Kybernetika. Terms of use: Persistent URL: Institute of Information Theory and Automation AS CR, 2014

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1 Kybernetika Li Ma; Meimei Xu; Ruting Jia; Hui Ye Exponential H filter design for stochastic Markovian jump systems with both discrete and distributed time-varying delays Kybernetika Vol. 5 (214) No Persistent URL: Terms of use: Institute of Information Theory and Automation AS CR 214 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 K Y B E R N E T I K A V O L U M E 5 ( ) N U M B E R 4 P A G E S EXPONENTIAL H FILTER DESIGN FOR STOCHASTIC MARKOVIAN JUMP SYSTEMS WITH BOTH DISCRETE AND DISTRIBUTED TIME-VARYING DELAYS Li Ma Meimei Xu Ruting Jia and Hui Ye This paper is concerned with the exponential H filter design problem for stochastic Markovian jump systems with time-varying delays where the time-varying delays include not only discrete delays but also distributed delays. First of all by choosing a modified Lyapunov Krasovskii functional and employing the property of conditional mathematical expectation a novel delay-dependent approach is developed to deal with the mean-square exponential stability problem and H control problem. Then a mean-square exponentially stable and Markovian jump filter is designed such that the filtering error system is mean-square exponentially stable and the H performance of estimation error can be ensured. Besides the derivative of discrete time-varying delay h(t) satisfies ḣ(t) η and simultaneously the decay rate can be finite positive value without equation constraint. Finally a numerical example is provided to illustrate the effectiveness of the proposed design approach. Keywords: stochastic systems distributed time-varying delay H filter linear matrix inequality Classification: 93E3 93B36 1. INTRODUCTION During the past decades the stability analysis and synthesis of stochastic Markovian jump systems with time delays have become very important research areas and considerable attention has been paid to these two subjects. The motivation for investigating this class of systems arises from the following three aspects. The first one is that time delay which is a primary source of instability and performance degradation in a dynamical system 11 is frequently encountered in engineering biology economy and other areas 12. The second one is that in real-time systems the signal transmission is usually a noisy process brought on by random fluctuations from probabilistic causes. Therefore stochastic modeling has been of vital importance in many branches of science such as biology economics and engineering applications. The abrupt phenomena such as random failures repairs of the components sudden environment changes and so on in practical systems is the last one. Usually Markov chain is used to model this class of systems. In 23 the DOI: /kyb

3 492 L. MA M. M. XU R. T. JIA AND H. YE author explains that jump systems have been emerging as an effective framework for various control problems in different fields such as target tracking manufacturing processes and fault-tolerant control systems. Many interesting results on stochastic systems with time delays have been presented e. g and references therein. On the other hand as pointed out in 18 3 when the number of summands in a system equation is increased and the differences between neighboring argument values are decreased the delay phenomena may not be simply considered as delays in the velocity terms and/or discrete delays in the states. This kind of delay phenomena is usually modeled as distributed time delays. Moreover there are important practical applications for systems including distributed time delays for example in the modeling of feeding systems and combustion chambers in a liquid monopropellant rocket motor with pressure-feeding as documented in Hence the theoretical and practical demand of taking distributed time delays into account is highly desirable. Until recently the stability analysis and synthesis problems for the systems with distributed time delays have received considerable attention. A great number of results on the above two problems have been reported. In terms of stability analysis a discretized Lypunov functional method to study the stability of systems with distributed delays is used in 9 1. However this method is complicated and is difficult to be extended to the synthesis problems. When the discrete time delays are also included and 3 investigate the asymptotical stability problem and the exponential stability problem for the neutral systems with both discrete and distributed delays by using a model transformation approach respectively. In order to further reduce the conservatism of the existed stability criteria by constructing a modified Lyapunov Krasovskii functional and employing free weighting matrices 18 studies the asymptotical stability of the neutral systems with discrete and distributed delays. For stochastic systems the exponential stability criteria are obtained for stochastic neural networks with discrete and distributed delays in Taking Makrovian jump into account in 1 2 the authors consider the stability problem for the stochastic Markovian jump neutral networks with both discrete and distributed delays. As far as the synthesis problem for the systems with distributed delays the robust H control problem is considered in 31 by employing the descriptor system approach and the H model reduction problem is addressed in 16. The references investigate the H filter design problem for systems with both discrete and distributed delays in which the discrete and distributed delays are both constant time delays. Based on the above results the H filter design approaches for systems with discrete and distributed time-varying delays are presented in It can be clearly observed that the above studies focus on the deterministic systems. For stochastic systems with discrete and distributed delays 19 and 34 consider the H control problem and the exponential output feedback controller design respectively. In addition for Markovian jump systems with discrete and distributed delays a delay-dependent H control approach is developed in 27. From the foregoing analysis it can be seen that there are few stability analysis and synthesis results available for stochastic Markovian jump systems with both discrete and distributed time-varying delays. Moreover when dealing with the exponential sta-

4 Exponential H filter design for stochastic Markovian jump systems 493 bility problem for systems with discrete and distributed time-varying delays there is a restrictive condition i. e. the derivative of the discrete time-varying delay is less than one In order to remove this restriction a weaker assumption (i. e. the derivative of time-varying delay is only bounded and not required to be less than one) is imposed see for example 44. However the decay rate must satisfy one transcendental equation or inequality for obtaining exponential stability criteria 44. To the best of the authors knowledge the exponential H filter design problem for stochastic Markovian jump systems with both discrete and distributed time-varying delays has not been investigated exactly. Motivated by the above observations in this paper we consider the exponential H filter design problem for stochastic Markovian jump systems with both discrete and distributed time-varying delays. The purpose is to design an exponential stability H filter such that the filtering error system is mean-square exponentially stable and the L 2 - induced norm from the noise signal to the estimation error is less than a prescribed level. By employing stochastic analysis technique and constructing a novel Lyap-Krasovskii functional a new approach is proposed to obtain the delay-dependent sufficient conditions for the solvability of the H filtering problem. The decay rate is not required to satisfy one transcendental equation and simultaneously the derivative of time-varying delay is not needed to be less than one. When the sufficient condition in terms of LMIs is feasible the explicit expression of the desired filter can be given. Finally a numerical example is provided to verify the effectiveness of the obtained result. 2. PROBLEM FORMULATION Consider the following stochastic Markovian jump systems with both discrete and distributed time-varying delays: dx(t) = (A 1 (r t )x(t) + A 1d (r t )x(t h(t)) + E 1 (r t ) x(s) ds + D 1 (r t )v(t)) dt + (B 1 (r t )x(t) + B 1d (r t )x(t h(t))) dω(t) t > dy(t) = (A 2 (r t )x(t) + A 2d (r t )x(t h(t)) + E 2 (r t ) + (B 2 (r t )x(t) + B 2d (r t )x(t h(t))) dω(t) t > z(t) = L(r t )x(t) + L d (r t )x(t h(t)) t > x(s) ds + D 2 (r t )v(t)) dt x(t) = φ(t) t τ (1) where x(t) R n is the state v(t) R m is the noise signal which is assumed to be an arbitrary signal in L 2 ) ω(t) is a one-dimensional Brownian motion and satisfies Eω(t) = E(ω 2 (t)) = t; y(t) R q is the measurement z(t) R p is the signal to be estimated h(t) and τ(t) denote the time-varying discrete delay and distributed delay respectively. They satisfy < h(t) h ḣ(t) η < τ(t) τ where h η τ are constants. Let τ = max{ h τ} φ(t) CF b ( τ ; R n ) be the initial condition. {r t } t is a continuous-time Markov chain taking values in a finite set S = { N}. Let Π = {π ij : i j S} be the density matrix of {r t } t. Thus π ij for i j and π ii = N j=1j i π ij. Furthermore the transition probability from mode i at time t to

5 494 L. MA M. M. XU R. T. JIA AND H. YE mode j at time t + can be described as { πij + o( ) i j P {r t+ = j r t = i} = 1 + π ii + o( ) i = j where > and lim (o( )/ ) =. In addition we assume that the Markov chain {r t } t is independent of the Brownian motion ω(t). The matrices A 1 (r t ) A 1d (r t ) E 1 (r t ) D 1 (r t ) B 1 (r t ) B 1d (r t ) A 2 (r t ) A 2d (r t ) E 2 (r t ) D 2 (r t ) B 2 (r t ) B 2d (r t ) L(r t ) and L d (r t ) are known real constant matrix functions of r t with appropriate dimensions. In order to avoid the notations becoming too complicated for each possible r t = i i S a matrix N(r t ) will be denoted by N i. It is well known that {x t r t } t is a C( τ ; R n ) S-valued Markov process (24). Its weak infinitesimal generator L acting on functional V ( ) : C( τ ; R n ) S R + R + is defined by the following formula: LV (x t r t t) = 1 lim + {EV (x t+ r t+ t + ) x t r t V (x t r t t)}. Now for stochastic time-varying delay system (1) and each i S the attention of this paper is focused on the design of a mean-square exponentially stable filter with the following form: { dˆx(t) = Afiˆx(t) dt + B fi dy(t) t > (2) ẑ(t) = C fiˆx(t) ˆx() = where ˆx R n and ẑ R p. A fi B fi C fi are filter matrices with appropriate dimensions which will be determined later. Denote ξ(t) = x T (t) ˆx T (t) T and e(t) = z(t) ẑ(t). Then from (1) and (2) for each i S we obtain the filtering error system as follows: dξ(t) = (Ãiξ(t) + ÃdiKξ(t h(t)) + ẼiK + ( B i ξ(t) + B di Kξ(t h(t)) dω(t) t > e(t) = L i ξ(t) + L di Kξ(t h(t)) t > ξ(t) = φ(t) = φ T (t) T t τ ξ(s) ds + D i v(t)) dt (3) where ( A1i à i = B fi A 2i B di = ( B1di B fi B 2di A fi ) ( A1di Ãdi = B fi A 2di ) Ẽi = ( E1i B fi E 2i ) D i = ) ( B B1i i = B fi B 2i ( D1i B fi D 2i ) K = (I ) ) L i = (L i C fi ). Based on the above discussion the exponential H filtering problem to be addressed in this paper can be stated as follows. Given scalars > γ > and the stochastic time-varying delay system (1) we will design an exponential filter in the form of (2) such that the filtering error system (3) is mean-square exponentially stable and the H performance e(t) E2 < γ v(t) 2 (4)

6 Exponential H filter design for stochastic Markovian jump systems 495 can be guaranteed for all nonzero v(t) L 2 ) under zero-initial condition i. e. ξ(t) = for t τ where e(t) E2 = {E e(t) 2 dt} 1 2 = { E e(t) 2 dt} 1 2 and 2 represents the usual L 2 ) norm i. e. v(t) 2 = { v(t) 2 dt} 1 2. Before investigating the main problems we first introduce the definition of meansquare exponential stability for the filtering error system (3). Definition 2.1. The filtering error system (3) is said to achieve mean-square exponential stability if for v(t) = any φ(t) C b F ( τ ; R 2n ) and initial mode r S there exist constant scalars d > and > such that: E{ ξ(t φ r ) 2 } d sup τ θ φ(θ) 2 e t where ξ(t φ r ) denotes the solution of system (1) at time t under the initial conditions φ( ) and r and is called the decay rate. 3. H PERFORMANCE ANALYSIS In this section we first give the following lemmas that are useful in deriving the LMIbased stability criterion. Lemma 3.1. (Gu 8) For any constant matrix M R n n M = M T > scalars b > a and vector function f( ) : a b R n such that the integrations in the following are well defined then the following inequality holds: { T { b } b b f(s) ds} M f(s) ds (b a) f T (s)mf(s) ds. a a Lemma 3.2. (Chung 4) For any random variable ξ F satisfying E ξ < + and Borel subfield G F there always has E(E(ξ G)) = Eξ. Lemma 3.3. (Evans 17) For any t R and f(s) L 2 ( t) we have { t ( t ) T ( E f(s) dω(s) = E f(s) dω(s) f(s) dω(s)) } = E a f(s) 2 ds. Lemma 3.4. (Gronwall 7) Let u(t) be nonnegative continuous functions defined on t t 1 and a b denote two constants. If u(t) b t u(s) ds+a then u(t) ae b(t t) t t t 1. In the following a delay-dependent and decay-rate-dependent sufficient condition which ensures the mean-square exponential stability and the H performance for the filtering error system (3) will be proposed. Theorem 3.5. For some given scalars h > τ > > and η if there exist matrices P i > Q > R 1 > R 2 > N ji and M ji j = for each i S such that the following LMI holds: ( ) Ω1i Ω 2i Ω T < (5) 2i Ω 3i

7 496 L. MA M. M. XU R. T. JIA AND H. YE where Θ 1i Θ 2i K T N3i T KT M 1i P i Ẽ i Ω 1i = Θ 3i N3i T M 2i + M3i T M 3i M3i T 1 τ R 2 Θ 1i = P i + N j=1 π ijp j + P i à i + ÃT i P i + K T (Q + e τ 1R 2 + N 1i + N1i T )K Θ 2i = P i à di K T (N 1i + N T 2i + M 1i) Θ 3i = ( (1 η)) ( (1 η)e h)q N 2i N2i T + M 2i + M2i T K T N 1i K T M 1i P i Di à T i KT R 1 BT i P i LT i Ω 2i = N 2i M 2i à T di KT R 1 BT di P i L T di N 3i M 3i Ẽi T KT R 1 Ω 3i = 1 hr 1 1 hr 1 γ 2 I DT i K T R 1 e h 1 R 1 P i I then the filtering error system (3) is mean-square exponentially stable with decay rate and the H performance (4) is satisfied under zero-initial condition for all nonzero v(t) L 2 ). P r o o f. Let ϕ(t) = Ãiξ(t) + ÃdiKξ(t h(t)) + Ẽi Kξ(s) ds ψ(t) = B i ξ(t) + B di Kξ(t h(t)). Considering the filtering error system (3) with v(t) the first equation in (3) becomes dξ(t) = ϕ(t) dt + ψ(t) dω(t) t >. (6) Firstly for t τ we choose a Lyapunov Krasovskii functional candidate as follows: V (ξ t r t t) = V 1 (ξ t r t ) + e t V 2 (ξ t r t t) + e t V 3 (ξ t r t t) + e t V 4 (ξ t r t t) (7) where V 1 (ξ t r t ) = ξ T (t)p (r t )ξ(t) V 2 (ξ t r t t) = V 3 (ξ t r t t) = V 4 (ξ t r t t) = t h(t) h t+θ τ t+θ e s ξ T (s)k T QKξ(s) ds e (s θ) ϕ T (s)k T R 1 Kϕ(s) dsdθ e (s θ) ξ T (s)k T R 2 Kξ(s) dsdθ.

8 Exponential H filter design for stochastic Markovian jump systems 497 By Itô s formula for each r t = i i S we obtain the stochastic differential of V (ξ t r t t) as follows: with LV 1 (ξ t i) = dv (ξ t i t) = LV (ξ t i t) dt + 2ξ T (t)p i ψ(t) dω(t) = (LV 1 (ξ t i) + L(e t V 2 (ξ t i t)) + L(e t V 3 (ξ t i t)) + L(e t V 4 (ξ t i t))) dt + 2ξ T (t)p i ψ(t) dω(t) N π ij ξ T (t)p j ξ(t) + 2ξ T (t)p i ϕ(t) + ψ T (t)p i ψ(t) j=1 L(e t V 2 (ξ t i t)) = ξ T (t)k T QKξ(t) (1 ḣ(t))e h(t) ξ T (t h(t))k T QKξ(t h(t)) e t V 2 (ξ t i t) L(e t e h 1 V 3 (ξ t i t)) = ϕ T (t)k T R 1 Kϕ(t) e t V 3 (ξ t i t) L(e t e τ 1 V 4 (ξ t i t)) = ξ T (t)k T R 2 Kξ(t) e t V 4 (ξ t i t). By a simple calculation we have So we obtain LV (ξ t i t) t h t τ ϕ T (s)k T R 1 Kϕ(s) ds ξ T (s)k T R 2 Kξ(s) ds (1 ḣ(t))e h(t) { (1 η)e h(t) (1 η)e h η 1 (1 η) η > 1 = ( (1 η)) ( (1 η)e h). N π ij ξ T (t)p j ξ(t) + 2ξ T (t)p i ϕ(t) + ψ T (t)p i ψ(t) + ξ T (t)k T QKξ(t) j=1 +( (1 η)) ( (1 η)e h)ξ T (t h(t))k T QKξ(t h(t)) e h 1 t + ϕ(t) T K T R 1 Kϕ(t) ϕ(s) T K T R 1 Kϕ(s) ds e τ h 1 t + ξ T (t)k T R 2 Kξ(t) ξ T (s)k T R 2 Kξ(s) ds t τ e t V 2 (ξ t i t) e t V 3 (ξ t i t) e t V 4 (ξ t i t). (9) It follows from Lemma 3.1 that { t } T { ϕ T (s)k T R 1 Kϕ(s) ds 1 h t h(t) } t h(t) Kϕ(s) ds R 1 Kϕ(s) ds t h t h t h { } T { 1 h t } t Kϕ(s) ds R 1 Kϕ(s) ds. t h(t) t h(t) (1) (8)

9 498 L. MA M. M. XU R. T. JIA AND H. YE Similarly we also have { t } T { ξ T (s)k T R 2 Kξ(s) ds 1 τ } Kξ(s) ds R 2 Kξ(s) ds t τ t τ t τ { } T { 1 τ t } t Kξ(s) ds R 2 Kξ(s) ds. (11) From (6) and the Newton Leibniz formula one has (ξ T (t)k T N 1i + ξ T (t h(t))k T N 2i + ξ T (t h)k T N 3i )K(ξ(t) ξ(t h(t)) t h(t) ϕ(s) ds t h(t) ψ(s) dω(s) = (ξ T (t)k T M 1i + ξ T (t h(t))k T M 2i + ξ T (t h)k T M 3i )K(ξ(t h(t)) h(t) h(t) ξ(t h) ϕ(s) ds ψ(s) dω(s)) =. (13) t h t h Substituting (1) (13) into (9) yields { } T { } LV (ξ t i t)+v (ξ t i t) ζ T (t)γ i ζ(t) 1 τ Kξ(s) ds R 2 Kξ(s) ds t τ t τ 2(ξ T (t)k T N 1i + ξ T (t h(t))k T N 2i + ξ T (t h)k T N 3i )K ψ(s) dω(s) t h(t) h(t) 2(ξ T (t)k T M 1i + ξ T (t h(t))k T M 2i + ξ T (t h)k T M 3i )K ψ(s) dω(s) t h (14) where ζ(t) = ξ T (t) (Kξ(t h(t))) T (Kξ(t h)) ( ) T ( T T Kξ(s) ds Kϕ(s) ds) t h(t) ( T t h(t) T Kϕ(s) ds) t h Θ 1i Θ2i K T (N3i T M 1i) P i Ẽ i + e h 1 à i KT R 1 KẼi K T N 1i K T M 1i Θ3i N3i T M 2i + M T e h 1 3i à di KT R 1 KẼi N 2i M 2i M 3i M T 3i N 3i M 3i Γ i = e 1 Ẽ i T KT R 1 KẼi 1 τ R 2 1 h R 1 1 h R 1 Θ 1i = Θ 1i + B i T P B e h 1 i i + à T i K T R 1 KÃi Θ 2i = Θ 2i + B i T P B e h 1 i di + à T i K T R 1 KÃdi Θ 3i = Θ 3i + B di T P B e h 1 i di + à T dik T R 1 KÃdi. (12)

10 Exponential H filter design for stochastic Markovian jump systems 499 Clearly it follows from (5) and Schur complement that Γ i <. Hence we have E(LV (ξ t i t)) + EV (ξ t i t) < for each i S. Then from the definition of weak infinitesimal generator L and Lemma 3.2 it is not difficult to verify that 1 E(LV (ξ t r t t)) = lim + E{EV (ξ t+ r t+ t + ) ξ t r t V (ξ t r t t)} 1 = lim + {EV (ξ t+ r t+ t + ) EV (ξ t r t t)} = de(v (ξ t r t t)). dt Consequently we obtain dev (ξ t r t t) EV (ξ t r t t) By integrating (15) from τ to t t > τ it can be testified that + dt <. (15) ln EV (ξ t r t t) + (t τ) <. (16) EV (ξ τ r τ τ) In addition by taking exponent of both sides of (16) we obtain EV (ξ t r t t) EV (ξ τ r τ τ)e (t τ) τ = E{ξ T (τ)p (r τ )ξ(τ) + e τ e s ξ T (s)k T QKξ(s) ds + e τ + e τ τ h τ+θ τ τ τ+θ τ h(τ) e (s θ) ϕ T (s)k T R 1 Kϕ(s) dsdθ e (s θ) ξ(s)k T R 2 Kξ(s) dsdθ}e (t τ). (17) Let µ 1 = max i S { Ãi } µ 2 = max i S { B i } µ 3 = max i S { ÃdiK } µ 4 = max i S { B di K } µ 5 = max i S { ẼiK }. From Lemma 3.3 and (6) for < t τ it is easy to check that 2 E ξ(t) 2 = E ξ() + ϕ(s) ds + ψ(s) dω(s) t 2 sup φ(θ) 2 + 3τE ( Ã(r s)ξ(s) + Ãd(r s )Kξ(s h(s)) τ θ s + Ẽ(r s)k Notice that s τ(s) ξ(θ) dθ ) 2 ds + 2E Ẽ(r s)k s s τ(s) ξ(θ) dθ ( B(r s )ξ(s) + B d (r s )Kξ(s h(s)) ) 2 ds. 2 ds µ 2 5τ τ ξ(θ) 2 dθ. (18)

11 5 L. MA M. M. XU R. T. JIA AND H. YE Then combined with (18) we have E ξ(t) 2 2(1 + 3τ 2 µ τµ 2 4) sup φ(θ) (3τµ τ 2 µ µ 2 2)E ξ(s) 2 ds. τ θ τ (19) Using Lemma 3.4 for t τ it follows from (19) that E ξ(t) 2 d 1 sup τ θ where d 1 = 2(1 + 3τ 2 µ τµ 2 4)exp{4τ(3τµ τ 2 µ µ 2 2)}. Consequently from (17) and (2) we have EV (ξ t r t t) < e τ { max i S { P i }d 1 + d 1 K T QK 1 e τ K T R 1 K eτ 1 τ 2 Notice that This together with (21) implies for t > τ + 2(µ 2 2d 1 + µ 2 4) K T R 2 K eτ 1 τ } 2 1 EV (ξ t r t t) 1 max{ Pi } E ξ(t) 2. i S φ(θ) 2 (2) + 2(µ 2 1d 1 + µ µ 2 5τd 1 ) sup τ θ φ(θ) 2 e t. (21) E ξ(t) 2 1 max{ Pi }e τ d 2 sup φ(θ) 2 e t (22) i S τ θ where d 2 = max { P i }d 1 +d 1 K T QK 1 e τ i S + 2(µ 2 2d 1 + µ 2 4) K T R 2 K eτ 1 τ. 2 On the other hand for < t τ there has It implies e t E ξ(t) 2 e τ d 1 E ξ(t) 2 e t e τ d 1 +2(µ 2 1d 1 +µ 2 3+µ 2 5τd 1 ) K T R 1 K eτ 1 τ 2 sup τ θ sup τ θ φ(θ) 2. φ(θ) 2. (23) Evidently for t > taking into account (22) and (23) the following estimate holds E ξ(t) 2 e t d sup τ θ φ(θ) 2 where d = max{e τ max i S { P 1 i }d 2 e τ d 1 }. Then according to Definition 2.1 the filtering error system (3) is mean-square exponentially stable with decay rate. Next under the zero initial condition i. e. φ(t) = we will prove the H performance of the filtering error system (3) is satisfied for all nonzero v(t) L 2 ). We choose another Lyapunov Krasovskii functional candidate as follows: ˆV (ξ t r t t) = V 1 (ξ t r t ) + e t V 2 (ξ t r t t) + e t ˆV3 (ξ t r t t) + e t V 4 (ξ t r t t)

12 Exponential H filter design for stochastic Markovian jump systems 51 where V 1 V 2 V 4 are defined in the line below (7) and For any T > let ˆV 3 (ξ t r t t) = ˆϕ(t) = J(T ) = E h t+θ e (s θ) ˆϕ T (s)k T R 1 K ˆϕ(s) dsdθ { ϕ(t) + D(rt )v(t) t > τ t. T (e T (t)e(t) γ 2 v T (t)v(t)) dt. (24) For t τ ) we let v(t) =. Similar to the proof of (14) using Itô s formula for r t = i i S we have { } T { L ˆV (ξ t i t) + ˆV (ξ t i t) ˆζ T 1 } (t)ˆγ i ˆζ(t) Kξ(s) ds R 2 Kξ(s) ds τ 2(ξ T (t)k T N 1i + ξ T (t h(t))k T N 2i + ξ T (t h)k T N 3i )K 2(ξ T (t)k T M 1i + ξ T (t h(t))k T M 2i + ξ T (t h)k T M 3i )K where t τ t h(t) h(t) t h t τ ψ(s) dω(s) ψ(s) dω(s) ˆζ(t) = ξ T (t) (Kξ(t h(t))) T (Kξ(t h)) T ( Kξ(s) ds)t ( t h(t) K ˆϕ(s) ds)t ( h(t) K ˆϕ(s) ds) T v T (t) T t h ( ) Γi Ψ i ˆΓ i = Ψ i = Ψ T i e h 1 (P i Di + e h 1 D T i KT R 1 K D i à T i KT R 1 K D i ) T ( e h 1 à T di KT R 1 K D i ) T ( e h 1 Ẽi T KT R 1 K D i ) T ( e h 1 D i T KT R 1 K D T i ) T. Then it can be deduced from (24) that J(T ) = E T (e T (t)e(t) γ 2 v T (t)v(t)) dt + E E ˆV (ξ T r T T ) + E ˆV (ξ r ). T L ˆV (ξ t r t t) dt Note that E ˆV (ξ T r T T ) and E ˆV (ξ r ) =. This implies for r t = i i S J(T ) E E = E T T T (e T (t)e(t) γ 2 v T (t)v(t)) dt + E (e T (t)e(t) γ 2 v T (t)v(t)) dt + E ˆζ T (t)λ(r t )ˆζ(t) dt T T L ˆV (ξ t r t t) dt L ˆV (ξ t r t t) dt + E T ˆV (ξ t r t t) dt

13 52 L. MA M. M. XU R. T. JIA AND H. YE where for r t = i Λ i = ˆΓ i + L T L i i LT i L di L T di L di γ 2 I. Using (5) and Schur complement we have Λ i <. Hence it follows that e(t) E2 < γ v(t) 2. Remark 3.6. During the proof of Theorem 3.5 in order to prove the mean-quare exponential stability a novel Lyapunov Krasovskii functional is constructed. Then combining with the property of conditional mathematical expectation and the stochastic analysis technique a new approach to achieving the delay-dependent mean-square exponential stability criterion is presented. This approach makes use of the knowledge of stochastic process sufficiently and is different from the approach proposed in 21 where the mean-square exponential stability problem is also solved but the property of conditional mathematical expectation has not been used. On the other hand we also consider the exponential H filter design problem in 22. However Ref. 22 in which the measurement missing phenomenon is taken into account studies the exponential H filter design problem for discrete stochastic systems while this paper studies the exponential H filter design problem for continuous case. Remark 3.7. To the best of the authors knowledge the decay rate in many literatures is a fixed value computed by solving a transcendental equation or inequality for example Moreover these results can not tell us whether a system can possess a larger decay rate than the computed value. Here the decay rate in Theorem 1 is a constant value without the above-mentioned constraint. This will introduce more flexibility in the analysis of systems. In addition the LMIs-based condition (5) is not only delay-dependent but also decay-rate-dependent. The delay-dependence of the sufficient condition makes the result less conservative while the decay-rate-dependence enables that a suboptimal upper bound of the decay rate can be computed by convex optimization algorithms. At the same time by using the inequality (8) the restriction on η that is η < 1 has been removed and a more general result is obtained. 4. EXPONENTIAL H FILTER DESIGN In this section based on Theorem 3.5 we aim to design the filter matrices A fi B fi and C fi for each i S. Theorem 4.1. For given constants > γ > η > τ > and h > the H filtering problem is solvable if there exist matrices Q > R 1 > R 2 > X i > Y i > V i Z i W i and any appropriately dimensional matrices M ji N ji j = for each

14 Exponential H filter design for stochastic Markovian jump systems 53 i S such that the following LMI holds: ( ) Ω1i Ω2i Ω T < (25) 2i Ω 3i where Ω 1i = Θ 11i Θ12i Θ21i N T 3i M 1i Y i E 1i Θ13i Θ22i (Y i X i )E 1i + V i E 2i Θ 3i N T 3i M 2i+M T 3i M 3i M T 3i 1 τ R 2 Θ 11i = N j=1 π ijy j + Y i + Y i A 1i + A T 1i Y i + e τ 1 R 2 + Q + N 1i + N T 1i Θ 12i = A T 1i (Y i X i ) + A T 2i V T i + Z T i Θ 13i = N j=1 π ij(x j Y j ) + (X i Y i ) + Z i + Z T i Θ 21i = Y i A 1di + N T 2i N 1i + M 1i Θ22i = (Y i X i )A 1di + V i A 2di Θ 3i = ( (1 η)) ( (1 η)e h)q N 2i N T 2i + M 2i + M T 2i Ω 2i = N 1i M 1i Y i D 1i A T 1i R 1 B T 1i Y i (Y i X i )D 1i + V i D 2i N 2i M 2i A T 1di R 1 B T 1di Y i N 3i M 3i E 1i R 1 B1i T (Y i X i ) + B2i T V i T L T i Wi T Wi T B T 1di (Y i X i ) + B T 2di V T i L T di 1 hr 1 1 hr 1 Ω 3i = γ 2 I D1i T R 1 e τ 1 R. 1 Y i Y i X i I Moreover for each i S a desired filter is given in the form of (2) with filter matrices as follows: A fi = (X i Y i ) 1 Z i B fi = (X i Y i ) 1 V i C fi = W i. (26)

15 54 L. MA M. M. XU R. T. JIA AND H. YE P r o o f. Firstly from (25) we can obtain X i Y i > by Schur complement. Consequently X i Y i is invertible. For each i S let ( ) X P i = i Y i X i. (27) Y i X i X i Y i By using Schur complement again we have P i >. For each i S let Ȳi = Y 1 i. Pre- and post-multiplying (25) by Σ 1i = diag{ȳi I I I I I I I I Ȳi I I} and Σ T 1i respectively we have ˆΘ 11i ˆΘ12i ˆΘ21i Ȳ i (N T 3i M 1i) E 1i ȲiN 1i ȲiM 1i Θ13i Θ22i (Y i X i )E 1i + V i E 2i Θ 3i N T 3i M 2i+M T 3i N 2i M 2i M 3i M T 3i N 3i M 3i 1 τ R 2 1 hr 1 1 hr 1 D 1i Ȳ i A T 1i R 1 Ȳ i B1i T Ȳ i (B1i T (Y i X i )+B2i T V i T ) Ȳ i (L T i W i T ) (Y i X i )D 1i + V i D 2i Wi T < where ˆΘ 11i = j=1 A T 1di R 1 B T 1di B T 1di (Y i X i )+B T 2di V T i E 1i R 1 γ 2 I D1i T R 1 e τ 1 R 1 Ȳi Y i X i I N ( e π ij Ȳ i Y j Ȳ i + Ȳi + A 1i Ȳ i + ȲiA T τ ) 1 1i + Ȳi R 2 + Q + N 1i + N1i T Ȳ i ˆΘ 12i = ȲiA T 1i(Y i X i ) + ȲiA T 2iV T i + ȲiZ T i ˆΘ 21i = A 1di + Ȳi(N T 2i N 1i + M 1i ). L T di (28)

16 Exponential H filter design for stochastic Markovian jump systems 55 ( ) Ȳi If we define Λ i = i S one can conclude from (26) and (27) that (28) Ȳ i I can be rewritten as: Λ T i Θ 1iΛ i Λ T i Θ 2i Λ T i KT (N3i T M 1i) Λ T i P iẽi Λ T i KT N 1i Λ T i KT M 1i Θ 3i N 3i T M 2i + M3i T N 2i M 2i M 3i M3i T N 3i M 3i 1 τ R 2 1 hr 1 1 hr 1 Λ T i P D i i Λ T i ÃT i KT R 1 Λ T i B i T P iλ i Λ T L i T i à T di KT R 1 BT di P i Λ i L T di Ẽi T KT R 1 γ 2 I DT i K T R 1 e h 1 R 1 Λ T i P iλ i I Now pre- and post-multiplying the above LMI by Σ 2i = diag{(λ T i ) 1 I I I I I I I (Λ T i ) 1 I} and Σ T 2i respectively we can obtain from Theorem 3.5 that the filter in (2) assures the solvability of H filtering problem for the filtering error system (3). Remark 4.2. When let E 1i = E 2i = for each i S the stochastic time-varying delay system (1) becomes the systems investigated in 21 in which the exponential H filter design problem is also considered and the decay rate has no equation constraint. However the derivative of time-varying delay is required to less than one. Here this confined condition is eliminated. The sufficient criteria obtained in Theorem 3.5 and Theorem 4.1 are also valid when the derivative of discrete time-varying delay is not less than one. Hence the results obtained in this paper are more general than the ones in NUMERICAL EXAMPLE Example 5.1. In the following example we consider the stochastic Markovian jump systems (1) with two modes that is S = {1 2}. The systems data are shown as follows: In mode 1: A 11 = A 1d1 = E 11 = 1.2 D 11 =.3

17 56 L. MA M. M. XU R. T. JIA AND H. YE B 11 = B 1d1 =.1.3 A 21 =.2.5 A 2d1 =.3 E 21 = 1 D 21 =.5 B 21 =.5 B 2d1 =.1.6 L 1 =.3.5 In mode 2:.5 A 12 =.1.2 B 12 = L d1 = A 1d2 = B 1d2 = 1 1 E 12 = 1.1 D 12 =.4 A 22 = 1.2 A 2d2 =.1.1 E 22 = 1 D 22 = 1 B 22 =.1 B 2d2 =.5 L 2 =.1 L d2 =.1. The density matrix Π = {π ij } is given by.5.5 Π = 1 1 The suboptimal upper bound of decay rate can be computed by Matlab LMI toolbox. With above parameters and h = 1.2 τ =.4 η = 1.2 γ =.5 from (25) we can obtain the suboptimal upper bound of decay rate is For demonstration purpose the value of is fixed as.1. By using the Matlab LMI Toolbox to solve LMI (25) a set of feasible solution is obtained as follows: X 1 = X 2 = Y 2 = Q = R 2 = V = V = Z 1 = Z = W 2 = M 11 = M 21 = M = M 32 = N = N 21 = N = Y 1 = R 1 = W 1 = M 12 = M 31 = N 12 = N 31 =

18 Exponential H filter design for stochastic Markovian jump systems 57 N 32 = So according to Theorem 4.1 the H filtering problem is solvable. Then from (26) the desired filter matrices can be given as: A f1 = B f1 = C f1 = A f2 = B f2 = C f2 = If we assume that the discrete time-varying delay h(t) =.5 sin 2 (t) the distributed time-varying delay τ(t) =.4 cos 2 (t) and the noise signal v(t) = 1 1+2t which belongs to L 2 ) the simulation results of estimation error e(t) are shown in Figures 1 2. From these two figures we can verify that the designed filter is effective for exponentially stabilizing the filter error system (3) with decay rate Estimation error e(t) times Fig. 1. The estimation error e(t) at mode 1. Estimation error e(t) times Fig. 2. The estimation error e(t) at mode CONCLUSION In this paper we consider the stochastic Markovian jump systems with time-varying delays in which the discrete and distributed time-varying delays are both included. By

19 58 L. MA M. M. XU R. T. JIA AND H. YE constructing a suitable Lyapunov Krasovskii functional and employing the property of conditional mathematical expectation sufficient condition that guarantees the meansquare exponential stability and the H performance satisfied for the filtering error system is established. Then based on obtained sufficient condition the exponential H filter is designed. There is no conventional constraint on the decay rate and the derivative of discrete time-varying delay is only required to satisfy ḣ(t) < η with any constant η. A numerical example is given to demonstrate the effectiveness of the proposed result in this paper. ACKNOWLEDGEMENT This work was partially supported by Natural Science Foundation of China (612354) the Priority Academic Program Development of Jiangsu Higher Education Institutions Initial Research Fund of Highly Specialized Personnel from Jiangsu University (No 11JDG13) Open Foundation of Key Laboratory of Measurement and Control of Complex Systems of Engineering Ministry of Education (No MCCSE214A1) and Specialized Research Fund for the Doctoral Program of Higher Education of China ( ). (Received January ) R E F E R E N C E S 1 P. Balasubramaniam and R. Rakkiyappan: Delay-dependent robust stability analysis for Markovian jumping stochastic Cohen-Grossberg neural networks with discrete interval and distributed time-varying delays. Nonlinear Anal. Hybrid Syst. 3 (29) Y. Y. Cao J. Lam and L. Hu: Delay-dependent stochastic stability and H analysis for time-delay systems with Markovian jumping parameters. J. Franklin Inst. 34 (23) W. H. Chen and W. Zheng: Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43 (27) K. L. Chung: A Course In Probability Theory. Academic Press London Y. C. Ding H. Zhu S. M. Zhong and Y. P. Zhang: L 2 L filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities. Commun. Nonlinear Sci. Numer. Simul. 17 (212) Y. A. Fiagbedzi and A. E. Pearson: A multistage reduction technique for feedback stabilizing distributed time-lag systems. Automatica 23 (1987) T. H. Gronwall: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 2 (1919) K. Gu: An integral inequality in the stability problem of time-delay systems. In: Proc. 39th IEEE Conference on Decision and Control Sydney 2 pp K. Gu: An improved stability criterion for systems with distributed delays. Int. J. Robust Nonlinear Control 13 (23) K. Gu Q. L. Han C. J. Albert and S. I. Niculescu: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74 (21) J. K. Hale: Theory Of Functional Differential Equations. Springer New York 1977.

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22 Exponential H filter design for stochastic Markovian jump systems 511 Ruting Jia Department of Electrical Engineering and Computer Science McNeese State University Lake Charles Louisiana 765. U. S. A. rja@mcneese.edu Hui Ye School of Mathematics and Physics Jiangsu University of Science and Technology Zhenjiang 212. P. R. China. yehui 1978@163.com