Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

Delay-dependent H∞ filtering for stochastic systems with Markovianswitching and mixed mode-dependent delaysI

Hao Shen a, Shengyuan Xu a,∗, Xiaona Song a, Yuming Chu ba School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR Chinab Department of Mathematics, Huzhou Teacher’s College, Huzhou, Zhejiang 313000, PR China

a r t i c l e i n f o

Article history:Received 13 May 2008Accepted 2 September 2009

Keywords:H∞ filteringMarkovian switchingExponential mean-square stabilityMixed mode-dependent delaysDelay-dependent criteria

a b s t r a c t

This paper studies the problem ofH∞ filtering for a class of stochastic systemswithMarko-vian switching and mixed mode-dependent time-varying delays. By introducing slackmatrix variables and using Markovian switching Lyapunov functionals, we obtain delay-dependent sufficient conditions which guarantee the existence of H∞ filters such that thefiltering error system is exponentially mean-square stable and satisfies a prescribed H∞performance level; the conditions are in terms of linear matrix inequalities (LMIs). Whenthese LMIs are feasible, an explicit expression of a desired H∞ filter is given. Numericalexamples are provided to demonstrate the effectiveness and the reduced conservatism ofour results.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The abrupt phenomena, such as sudden environmental changes, random failures and repairs of the components andchanges in the interconnections of subsystems, can be modeled by a class of hybrid systems termed as Markovian jumpsystems (MJS) [1–4]. The study of such systems has been attracting considerable attention over the past several decades.For instance, the mean square stochastic stability of linear time-delay MJS was investigated in [5], while the problems ofH∞ control, variable structure control, moment decay rates and stabilization for MJS were studied in [6–9], respectively.Furthermore, taken the uncertainty caused by stochastic environment into account, the stability and stabilization problemsfor MJS were addressed in [10–12], and the sliding mode control problem was considered in [13].It is known that the celebrated Kalman filtering is not applicable when a prior information on the external noise is

not precisely known. In this case, the H∞ filtering scheme has been proposed to deal with the filtering problem [14–17].Since time delays are ubiquitous in many physical systems, the H∞ filtering problem of delayed systems has been widelyinvestigated. For example, delay-independent LMI-based conditions for the existence of H∞ filters were presented in [18,19], while delay-dependent results were reported in [20–23]. On the other hand, the mode-dependent time delays (MDTD)aremore natural and general thanmode-independent time delays inMJS. Recently, the problems of robust stabilization andguaranteed cost control for MJS withMDTD have been studied in [24,25]. By the LMI approach, theH∞ filtering problem hasbeen investigated in [26,27], where themode-dependent time-varying delays are assumed to have a common upper bound.These results, however, still can be improved by choosing Lyapunov functionals appropriately.Considering above, this paper investigates the problem of delay-dependent H∞ filtering for stochastic systems with

Markovian switching and mixed MDTD. The time delays we consider are discrete delays combined with distributed

I This work is supported by the Natural Science Foundation of Jiangsu Province under Grant BK2008047, and the National Natural Science Foundationof PR China under Grant 60850005.∗ Corresponding author.E-mail address: syxu02@yahoo.com.cn (S. Xu).

1751-570X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2009.09.001

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 123

delays, all of which are time-varying and related to the mode. Our purpose is to design a filter, which guarantees thatthe filtering error system is not only exponentially mean-square stable, but also satisfies a prescribed H∞ performancelevel. By employing a Markovian switching Lyapunov functional and introducing slack matrices, delay-dependent sufficientconditions for the solvability of the H∞ filtering problem are obtained in terms of LMIs. Numerical examples are given toillustrate the effectiveness and advantages of our results.

Notation. Throughout this paper, for symmetric matrices X and Y , the notation X ≥ Y (respectively, X > Y ) means that thematrix X − Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension.The notation MT represents the transpose of the matrix M . E · denotes the expectation operator with respect to someprobability measure P .L2 [0,∞) is the space of square-integrable vector functions over [0,∞); |·| refers to the Euclideanvector norm, while ‖·‖2 stands for the usual L2[0,∞) norm. ‖·‖E2 denotes the norm in L2 ((Ω,F ,P ) , [0,∞)) , where(Ω,F ,P ) is a probability space.Ω is the sample space, and F is the σ -algebra of subsets of the sample space andP is theprobability measure on F . Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symbol ∗ isused to denote a matrix which can be inferred by symmetry.

2. System description and definitions

Consider the following class of stochastic Markovian jump systems with mixed MDTD (Σ):

dx (t) =

[A (rt) x(t)+ A1 (rt) x

(t − hrt (t)

)+ A2 (rt)

∫ t

t−hrt (t)x (s) ds+ B1 (rt) v (t)

]dt

+

[H (rt) x(t)+ H1 (rt) x

(t − hrt (t)

)+ H2 (rt)

∫ t

t−hrt (t)x (s) ds+ B2 (rt) v (t)

]dω (t) , (1)

dy (t) =

[C (rt) x(t)+ C1 (rt) x

(t − hrt (t)

)+ C2 (rt)

∫ t

t−hrt (t)x (s) ds+ B3 (rt) v (t)

]dt

+

[D (rt) x(t)+ D1 (rt) x

(t − hrt (t)

)+ D2 (rt)

∫ t

t−hrt (t)x (s) ds+ B4 (rt) v (t)

]dω (t) , (2)

z (t) = L (rt) x (t) , (3)x (t) = φ (t) , ∀t ∈ [−h, 0], (4)

where x (t) ∈ Rn is the state vector; v (t) ∈ Rp is the noise signal which is assumed to be in L2 [0,∞); y (t) ∈ Rq is themeasurement; z (t) ∈ Rm is the signal to be estimated; rt is a continuous-time Markovian process with right continuoustrajectories taking values in a finite set S = 1, 2, . . . ,N with transition probabilities given by

Pr r(t +∆) = j|r(t) = i =

πij∆+ o (∆) i 6= j,1+ πii∆+ o (∆) i = j, (5)

where ∆ > 0, lim∆→0 (o (∆) /∆) = 0, and πij ≥ 0, for j 6= i, is the transition rate from mode i at time t to mode j at timet +∆ and

πii = −

s∑j=1,j6=i

πij. (6)

In system (Σ), ω (t) is a zero-mean real scalar Wiener process on (Ω,F ,P ) relative to an increasing family (Ft)t∈[0,∞) ofσ -algebras Ft ⊂ F satisfying

E dω (t) = 0, Edω (t)2

= dt. (7)

To simplify the notation, we denote Ai = A (rt) for each rt = i ∈ S, and the other symbols are similarly denoted. The MDTDhi (t) satisfy 0 ≤ hi (t) ≤ hi ≤ h, hi (t) ≤ µ, for each rt = i ∈ S; φ (t) is the initial condition, Ai, A1i,A2i,Hi,H1i,H2i, B1i, B2i,B3i, B4i, Ci, C1i, C2i,Di,D1i,D2i and Li are known real constant matrices representing the nominal system for each i ∈ S.Now, we consider the following filter for the estimate of z (t):(

Σf): dx (t) = Afix (t) dt + Bfidy (t) , (8)

z (t) = Cfix (t) , (9)

where x (t) ∈ Rn, and z (t) ∈ Rm. Denote

ξ (t) =[xT (t) xT (t)

]T, z (t) = z (t)− z (t) . (10)

124 H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

Then, the filtering error dynamics from the systems (Σ) and(Σf)are described by(

Σf): dξ (t) = A (t) dt +H (t) dω(t), (11)

z (t) = Liξ (t) , (12)

where

A (t) = Aiξ (t)+ A1iGξ (t − hi (t))+ A2iG∫ t

t−hi(t)ξ (s) ds+ B1iv (t) ,

H (t) = Hiξ (t)+ H1iGξ (t − hi (t))+ H2iG∫ t

t−hi(t)ξ (s) ds+ B2iv (t) ,

Ai =[Ai 0BfiCi Afi

], A1i =

[A1iBfiC1i

], A2i =

[A2iBfiC2i

], B1i =

[B1iBfiB3i

],

Hi =[Hi 0BfiDi 0

], H1i =

[H1iBfiD1i

], H2i =

[H2iBfiD2i

], B2i =

[B2iBfiB4i

],

G =[I 0

], Li =

[Li −Cfi

].

Throughout the paper we shall use the following definitions.

Definition 1 ([28]). The linear stochastic system with Markovian jump parameters in (Σ) with v (t) = 0 is said to beexponentially mean-square stable if there exist constant scalars c > 0 and ρ > 0 such that

E |x (t)|2 ≤ ce−ρt sup−h≤σ≤0

E |φ (σ)|2 . (13)

Definition 2. Given a scalar γ > 0, the filtering error system(Σf)is said to be exponentially mean-square stable with an

H∞ performance γ if it is exponentially mean-square stable when v (t) = 0 and the filtering error z (t) = z (t) − z (t)satisfies

‖z (t)‖E2 < γ ‖v (t)‖2 , (14)

under zero-initial condition for any nonzero v (t) ∈ L2 [0,∞).

3. H∞ performance analysis

In this section, the H∞ performance analysis for the filtering error dynamics (Σf ) will be carried out. New delay-dependent H∞ performance criteria are established. To obtain these results, we need the following lemma.

Lemma 1 ([29]). For any constant symmetric matrix M ∈ Rn×n, M = MT > 0, vector function ω : [0, r] → Rn such that theintegrations in the following are well defined, then

r∫ r

0ωT(s)Mω(s)ds >

(∫ r

0ω(s)ds

)TM(∫ r

0ω(s)ds

).

Theorem 1. For given scalars hi > 0, for each i ∈ S, h > 0 and µ, the filtering error system (Σf ) is exponentially mean-square stable with an H∞ performance γ , if there exist matrices Pi > 0,Q1 > 0,Q2 > 0,Q3 > 0,U > 0,W > 0, Rmi andSmi,m = 1, . . . , 4, such that the following LMIs hold for i ∈ S,

Ω1i Γ1i PiA2i Γ2i −GTS1i GTRT4i Γ5i∗ Ω2i 0 Γ3i −S2i Γ4i Γ6i∗ ∗ Ω3i 0 0 0 Γ7i∗ ∗ ∗ −γ 2I −S3i 0 Γ8i∗ ∗ ∗ ∗ −Q2 −ST4i 0∗ ∗ ∗ ∗ ∗ −Q3 Γ9i∗ ∗ ∗ ∗ ∗ ∗ Ω4i

< 0, (15)

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 125

where

Ω1i = PiAi + ATi Pi +∑j∈S

πijPj + GT (Q1 + Q2 + Q3)G+ πhGT (Q1 + Q2)G+ GTR1iG+ GTRT1iG,

π = max |πii|, i ∈ S , Ω2i = − (1− µ)Q1 − R2i − RT2i + S2i + ST2i,

Γ1i = PiA1i − GTR1i + GTRT2i + GTS1i, Γ2i = PiB1i + GTRT3i, Γ3i = −RT3i + S

T3i,

Γ4i = −RT4i + ST4i, Ω4i = diag

−1hiU,−

1hiU,−U,−W ,−Pi,−W ,−W ,−I

,

Γ5i =[−GTR1i −GTS1i ATi G

TU HTi GTW HTi Pi GTR1i GTS1i LTi

],

Γ6i =[−R2i −S2i AT1iG

TU HT1iGTW HT1iPi R2i S2i 0

], W =

(hi +

12πh2

)W ,

Γ7i =[0 0 AT2iG

TU HT2iGTW HT2iPi 0 0 0

], U =

(hi +

12πh2

)U,

Γ8i =[−R3i −S3i BT1iG

TU BT2iGTW BT2iPi R3i S3i 0

],

Γ9i =[−R4i −S4i 0 0 0 R4i S4i 0

], Ω3i =

πii

hi(Q1 + Q2) .

Proof. Define a new process (ξt , i) , t ≥ 0 by

ξt (s) = ξ (t + s) , −h ≤ s ≤ 0.

Then choose a Markovian switching Lyapunov functional candidate as

V (ξt , i, t) = ξ T (t) Piξ (t)+∫ t

t−hi(t)ξ T (s)GTQ1Gξ (s) ds+

∫ t

t−hiξ T (s)GTQ2Gξ (s) ds

+

∫ 0

−hi

∫ t

t+βHT (s)GTWGH (s) dsdβ + π

∫ h

0

∫ t

t−β(s− t + β)HT (s)GTWGH (s) dsdβ

+

∫ 0

−hi

∫ t

t+βAT (s)GTUGA (s) dsdβ + π

∫ h

0

∫ t

t−β(s− t + β)AT (s)GTUGA (s) dsdβ

+

∫ t

t−hξ T (s)GTQ3Gξ (s) ds+ π

∫ 0

−h

∫ t

t+βξ T (s)GT (Q1 + Q2)Gξ (s) dsdβ. (16)

Let L be the weak infinitesimal generator of the random process ξt , i. Then, by employing the generalized Itô’s formulain [10,30] and considering the filtering error system (Σf ), it can be obtained that the stochastic differential dV (ξt , i, t)as [10,30]:

dV (ξt , i, t) = LV (ξt , i, t) dt + 2ξ T (t) PiH (t) dw (t) , (17)

where

LV (ξt , i, t) = 2ξ T(t)PiA (t)+H (t)T PiH (t)+∑j∈S

πijξT (t) Pjξ (t)+ ξ T (t)GT (Q1 + Q2 + Q3)Gξ (t)

−(1− hi (t)

)ξ T (t − hi (t))GTQ1Gξ (t − hi (t))− ξ T (t − hi)GTQ2Gξ (t − hi)

− ξ T (t − h)GTQ3Gξ (t − h)+∑j∈S

πij

∫ t

t−hj(t)ξ T (s)GTQ1Gξ (s) ds

+

∑j∈S

πij

∫ t

t−hjξ T (s)GTQ2Gξ (s) ds+ πhξ T (t)GT (Q1 + Q2)Gξ (t)

− π

∫ t

t−hξ T (s)GT (Q1 + Q2)Gξ (s) ds+ hiAT (t)GTUGA (t)

−

∫ t

t−hiAT (s)GTUGA (s) ds+

∑j∈S

πij

∫ 0

−hj

∫ t

t+βAT (s)GTUGA (s) dsdβ

+12πh2AT (t)GTUGA (t)− π

∫ h

0

∫ t

t−βAT (s)GTUGA (s) dsdβ + hiHT (t)GTWGH (t)

126 H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

−

∫ t

t−hiHT (s)GTWGH (s) ds+

∑j∈S

πij

∫ 0

−hj

∫ t

t+βHT (s)GTWGH (s) dsdβ

+12πh2HT (t)GTWGH (t)− π

∫ h

0

∫ t

t−βHT (s)GTWGH (s) dsdβ.

According to Lemma 1, we have

∑j∈S

πij

∫ t

t−hj(t)ξ T (s)GTQ1Gξ (s) ds =

∑j∈S, j6=i

πij

∫ t

t−hj(t)ξ T (s)GTQ1Gξ (s) ds+ πii

∫ t

t−hi(t)ξ T (s)GTQ1Gξ (s) ds

≤ π

∫ t

t−hξ T (s)GTQ1Gξ (s) ds−

(−πii)

hi

∫ t

t−hi(t)ξ T (s)GTdsQ1

∫ t

t−hi(t)Gξ (s) ds, (18)

∑j∈S

πij

∫ t

t−hjξ T (s)GTQ2Gξ (s) ds ≤ π

∫ t

t−hξ T (s)GTQ2Gξ (s) ds−

(−πii)

hi

∫ t

t−hi(t)ξ T (s)GTdsQ2

∫ t

t−hi(t)Gξ (s) ds, (19)

∑j∈S

πij

∫ 0

−hj

∫ t

t+βAT (s)GTUGA (s) dsdβ ≤ π

∫ h

0

∫ t

t−βAT (s)GTUGA (s) dsdβ, (20)

and

∑j∈S

πij

∫ 0

−hj

∫ t

t+βHT (s)GTWGH (s) dsdβ ≤ π

∫ h

0

∫ t

t−βHT (s)GTWGH (s) dsdβ. (21)

We define

ς (t, s) =[ξ T (t) xT (t − hi (t))

∫ t

t−hi(t)xT (s) ds vT (t) xT (t − hi) xT (t − h) ς1 (t, s)

]T,

ς1 (t, s) =[∫ t

t−hi(t)AT (s)GTds

∫ t−hi(t)

t−hiAT (s)GTds

].

Noting that the following relationship holds for any matrices Rmi,m = 1, 2, 3, 4, and Smi, for each i ∈ S,

2ςT (t, s)Ri[Gξ (t)− x (t − hi (t))−

∫ t

t−hi(t)Gdξ (s)

]= 0,

2ςT (t, s) Si[x (t − hi (t))− x (t − hi)−

∫ t−hi(t)

t−hiGdξ (s)

]= 0,

Ri =[RT1iG RT2i 0 RT3i 0 RT4i 0 0

]T, Si =

[ST1iG ST2i 0 ST3i 0 ST4i 0 0

]T.

It can be shown readily that for each rt = i ∈ S,

−2ςT (t, s)Ri∫ t

t−hi(t)GH (s) dω(s)

≤ ςT (t, s)RiW−1RTi ς (t, s)+(∫ t

t−hi(t)GH (s) dω (s)

)TW∫ t

t−hi(t)GH (s) dω (s) ,

−2ςT (t, s) Si∫ t−hi(t)

t−hiGH (s) dω (s)

≤ ςT (t, s) SiW−1STi ς (t, s)+(∫ t−hi(t)

t−hiGH (s) dω (s)

)TW∫ t−hi(t)

t−hiGH (s) dω (s) .

Then, we have

Ez (t)T z (t)− γ 2v (t)T v (t)+LV (ξt , i, t)

≤ E

ςT (t, s)∆iς (t, s)

, (22)

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 127

where

∆i =

Ω1i + LTi Li Γ1i PiA2i Γ2i −S1i RT4i −R1i −S1i∗ Ω2i 0 Γ3i −S2i Γ4i −R2i −S2i∗ ∗ Ω3i 0 0 0 0 0∗ ∗ ∗ −γ 2I −S3i 0 −R3i −S3i∗ ∗ ∗ ∗ −Q2 −ST4i 0 0∗ ∗ ∗ ∗ ∗ −Q3 −R4i −S4i

∗ ∗ ∗ ∗ ∗ ∗ −1hiU 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −1hiU

+HTi PiHi + ATi G

TUGAi + HTi GTWGHi + RiW−1RTi + SiW−1STi , (23)

Ai =[Ai A1i A2i B1i

], Hi =

[Hi H1i H2i B2i

].

On the other hand, under the zero initial condition, we introduce

JΥ = E

∫ Υ

0

[z (t)T z (t)− γ 2v (t)T v (t)

]dt,

where Υ > 0. It can be seen that for each i ∈ S,

JΥ ≤ E

∫ Υ

0

[z (t)T z (t)− γ 2v (t)T v (t)+LV (ξt , i, t)

]dt

≤ E

∫ Υ

0ςT (t, s)∆iς (t, s) dt

.

By using (15) and the Schur complement, it is easy to observe that the right-hand side of (23) is negative definite, whichimplies JΥ < 0 for any nonzero v (t) ∈ L2 [0,∞), and the inequality in (14) holds. When the system

(Σf)with v (t) = 0 is

considered, it is easy to obtain that there exists a scalar δ > 0 such that for ∀ς (t, s) 6= 0 and each i ∈ S,

E LV (ξt , i, t) ≤ −δ |ξ (t)|2 . (24)

Considering that for t − hi ≤ s ≤ t, s− hi ≤ θ ≤ s,∫ t

t−hi

∫ s

s−hi

|ξ (θ)|2 dθds ≤ hi

∫ t−hi

t−2hi

|ξ (θ)|2 dθ + hi

∫ t

t−hi

|ξ (θ)|2 dθ

≤ hi

∫ t

t−2h|ξ (θ)|2 dθ,

then there exists a scalarm > 0 such that∫ 0

−hi

∫ t

t+βAT (s)GTUGA (s) dsdβ ≤ mhiλmax

(GTUG

) ∫ t

t−hi

(|ξ (s)|2 + |ξ (s− hi (s))|2 +

∫ s

s−hi(s)|ξ (θ)|2 dθ

)ds

≤ m(hi + h2i

)λmax

(GTUG

) ∫ t

t−2h|ξ (s)|2 ds.

Similarly, there exists a scalar n > 0 such that∫ 0

−hi

∫ t

t+βHT (s)GTWGH (s) dsdβ ≤ n

(hi + h2i

)λmax

(GTWG

) ∫ t

t−2h|ξ (s)|2 ds.

Hence, we have

V (ξt , i, t) ≤ λmax (Pi) |ξ (t)|2 + πm(h2 + h3

)λmax

(GTUG

) ∫ t

t−2h|ξ (s)|2 ds

+ πhλmax(GT (Q1 + Q2)G

) ∫ t

t−h|ξ (s)|2 ds+m

(hi + h2i

)λmax

(GTUG

) ∫ t

t−2h|ξ (s)|2 ds

+ λmax(GT (Q1 + Q2 + Q3)G

) ∫ t

t−h|ξ (s)|2 ds

+ πn(h2 + h3

)λmax

(GTWG

) ∫ t

t−2h|ξ (s)|2 ds+ n

(hi + h2i

)λmax

(GTWG

) ∫ t

t−2h|ξ (s)|2 ds

128 H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

≤ λmax (Pi) |ξ (t)|2 +[λmax

(GT (Q1 + Q2 + Q3)G

)+m

[(hi + h2i

)+ π

(h2 + h3

)]λmax

(GTUG

)+ n

[(hi + h2i

)+ π

(h2 + h3

)]λmax

(GTWG

)+ πhλmax

(GT (Q1 + Q2)G

)] ∫ t

t−2h|ξ (s)|2 ds

≤ α |ξ (t)|2 + α∫ t

t−2h|ξ (s)|2 ds, (25)

where

α = maxλmax (Pi) , λmax

(GT (Q1 + Q2 + Q3)G

)+m

[(hi + h2i

)+ π

(h2 + h3

)]λmax

(GTUG

)+ n

[(hi + h2i

)+ π

(h2 + h3

)]λmax

(GTWG

)+ πhλmax

(GT (Q1 + Q2)G

).

For any scalar ρ > 0, it can be verified that

d(eρtV (ξt , i, t)

)≤ eρt

[(−δ + ρα) |ξ (t)|2 + ρα

∫ t

t−2h|ξ (s)|2 ds

]+ 2eρtξ T (t) PiH (t) dω (t) . (26)

Integrating both sides of (26) from 0 to Υ > 0 and taking the expectation, we have

eρΥ EV (ξΥ , i,Υ ) = V (ξ0, r0, 0)+ (−δ + ρα)∫ Υ

0eρtE |ξ (t)|2 dt + ρα

∫ Υ

0

∫ t

t−2heρtE |ξ (s)|2 dsdt.

Note that∫ Υ

0

∫ t

t−2heρtE |ξ (s)|2 dsdt ≤ 2h

∫ Υ

−2heρ(t+2h)E |ξ (t)|2 dt.

Then

eρΥ EV (ξΥ , i,Υ ) ≤ V (ξ0, r0, 0)+(−δ + ρα + 2ραhe2ρh

) ∫ Υ

0eρtE |ξ (t)|2 dt + 2ραhe2ρh

∫ 0

−2heρtE |ξ (t)|2 dt.

Now, for each i ∈ S, we choose ρ > 0 to satisfy

−δ + ρα + 2ραhe2ρh = 0.

It is easy to see that there exists a scalar z > 0 such that

eρtE |ξ(t)|2 ≤ z(α + 2αh+ 4ραh2e2ρh

)sup−h≤θ≤0

E |ξ (θ)|2 .

Let

c = z(α + 2αh+ 4ραh2e2ρh

).

Then

E |ξ(t)|2 ≤ ce−ρt sup−h≤σ≤0

E |ξ (σ )|2 .

Therefore, it can be readily observed that the filtering error system is exponentially mean-square stable by Definition 1. Thiscompletes the proof.

Remark 1. It is noted that delay-dependent stability criteria forMarkovian jump systemswithMDTDwere proposed in [24,26,27]. However, in this paper, an improved Markovian switching Lyapunov functional has been employed, and the terms∫ tt−hi(t)

AT (s)GTds and∫ t−hi(t)t−hi

AT (s)GTds have been introduced, which can reduce the potential conservatism in [24,26,27].In Section 5, numerical examples are provided to demonstrate that our results are less conservative than those in [24,26,27].

If there is only one mode in operation, that is, S = 1, then system (Σf ) reduces to a stochastic delayed systemwithout

Markovian jump parameters described by(Σ

):

dξ (t) = [Aξ (t)+ A1Gξ (t − h (t))+ A2G∫ t

t−h(t)ξ (s) ds+ B1v (t)]dt

+ [Hξ (t)+ H1Gξ (t − h (t))+ H2G∫ t

t−h(t)ξ (s) ds+ B2v (t)]dω (t) ,

z (t) = Lξ (t) .

We can establish the H∞ performance criterion for the system(Σ

)in the following corollary.

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 129

Corollary 1. Given scalars γ > 0, h > 0 and µ, the system(Σ

)is exponentially mean-square stable with an H∞ performance

level γ if there exist matrices P > 0,Q1 > 0,Q2 > 0,U1 > 0,U > 0,W > 0, Rm, m = 1, 2, 3 and Sm such that the followingLMI holds:

Υ1 Υ2 hPA2 PB1 + GTRT3 −GTS1 Γ5

∗ Υ3 0 −RT3 + ST3 −S2 Γ6

∗ ∗ −hU1 0 0 Γ7∗ ∗ ∗ −γ 2I −S3 Γ8∗ ∗ ∗ ∗ −Q2 0∗ ∗ ∗ ∗ ∗ Ω4

< 0, (27)

where

Υ1 = PA+ ATP + GT(Q1 + Q2 + hU1 + RT1 + R1)G,

Υ2 = PA1 + GTS1 − GTR1 + GTRT2,

Υ3 = −(1− µ)Q1 − R2 − RT2 + S2 + ST2,

Γ5 =[−GTR1 −GTS1 hATGTU hHTGTW HTP GTR1 GTS1 LT

],

Γ6 =[−R2 −S2 hAT1G

TU hHT1GTW HT1P R2 S2 0

],

Γ7 =[0 0 hAT2G

TU hHT2GTW HT2P 0 0 0

],

Γ8 =[−R3 −S3 hBT1G

TU hBT2GTW BT2P R3 S3 0

],

Ω4 = diag−1hU,−

1hU,−hU,−hW ,−P,−W ,−W ,−I

.

4. H∞ filter design

In this section, we present a solution to the H∞ filter design problem.

Theorem 2. Consider the stochastic system with Markovian switching and mixed MDTD (Σ) and let γ > 0 be a prescribedscalar. For given scalars hi > 0, for each i ∈ S, h > 0 and µ, there exists an admissible H∞ filter in the form of (8) and (9) suchthat the filtering error system

(Σf)is exponentially mean-square stable with an H∞ performance level γ , if there exist matrices

Xi > 0, Yi > 0, Rmi, Smi, m = 1, 2, 3, 4,Λi,Φi,Ψi, i ∈ S, Q1 > 0,Q2 > 0,Q3 > 0,W > 0 and U > 0 such that the followingLMIs hold for i ∈ S,

Ω1i Σ1i Σ2i Σ3i 0 0 Σ4i Ξ1i∗ Ω2i 0 Γ3i −S2i Γ4i Σ5i Ξ2i∗ ∗ Ω3i 0 0 0 Σ6i 0∗ ∗ ∗ −γ 2I −S3i 0 Σ7i Ξ3i∗ ∗ ∗ ∗ −Q2 −ST4i 0 Ξ4i∗ ∗ ∗ ∗ ∗ −Q3 Σ8i Ξ5i∗ ∗ ∗ ∗ ∗ ∗ Σ9i Ξ6i∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ7i

< 0, (28)

where

Ω1i = Ω1i + ΩT1i + Ω2i,

Ω1i =

[AiYi AiΛi XiAi + ΦiCi

],

Ω2i =

πiiYi πiiIπiiI

∑j∈S

πijXj +∑j∈S ,j6=i

πijYj −∑j∈S ,j6=i

πij2I

,Σ1i =

[A1i

XiA1i + ΦiC1i

], Σ2i =

[A2i

XiA2i + ΦiC2i

], Σ3i =

[B1i

XiB1i + ΦiB3i

],

Σ4i =

[0 0 Y Ti A

Ti Y Ti H

Ti 0 0 0 0 YiLTi − Ψ

Ti

0 0 ATi HTi 0 0 0 0 LTi

],

Σ5i =[−R2i −S2i AT1i HT1i HT1i HT1iXi + D

T1iΦ

Ti R2i S2i 0

],

130 H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

Σ6i =[0 0 AT2i HT2i HT2i HT2iXi + D

T2iΦ

Ti 0 0 0

],

Σ7i =[−R3i −S3i BT1i BT2i BT2i BT2iXi + B

T4iΦ

Ti R3i S3i 0

],

Σ8i =[−R4i −S4i 0 0 0 0 R4i S4i 0

], Ξ5i =

[R4i 0 0

],

Σ9i = diag−1hiU,−

1hiU, U − 2I, W − 2I,−

[Yi II Xi

],−W ,−W ,−I

,

Ξ1i =

[0 Yi

√πi1Yi . . .

√πi,i−1Yi

√πi,i+1Yi . . .

√πiN Yi

0 I√πi1I . . .

√πi,i−1I

√πi,i+1I . . .

√πiN I

],

Ξ2i =[−RT1i + R2i + S

T1i 0 0

], Ξ3i =

[R3i 0 0

], Ξ4i =

[−ST1i 0 0

],

Ξ6i =

−R1i −S1i 0 0 HTi HTi Xi + DTiΦTi R1i S1i 0

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

T ,Ξ7i = diag

−I,Q1 + Q2 + Q3 + πh (Q1 + Q2)+ R1i + RT1i − I,Ξ8i

,

Ξ8i = diag −Y1, . . . ,−Yi−1,−Yi+1, . . . ,−YN .

In this case, a desired H∞ filter in the form of (8) and (9) can be chosen with parameters as follows:

Afi =(Y−1i − Xi

)−1(Λi − XiAiYi − ΦiCiYi) Y−1i , (29)

Bfi =(Y−1i − Xi

)−1Φi, Cfi = ΨiY−1i , i ∈ S. (30)

Proof. From (28), it is easy to be seen that I − YiXi is invertible. Now, we define

Π1i =

[Yi IYi 0

], Π2i =

[I Xi0 Y−1i − Xi

],

and set

Pi = Π2iΠ−11i , i ∈ S.

Note that(hi +

12πh2

)U +

((hi +

12πh2

)U)−1− 2I

=

((hi +

12πh2

)U − I

)T ((hi +

12πh2

)U)−1 ((

hi +12πh2

)U − I

)> 0,

which implies

−

((hi +

12πh2

)U)−1≤

(hi +

12πh2

)U − 2I.

Applying this to (28) results in

Ω1i Σ1i Σ2i Σ3i 0 0 Σ4i Ξ1i∗ Ω2i 0 Γ3i −S2i Γ4i Σ5i Ξ2i∗ ∗ Ω3i 0 0 0 Σ6i 0∗ ∗ ∗ −γ 2I −S3i 0 Σ7i Ξ3i∗ ∗ ∗ ∗ −Q2 −ST4i 0 Ξ4i∗ ∗ ∗ ∗ ∗ −Q3 Σ8i Ξ5i∗ ∗ ∗ ∗ ∗ ∗ Σ9i Ξ6i∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ7i

< 0, (31)

where

Ω1i = Ω1i + ΩT1i + Ω2i,

Ω2i =

πiiYi πiiIπiiI

∑j∈S

πijXj −∑j∈S,j6=i

πijY−1j

,

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 131

Σ9i = diag−1hiU,−

1hiU,−U−1,−W−1,−

[Yi II Xi

],−W ,−W ,−I

,

Ξ7i = diag−I,−

(Q1 + Q2 + Q3 + πh (Q1 + Q2)+ R1i + RT1i + I

)−1,Ξ8i

.

Then, by (28), it can be verified that Pi > 0. Noting the parameters in (29) and (30), using the Schurcomplements formula to (31) and pre- and post-multiplying by diag

Π−T1i , I, I, I, I, I, I, I, I, I,Π

−T2i , I, I, I, I

and

diagΠ−11i , I, I, I, I, I, I, I, I, I,Π

−12i , I, I, I, I

, respectively, we can have (15). Therefore, it can be easily concluded from

Theorem 1 that the filtering error system(Σf)is exponentially mean-square stable with an H∞ performance γ . This

completes the proof.

5. Numerical examples

To demonstrate the effectiveness of the proposed method in this paper, we provide the following numerical examples.

Example 1. Consider a stochastic system with Markovian switching and mixed MDTD (Σ)with the following parameters:

A1 =[−2.25 0.19−0.55 −1.22

], A11 =

[−0.5 00 −0.2

], A21 =

[0.12 −0.060 0

],

B11 =[0 −0.150 0

], H1 =

[−0.10 −0.080 0

], H11 =

[0.01 00 −0.08

],

H21 =[−0.01 0.220 −0.57

], B21 =

[0.12 0−0.09 0

], C1 =

[−0.12 00.21 0.17

],

C11 =[−0.28 00.12 0.42

], C21 =

[−0.51 00.19 0.70

], B31 =

[0.09 0−0.10 0

],

D1 =[−0.81 00.29 0.19

], D11 =

[−0.29 00.21 0.36

], D21 =

[−0.79 00.21 0.68

],

B41 =[0.19 0−0.16 0

], A2 =

[−1.81 0.240 −0.85

], A12 =

[−0.57 00 0

],

A22 =[−0.09 00 0

], B12 =

[−0.25 0−0.01 0

], H2 =

[−0.02 0−0.01 0

],

H12 =[0 −0.010 −0.01

], H22 =

[0 0.48−0.41 0.11

], B22 =

[0.23 0−0.19 0

],

C2 =[−0.43 00.59 0.23

], C12 =

[0.30 0.390.31 −0.14

], C22 =

[0 0.51−0.36 0.10

],

B32 =[0.24 0−0.29 0

], D2 =

[−0.12 00.59 0.11

], D12 =

[0.34 0.200.28 −0.34

],

D22 =[0 0.48−0.35 0.16

], B42 =

[0.18 0−0.19 0

],

L1 =[−0.84 00 −0.15

], L2 = −0.12I, πij2×2 =

[−0.8 0.80.2 −0.2

].

Now, we assume γ = 1, h1 = 0.5, h2 = 0.2, h = 1.9, µ = 0.1. By using the Matlab LMI control Toolbox to solve the LMIsin (28), we can obtain a desired H∞ filter in (8) and (9) with parameters as

Af1 =[0.9150 0.46232.6436 0.0814

], Bf 1 =

[0.0749 0.14350.0321 0.0152

], Cf 1 =

[−0.8401 0.00000.0002 −0.1500

],

Af 2 =[−0.0252 0.14930.2806 −0.0397

], Bf 2 =

[0.0104 0.05580.0462 −0.0446

], Cf 2 =

[−0.1202 −0.0001−0.0002 −0.1200

].

Example 2. Consider a Markovian jump delayed system with only one mode:

x (t) = Ax (t)+ A1x (t − h (t))+ B1v (t) ,z (t) = Lx (t) ,

132 H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133

Table 1Comparisons of maximum allowed h for Example 1.

a a = −1.0 a = −1.1 a = −1.2 a = −1.3 a = −1.4 a = −1.5

h by [31,20] 0.99 0.90 0.83 0.76 0.71 0.66h by Corollary 1 1.44 1.30 1.18 1.09 1.00 0.93

Table 2The upper bound of time delay hi (t) for Example 2.

µ µ = 0 µ = 0.15 µ = 0.25 µ = 0.35 µ = 0.45

h by [24] 1.7474 – – – –h by [26,27] 1.7474 1.2353 0.8939 0.5525 0.2110h by Theorem 1 3.8524 3.0501 2.5398 2.0491 1.5793

which can be obtained by modifying the system(Σ

)with A2 = H = H1 = H2 = B2 = 0,G = I . The following parameters

are borrowed from [31,32]:

A =[−2 00 −0.9

], A1 =

[a 0−1 −1

],

B1 =[−0.5 1

]T, L =

[1 0

].

To indicate the less conservatism of the result in Corollary 1, we set µ = 0 and γ = 0.3. The comparison results of themaximum allowable delay bounds h from [31,20] and our results are tabulated in Table 1 for different values of a. It isshown that our method generally produces less conservative results than those in [31,20].

Example 3. When the following Markovian jump system with MDTD is considered,

x (t) = Aix (t)+ A1ix (t − hi (t)) ,

which can be regarded as a special case of the system (Σ)with two modes with the following parameters:

A1 =[−0.5 −10 −3

], A11 =

[0.5 −0.20.2 0.3

],

A2 =[−2 1−1 0.2

], A12 =

[−0.3 0.40.4 −0.5

],

and the other parameters are set to 0. For different values of µ, the maximum allowable upper bound of MDTD can beobtained by solving the LMIs. When π11 = −0.6 and π22 = −0.4, the comparison results are listed in Table 2. It is readilyseen that our result is less conservative than those in [24,26,27].

6. Conclusions

By employing Markovian switching Lyapunov functionals, we have obtained delay-dependent conditions on thesolvability of the H∞ filtering problem for stochastic systems with Markovian switching and mixed MDTD. Slack matrixvariables are used to reduce the conservatism. Based on these conditions, an LMI-based approach has been developed to thedesign of an H∞ filter. Numerical examples have been provided to illustrate the effectiveness and benefits of the proposedapproach.

References

[1] O.L.V. Costa, J.B.R. Do Val, J.C. Geromel, Continuous-time state-feedbackH2-control of Markovian jump linear systems via convex analysis, Automatica35 (1999) 259–268.

[2] O.L.V. Costa, M.D. Fragoso, Discrete-time LQ-optimal control problems for infinite Markovian jump parameter systems, IEEE Trans. Automat. Control40 (1995) 2076–2088.

[3] M.D. Fragoso, J.B.R. Do Val, D.L. Pinto Jr., Jump linear H∞ control: The discrete-time case, Control Theory Adv. Tech. 10 (1995) 1459–1474.[4] M. Mariton, Control of nonlinear systems with Markovian parameter changes, IEEE Trans. Automat. Control 36 (1991) 233–238.[5] K. Benjelloun, E.K. Boukas, Mean square stochastic stability of linear time-delay system with Markovian jumping parameters, IEEE Trans. Automat.Control 43 (1998) 1456–1460.

[6] E.K. Boukas, Z.K. Liu, Robust H∞ control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Trans. Automat.Control 46 (2001) 1918–1924.

[7] L. Gao, Y. Wu, A design scheme of variable structure H-infinity control for uncertain singular Markovian switched systems based on linear matrixinequality method, Nonlinear Anal. Hybrid Syst. 1 (2007) 306–316.

[8] J. Luo, Moment decay rates of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Nonlinear Anal. Hybrid Syst.2 (2008) 28–37.

H. Shen et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 122–133 133

[9] J. Xiong, J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers, Automatica 42 (2006) 747–753.[10] X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Automat. Control 47 (2002) 1604–1612.[11] C. Yuan, J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian switching, Systems Control Lett. 54 (2005) 819–833.[12] C. Yuan, X. Mao, Robust stability and controllability of stochastic differential delay equations with with Markovian switching, Automatica 40 (2004)

343–354.[13] Y. Niu, D.W. Ho, X. Wang, Sliding mode control for Ito stochastic systems with Markovian switching, Automatica 43 (2007) 1784–1790.[14] C.E. de Souza, M.D. Fragoso, Robust H∞ filtering for uncertain Markovian jump linear systems, Int. J. Robust Nonlinear Control 12 (2002) 435–446.[15] G. Feng, Robust H∞ filtering of fuzzy dynamic systems, IEEE Trans. Aerosp. Electron. Syst. 41 (2005) 658–670.[16] E. Gershon, D.J.N. Limebeer, U. Shaked, I. Yaesh, Robust H∞ filtering of stationary continuous-time linear systems with stochastic uncertainties, IEEE

Trans. Automat. Control 46 (2001) 1788–1793.[17] S. Zhou, J. Lam, A. Xue, H∞ filtering of discrete-time fuzzy systems via bais-dependent Lyapunov function approach, Fuzzy Sets and Systems 158

(2007) 180–193.[18] D. Du, B. Jiang, P. Shi, S. Zhou, H∞ filtering of discrete-time switched systems with state delays via switched Lyapunov function approach, IEEE Trans.

Automat. Control 52 (2007) 1520–1525.[19] B. Zhang, S. Xu, Robust H∞ filtering for uncertain discrete piecewise time-delay systems, Internat. J. Control 80 (2007) 636–645.[20] E. Fridman, U. Shaked, A new H∞ filter design for linear time delay systems, IEEE Trans. Signal Process. 49 (2001) 2839–2843.[21] C. Lin, Q. Wang, T. Lee, Y. He, Fuzzy weight-dependent approach to H∞ filter design for time-delay fuzzy systems, IEEE Trans. Signal Processing 55

(2007) 2746–2751.[22] L. Wu, P. Shi, C. Wang, H. Gao, Delay-dependent robust H∞ and L2–L∞ filtering for LPV systems with both discrete and distributed delays, IEE Proc.

-Control Theory Appl. 153 (2006) 483–492.[23] S. Xu, J. Lam, T. Chen, Y. Zou, A delay-dependent approach to robust H∞ filtering for uncertain distributed delay systems, IEEE Trans. Signal Processing

53 (2005) 3764–3772.[24] E. Boukas, Z. Liu, Output feedback robust stabilization of jump linear system with mode-dependent time-delays, in: Proceedings of the American

Control Conference, Arlington, VA, June, 2001, pp. 4683–4688.[25] W.-H. Chen, J.-X. Xu, Z.-H. Guan, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Trans.

Automat. Control 48 (2003) 2270–2276.[26] H. Shao, Delay-range-dependent robust H∞ filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump

parameters, J. Math. Anal. Appl. 342 (2008) 1084–1095.[27] S. Xu, T. Chen, J. Lam, Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Trans. Automat. Control 48

(2003) 900–907.[28] S. Xu, Y. Chu, J. Lu, Y. Zou, Exponential dynamic output feedback controller design for stochastic neutral with distributed delays, IEEE Trans. Syst. Man

Cybern. Part A 36 (2006) 540–547.[29] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhäuser, Boston, 2003.[30] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.[31] E. Fridman, U. Shaked, Newbounded real lemma representations for time-delay systems and their applications, IEEE Trans. Automat. Control 46 (2001)

1973–1979.[32] U. Shaked, I. Yaesh, C.d. Souza, Bounded real criteria for linear time-delay systems, IEEE Trans. Automat. Control 43 (1998) 1016–1022.