

15

2

25

Mod

e

Time t (s)

Figure 1 Operation modes

0 5 10 15 20 25 3004

05

06

07

08

09

1

11

Time t (s)

120591(t)

Figure 2 Interval time-varying delay

we can get the fact that the maximum time delay will becomelarger with decreasing rates of 120591(119905) when other variables arefixed For example the maximum time delay is 120591

119872= 64072

for 120583 = 01 if other parameters did not changeThe corresponding state estimator gain matrices for 120583 =

05 are given by

1198661= [

[

07370

minus13432

minus34025

]

]

1198662= [

[

07847

02051

minus11228

]

]

(48)

To illustrate the performance of the designed state esti-mator choose the disturbance function as follows

120596 (119905) =

minus0425 5 lt 119905 lt 8

0375 13 lt 119905 lt 18

0 otherwise(49)

With this state estimator the simulation results are shown inFigures 1 2 and 3 which show the operation modes of theMJSs interval time-varying delay and estimated signal error120578(119905) = 119911(119905) minus (119905) respectively From Figures 1 2 and 3 it canbe showed that the designed state estimator performs well

10 Abstract and Applied Analysis

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

0 5 10 15 20 25 30

0

02

04

06

08

1

Time t (s)

minus02

minus04

minus06

minus08

minus1

120578(t)

Figure 3 Estimated signals error 120578(119905) = 119911(119905) minus (119905)

5 Conclusions

In this paper we established the design method of stateestimation problem for a class of time-delay systems withMarkov jump parameters and missing measurements Byemploying a new Lyapunov function method and using theconvexity property of the matrix inequality an LMI-basedsufficient condition for the existence of the desired stateestimator derived is proposed which can lead to much lessconservative analysis results Finally a numerical example hasbeen carried out to show the effectiveness of our obtainedresults of the proposed method

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the Natural ScienceFoundation of China (no 11226240) the Natural ScienceFoundation of Jiangsu Province of China (no BK2102469)National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035and a project funded by the Priority Academic Progr-am Department of Jiangsu Higher Education Institutions(PAPD)

References

[1] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005

[2] 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

[3] H Shao ldquoDelay-range-dependent robust 119867infin filtering for

uncertain stochastic systems with mode-dependent time delays

and Markovian jump parametersrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1084ndash1095 2008

[4] O Costa and M Fragoso ldquoA separation principle for the 1198672-

control of continuous-time infiniteMarkov jump linear systemswith partial observationsrdquo Journal ofMathematical Analysis andApplications vol 331 no 1 pp 97ndash120 2007

[5] S Xu T Chen and J Lam ldquoRobust119867infin filtering for uncertainMarkovian jump systems with mode-dependent time delaysrdquoIEEE Transactions on Automatic Control vol 48 no 5 pp 900ndash907 2003

[6] J Liu E Tian Z Gu andY Zhang ldquoState estimation forMarko-vian jumping genetic regulatory networks with random delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 7 pp 2479ndash2492 2014

[7] S He and F Liu ldquoUnbiased estimation of markov jump systemswith distributed delaysrdquo Signal Processing vol 100 pp 85ndash922014

[8] S He and F Liu ldquoControlling uncertain fuzzy neutral dynamicsystemswithMarkov jumpsrdquo Journal of Systems Engineering andElectronics vol 21 no 3 pp 476ndash484 2010

[9] S He and F Liu ldquoRobust peak-to-peak filtering for Markovjump systemsrdquo Signal Processing vol 90 no 2 pp 513ndash5222010

[10] O L Costa andW L de Paulo ldquoIndefinite quadratic with linearcosts optimal control of Markov jump with multiplicative noisesystemsrdquo Automatica vol 43 no 4 pp 587ndash597 2007

[11] H Zhang H Yan J Liu and Q Chen ldquoRobust 119867infin

filters forMarkovian jump linear systems under sampledmeasurementsrdquoJournal of Mathematical Analysis and Applications vol 356 no1 pp 382ndash392 2009

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

[13] Z Wang Y Liu L Yu and X Liu ldquoExponential stability ofdelayed recurrent neural networks with Markovian jumpingparametersrdquo Physics Letters A vol 356 no 4-5 pp 346ndash3522006

[14] ZWang J Lam andX Liu ldquoExponential filtering for uncertainMarkovian jump time-delay systems with nonlinear distur-bancesrdquo IEEE Transactions on Circuits and Systems II vol 51no 5 pp 262ndash268 2004

[15] 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

[16] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[17] L Hou G Zong W Zheng and Y Wu ldquoExponential 1198972minus

119897infin

control for discrete-time switching markov jump linearsystemsrdquo Circuits Systems and Signal Processing vol 32 no 6pp 2745ndash2759 2013

[18] M G Todorov and M D Fragoso ldquoA new perspective on therobustness ofMarkov jump linear systemsrdquoAutomatica vol 49no 3 pp 735ndash747 2013

[19] O Costa and G Benites ldquoRobust mode-independent filteringfor discrete-time Markov jump linear systems with multiplica-tive noisesrdquo International Journal of Control vol 86 no 5 pp779ndash793 2013

[20] O L V Costa M M D Fragoso and R P Marques Discrete-TimeMarkov Jump Linear Systems Springer London UK 2005

Abstract and Applied Analysis 11

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

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

[21] Y Liu ZWang J Liang and X Liu ldquoSynchronization and stateestimation for discrete-time complex networks with distributeddelaysrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 38 no 5 pp 1314ndash1325 2008

[22] Z Wang Y Liu and X Liu ldquoState estimation for jumpingrecurrent neural networks with discrete and distributed delaysrdquoNeural Networks vol 22 no 1 pp 41ndash48 2009

[23] J Liang Z Wang and X Liu ldquoState estimation for coupleduncertain stochastic networks with missing measurements andtime-varying delays the discrete-time caserdquo IEEE Transactionson Neural Networks vol 20 no 5 pp 781ndash793 2009

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

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

[25] A Doucet N J Gordon and V Krishnamurthy ldquoParticle filtersfor state estimation of jump Markov linear systemsrdquo IEEETransactions on Signal Processing vol 49 no 3 pp 613ndash6242001

[26] P Balasubramaniam S Lakshmanan and S Jeeva SathyaTheesar ldquoState estimation for Markovian jumping recurrentneural networks with interval time-varying delaysrdquo NonlinearDynamics vol 60 no 4 pp 661ndash675 2010

[27] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[28] J Liu R YanHHan SHu andYHu ldquo119867infinfiltering forMarko-

vian jump systems with time-varying delaysrdquo in Proceedings ofthe IEEE Chinese Control and Decision Conference (CCDC rsquo10)pp 1249ndash1254 May 2010

[29] C P Tan and C Edwards ldquoSliding mode observers for robustdetection and reconstruction of actuator and sensor faultsrdquoInternational Journal of Robust and Nonlinear Control vol 13no 5 pp 443ndash463 2003

[30] E Tian D Yue and C Peng ldquoBrief Paper reliable controlfor networked control systems with probabilistic sensors andactuators faultsrdquo IET ControlTheory and Applications vol 4 no8 pp 1478ndash1488 2010

[31] S Hu D Yue J Liu and Z Du ldquoRobust 119867infin

control for net-worked systems with parameter uncertainties and multiplestochastic sensors and actuators faultsrdquo International Journal ofInnovative Computing Information and Control vol 8 no 4 pp2693ndash2704 2012

[32] J Liu ldquoEvent-based reliable h control for networked controlsystem with probabilistic actuator faultsrdquo Chinese Journal ofElectronics vol 22 no 4 2013

[33] J Liu and D Yue ldquoEvent-triggering in networked systems withprobabilistic sensor and actuator faultsrdquo Information Sciencesvol 240 pp 145ndash160 2013

[34] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Boston Mass USA 2003

[35] D Yue E Tian Y Zhang and C Peng ldquoDelay-distribution-dependent stability and stabilization of T-S fuzzy systems withprobabilistic interval delayrdquo IEEETransactions on SystemsManand Cybernetics B vol 39 no 2 pp 503ndash516 2009

[36] X Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002

[37] E Boukas Z Liu and G Liu ldquoDelay-dependent robust stabilityand 119867

infincontrol of jump linear systems with time-delayrdquo

International Journal of Control vol 74 no 4 pp 329ndash340 2001

[38] S Xu J Lam and X Mao ldquoDelay-dependent 119867infin

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

[39] B Chen X Liu and S Tong ldquoDelay-dependent stability anal-ysis and control synthesis of fuzzy dynamic systems with timedelayrdquo Fuzzy Sets and Systems vol 157 no 16 pp 2224ndash22402006

[40] X Jiang and Q Han ldquoRobust119867infincontrol for uncertain Takagi-

Sugeno fuzzy systems with interval time-varying delayrdquo IEEETransactions on Fuzzy Systems vol 15 no 2 pp 321ndash331 2007

[41] E Tian and C Peng ldquoDelay-dependent stability analysis andsynthesis of uncertain T-S fuzzy systems with time-varyingdelayrdquo Fuzzy Sets and Systems vol 157 no 4 pp 544ndash559 2006

[42] H N Wu and H X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[43] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of