# RobustH∞ Filtering for General Nonlinear Stochastic State-Delayed Systems

Authors: Weihai Zhang; Gang Feng; Qinghua Li
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper studies the robustH∞ filtering problem of nonlinear stochastic systems
with time delay appearing in state equation, measurement, and controlled output, where the
state is governed by a stochastic Itô-type equation. Based on a nonlinear stochastic bounded
real lemma and an exponential estimate formula, an exponential (asymptotic) mean square H∞
filtering design of nonlinear stochastic time-delay systems is presented via solving a Hamilton-Jacobi inequality. As one corollary, for linear stochastic time-delay systems, a Luenberger-type filter is obtained by solving a linear matrix inequality. Two simulation examples are finally given
to show the effectiveness of our results.


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## Body

## 1. Introduction

RobustH∞ filtering has been studied extensively for more than two decades, which is very useful in signal processing and engineering applications; see [1–7] and the references therein. Compared with classical Kalman filter, one does not need to know the exact statistic information about the external disturbance in the H∞ filtering design. H∞ filtering requires one to design a filter such that the ℒ2-gain from the external disturbance to the estimation error is below a prescribed level γ>0. Stochastic Itô modelling has become more and more important in both theory and practical applications such as in mathematical finance and population models [8]. In recent years, the study of stochastic H∞ filtering for the systems governed by stochastic Itô-type equations has attracted a great deal of attention, for example, [2, 5, 9]. References [2, 5] presented approaches to linear stochastic delay-free and time delayed H∞ filtering design via linear matrix inequalities (LMIs), respectively. Reference [9] first solved the nonlinear stochastic delay-free H∞ filtering problem by means of a stochastic bounded real lemma derived in [10]. References [11, 12], respectively, solve the H∞ filtering and control of nonlinear stochastic time delayed systems, where time delay only happens in the state equation.It is well known that time delay phenomena are often encountered in many engineering applications such as network control and communication, and a study of time delay systems has been a popular research topic for a long time [13]. Stochastic time delay systems are ideal models in mathematical finance and population growth theory [8]. Recently, [14, 15] investigated the Kalman filter problem of linear stochastic time delay systems. Reference [5] presented an approach to stochastic H∞ filtering design for linear uncertain time delay systems via LMIs. Reference [11] first studied the H∞ design issue for a class of nonlinear stochastic time delayed systems under a stronger assumption (assumption 2.1 of [11]), for which only the state equation contains a time delay. Because, in practice, time delay often exists not only in a state equation but also in a measurement equation and a controlled output, it is necessary to study such a nonlinear stochastic H∞ filtering design.To our best knowledge, few works onH∞ filtering have been reported for general nonlinear stochastic time delay systems. The aim of this paper is to study the robust H∞ filtering design for nonlinear stochastic state-delayed systems, where the time delay appears in the state equation, measurement equation, and controlled output. Similar to Proposition  1 of [9], a nonlinear stochastic bounded real lemma for time delay systems is obtained, and then an exponential estimate formula is also presented. Finally, based on our developed nonlinear stochastic bounded real lemma and exponential estimate formula, we present a sufficient condition for exponential and asymptotic mean square H∞ filtering synthesis of nonlinear stochastic time delay systems via solving a constrained Hamilton-Jacobi inequality (HJI), respectively. Compared with the delay-free H∞ filtering [9], the current HJI depends on more variables due to the appearance of time delays. A key procedure to derive an exponential mean square H∞ filtering is to develop an exponential estimate formula (Lemma 2.3), which is very useful in its own right. In particular, in the case of linear time-invariant-delayed systems, if a quadratic Lyapunov function is chosen, then the HJI reduces to an LMI, which may be easily solved by the existing Matlab control toolbox [16].For convenience, we adopt the following traditional notations:A′: transpose of the matrix A; A≥0 (A>0): A is a positive semidefinite (positive definite) matrix; I: identity matrix. ∥x∥: Euclidean 2-norm of n-dimensional real vector x; ℒℱ2(ℛ+,ℛl): the space of nonanticipative stochastic processes y(t) with respect to filtration ℱt satisfying ∥y(t)∥L22:=E∫0∞∥y(t)∥2dt<∞; C2,1(U,T): class of functions V(x,t) twice continuously differentiable with respect to x∈U and once continuously differentiable with respect to t∈T except possibly at x=0; Vt(x,t):=(∂V(x,t))/∂t; Vx(x,t):=(∂V(x,t)/∂xi)n×1; Vxx(x,t):=(∂2V(x,t)/∂xi∂xj)n×n; C([-τ,0],ℛn): a vector space of all continuous ℛn-valued functions defined on [-τ,0].


## 2. Preliminaries

Consider the following nonlinear stochastic time delay system:(2.1)dx(t)=(f(x(t),x(t-τ),t)+g(x(t),x(t-τ),t)v(t))dt+(h(x(t),x(t-τ),t)+s(x(t),x(t-τ),t)v(t))dW(t),y(t)=l(x(t),x(t-τ),t)+k(x(t),x(t-τ),t)v(t),z(t)=m(x(t),x(t-τ),t),x(t)=ϕ(t)∈CF0b([-τ,0];Rn),
where x(t)∈ℛn is called the system state, y(t)∈ℛr is the measurement, W(·) is a standard one-dimensional Wiener process defined on a complete filtered space (Ω,ℱ,{ℱt}t∈ℛ+,𝒫) with a filtration {ℱt}t∈ℛ+ satisfying usual conditions, z(t)∈ℛm is the state combination to be estimated, v∈ℒℱ2(ℛ+,ℛnv) stands for the exogenous disturbance signal, which is a square integrable, ℱt-adapted stochastic process, and Cℱ0b([-τ,0];ℛn) denotes all ℱ0-measurable bounded C([-τ,0],Rn)-valued random variable ξ(s) with s∈[-τ,0]. We assume that f,h:ℛn×ℛn×ℛ+↦ℛn, g,s:ℛn×ℛn×ℛ+↦ℛn×nv, l:ℛn×ℛn×ℛ+↦ℛr, k:ℛn×ℛn×ℛ+↦ℛr×nv, and m:ℛn×ℛn×ℛ+↦ℛnz satisfy the local Lipschitz condition and the linear growth condition, which guarantee that the system (2.1) admits a unique strong solution; see [8]. In addition, suppose that f(0,0,t)=h(0,0,t)=l(0,0,t)≡0, so x≡0 is an equilibrium point of (2.1).Since this paper deals with the infinite horizon stochasticH∞ filtering problem, it is inevitably related to stochastic stability. Hence, we first present the following definition.Definition 2.1.
The nonlinear stochastic time delayed system(2.2)dx(t)=f(x(t),x(t-τ),t)dt+h(x(t),x(t-τ),t)dW(t),x(t)=ϕ(t)∈CF0b([-τ,0];Rn),
is called exponentially mean square stable if there are positive constants A and α such that
(2.3)E‖x(t)‖2≤A‖ϕ‖2e-αt,
where ∥ϕ∥2=Emax-τ≤t≤0∥ϕ(t)∥2.
Associated with (2.1) and V:ℛn×ℛ+↦ℛ+, we define an operator ℒ1V:ℛn×ℛn×ℛ+↦ℛ as follows:
(2.4)L1V(x,y,t)=Vt(x,t)+Vx′(x,t)[f(x,y,t)+g(x,y,t)v(t)]+12[h(x,y,t)+s(x,y,t)v(t)]′Vxx(x,t)[h(x,y,t)+s(x,y,t)v(t)].The following lemma is a generalized version of Proposition  1 in [9], which may be viewed as a nonlinear stochastic bounded real lemma for time delayed systems.Lemma 2.2.
Consider the following input-output system:(2.5)dx(t)=(f(x(t),x(t-τ),t)+g(x(t),x(t-τ),t)v(t))dt+(h(x(t),x(t-τ),t)+s(x(t),x(t-τ),t)v(t))dW(t)z(t)=m(x(t),x(t-τ),t),x(t)=ϕ(t)∈CF0b([-τ,0];Rn).,
If there exists a positive definite Lyapunov function V(x,t)∈C2,1(ℛn,ℛ+) solving the following HJI:
(2.6)Γ(x,y,t):=Vt(x,t)+Vx′(x,t)f(x,y,t)+12(Vx′(x,t)g(x,y,t)+h′(x,y,t)Vxx(x,t)s(x,y,t))×(γ2I-s′(x,y,t)Vxx(x,t)s(x,y,t))-1×(g′(x,y,t)Vx(x,t)+s′(x,y,t)Vxx(x,t)h(x,y,t))+12‖z(t)‖2+12h′(x,y,t)Vxx(x,t)h(x,y,t)<0γ2I-s′(x,y,t)Vxx(x,t)s(x,y,t)>0,∀(x,y,t)∈Rn×Rn×R+,V(0,0)=0
for some γ>0, then the following inequality:
(2.7)‖z(t)‖L22≤γ2‖v(t)‖L22,∀v∈LF2(R+,Rnv),v≠0,
holds with initial state x(s)=0, a.s., for all s∈[-τ,0].Proof.
See AppendixA.Lemma 2.3.
Consider the unforced system(2.8)dx(t)=f(x(t),x(t-τ),t)dt+h(x(t),x(t-τ),t)dW(t),x(t)=ϕ(t)∈CF0b([-τ,0];Rn).
If there exists a positive definite Lyapunov function V(x,t)∈C2,1(Rn,[-τ,∞)), c1,c2,c3,c4>0 with c1c3>c2c4 satisfying the following conditions:(i)
c1∥x∥2≤V(x,t)≤c2∥x∥2, for all (x,t)∈ℛn×[-τ,∞),(ii)
ℒ1V(x,y,t)|v=0≤-c3∥x∥2+c4∥y∥2, for all t>0,
then(2.9)E∥x(t)∥2≤{(c4c2/c1)τ+c2c1‖ϕ‖2e-(c3/c2)t,0≤t≤τ,(c4c2/c1)τ+c2c1∥ϕ∥2e-((c3/c2)-(c4/c1))t,t>τ,
that is, (2.8) is exponentially mean square stable.Proof.
See AppendixB.In what follows, we construct the following filtering equation for the estimation ofz(t): (2.10)dx̂(t)=f̂(x̂(t),x̂(t-τ),t)dt+Ĝ(x̂(t),x̂(t-τ),t)y(t)dtẑ(t)=m̂(x̂(t),x̂(t-τ),t),x̂(0)=0,
where f̂, Ĝ, and m̂ that are to be determined are matrices of appropriate dimensions. One may find that (2.10) is more general, which includes the following Luenberger-type filtering as a special form:
(2.11)dx̂(t)=f(x̂(t),x̂(t-τ),t)dt+G(x̂(t),x̂(t-τ),t)(y(t)-l(x̂(t),x̂(t-τ),t))dt,ẑ(t)=m(x̂(t),x̂(t-τ),t),x̂(0)=0.
Set η(t)=[x′(t)x̂′(t)]′, and let
(2.12)z̃(t)=z(t)-ẑ(t)=m(x(t),x(t-τ),t)-m̂(x̂(t),x̂(t-τ),t)
denote the estimation error; then we get the following augmented system:
(2.13)dη(t)=(fe(η(t))+ge(η(t))v(t))dt+(he(η(t))+se(η(t))v(t))dW(t),z̃(t)=z(t)-ẑ(t)=m(x(t),x(t-τ),t)-m̂(x̂(t),x̂(t-τ),t),η(t)=[ϕ(t)0],ϕ(t)∈CF0b([-τ,0];Rn),∀t∈[-τ,0],
where
(2.14)fe(η(t))=[f(x(t),x(t-τ),t)f̂(x̂(t),x̂(t-τ),t)+Ĝ(x̂(t),x̂(t-τ),t)l(x(t),x(t-τ),t)],ge(η(t))=[g(x(t),x(t-τ),t)Ĝ(x̂(t),x̂(t-τ),t)k(x(t),x(t-τ),t)],he(η(t))=[h(x(t),x(t-τ),t)0],se(η(t))=[s(x(t),x(t-τ),t)0].
In Section 3, we let ℒη denote the infinitesimal operator of system (2.13). According to different requirements for internal stability, we are in a position to define various of H∞ filters as follows.Definition 2.4 (exponential mean squareH∞ filtering).
Find the matricesf̂, Ĝ, and m̂ in (2.10), such that(i)
the equilibrium pointη≡0 of the augmented system (2.13) is exponentially mean square stable in the case v=0,(ii)
for a given disturbance attenuation levelγ>0, the following H∞ performance holds for x(t)≡0 on t∈[-τ,0]:
(2.15)‖z̃‖L22≤γ2‖v‖L22,∀v∈LF2(R+,Rnv),v≠0.Definition 2.5 (asymptotic mean squareH∞ filtering).
If in (i) of Definition2.4 the equilibrium point η≡0 of the augmented system (2.13) is asymptotically mean square stable, that is,
(2.16)lim⁡t→∞E‖η(t)‖2=0
and (2.15) holds, then (2.10) is called an asymptotic mean square H∞ filter.


## 3. Main Results

Our first main result is about exponential mean squareH∞ filter.Theorem 3.1.
Suppose that there exists a positive Lyapunov functionV(η,t)=V(x,x̂,t)∈C2,1(ℛ2n×[-τ,∞)), c1,c2,c3,c4>0 with c1c3>c2c4, such that
(3.1)c1(‖x‖2+‖x̂‖2)≤V(x,x̂,t)≤c2(‖x‖2+‖x̂‖2),∀(x,x̂,t)∈R2n×[-τ,∞),-12‖m(x,y,t)-m̂(x̂,ŷ,t)‖2≤-c3(‖x‖2+‖x̂‖2)+c4(‖y‖2+‖ŷ‖2),∀t>0.
For given disturbance attenuation level γ>0, if V(η,t) solves the following HJI:
(3.2)Γ(x,y,x̂,ŷ):=Vt+Vx′f(x,y,t)+Vx̂′(f̂(x̂,ŷ,t)+Ĝ(x̂,ŷ,t)l(x,y,t))+12Θ′(x,x̂,y,ŷ,t)(γ2I-s′(x,y,t)Vxxs(x,y,t))-1Θ(x,x̂,y,ŷ,t)+12‖m(x,y,t)-m̂(x̂,ŷ,t)‖2+12h′(x,y,t)Vxxh(x,y,t)<0,γ2I-s′(x,y,t)Vxxs(x,y,t)>0,∀(x,y,x̂,ŷ,t)∈Rn×Rn×Rn×Rn×R+,V(0,0)=0
for some matrices f̂,Ĝ, and m̂ of suitable dimensions, then the exponential mean square H∞ filtering is obtained by (2.10), where
(3.3)Θ′(x,x̂,y,ŷ,t)=Vx′g(x,y,t)+Vx̂′Ĝ(x̂,ŷ,t)k(x,y,t)+h′(x,y,t)Vxxs(x,y,t).Proof of Theorem3.1.
In Lemma2.2, we substitute V(x,x̂,t), z̃=m(x,y,t)-m̂(x̂,ŷ,t),
(3.4)fe=[f(x,y,t)f̂(x̂,ŷ,t)+Ĝ(x̂,ŷ,t)l(x,y,t)],ge=[g(x,y,t)Ĝ(x̂,ŷ,t)k(x,y,t)],he=[h(x,y,t)0],se=[s(x,y,t)0],
for V(x,t),z,f,g,h, and s, respectively; then, by a series of simple computations, (2.15) is obtained.
Next, we show the augmented system (2.13) to be exponential mean square stable for v≡0. Set
(3.5)Lηv=0V(x,x̂,t):=Vt+Vη′fe+12he′Vηηhe.
By (3.2),
(3.6)Lηv=0V(x,x̂,t)<-12‖m(x,y,t)-m̂(x̂,ŷ,t)‖2-12Θ′(x,x̂,y,ŷ,t)(γ2I-s′(x,y,t)Vxxs(x,y,t))-1Θ(x,x̂,y,ŷ,t)≤-12‖m(x,y,t)-m̂(x̂,ŷ,t)‖2≤-c3(‖x‖2+‖x̂‖2)+c4(‖y‖2+‖ŷ‖2).
Applying Lemma 2.3, we know that (2.13) is internally stable in exponential mean square sense. The proof of Theorem 3.1 is ended.Inequality (3.2) is a constrained HJI, which is not easily tested in practice. However, if in (2.1), s≡0, that is, only the state depends on noise, then the constraint condition γ2I-s′(x,y,t)Vxxs(x,y,t)>0 holds automatically, and HJI (3.2) becomes an unconstrained one.The following theorem is about asymptotic mean squareH∞ filter, which is weaker than the exponential mean square H∞ filter.Theorem 3.2.
Assume thatV(η,t)∈C2,1(ℛ2n,ℛ+) has an infinitesimal upper limit, that is,
(3.7)lim⁡∥η∥→∞inf⁡t>0V(η,t)=∞.
Additionally, one assume that V(η,t)>c∥η∥2 for some c>0. If V(η,t) solves HJI (3.2), then (2.10) is an asymptotic mean square H∞ filter.Proof.
Obviously, it only needs to show that (2.13) is asymptotically mean square stable while v=0. From (3.6), ℒηv=0V(x,x̂,t)<0, so (2.13) is globally asymptotically stable in probability 1 according to the result of [17].
By Itô’s formula and the property of stochastic integration, we have(3.8)EV(η(t),t)=EV(η(0),0)+E∫0tLηV(η(s),s)∣v=0ds+E∫0the′(η(s),s)Vη(η(s),s)dW(s)=EV(η(0),0)+E∫0tLηV(η(s),s)∣v=0ds≤EV(η(0),0)-12E∫0t‖m(x(s),x(s-τ),s)-m̂(x̂(s),x̂(s-τ),s)‖2ds≤EV(η(0),0)<∞.
Set ℱ̃t=ℱt∪σ(y(s),0≤s≤t); then (3.8) yields
(3.9)E[V(η(t),t)∣F̃s]≤V(η(s),s)a.s.,
which says that {V(η(t),t),ℱ̃t,0≤s≤t} is a nonnegative supermartingale with respect to {ℱ̃t}t≥0. By Doob’s convergence theorem [18] and the fact that limt→∞η(t)=0 a.s., it immediately yields V(η(∞),∞)=limt→∞V(η(t),t)=0a.s. Moreover, limt→∞EV(η(t),t)=EV(η(∞),∞)=EV(0,∞)=0. Because V(η,t)≥c∥η∥2 for some c>0, it follows that limt→∞E∥η(t)∥2=0. This theorem is proved.As one application of Theorem3.2, we concentrate our attention on linear stochastic time delay H∞ filtering design. Consider the following linear time-invariant stochastic time delay system: (3.10)dx(t)=(A0x(t)+A1x(t-τ)+Bv(t))dt+(C0x(t)+C1x(t-τ)+Dv(t))dW(t),y(t)=l0x(t)+l1x(t-τ)+Kv(t),z(t)=m0x(t)+m1x(t-τ),x(t)=ϕ(t)∈CF0b([-τ,0];Rn),
where, in (3.10), all coefficient matrices are assumed to be constant. Consider the following Luenberger-type filtering equation: (3.11)dx̂(t)=A0x̂(t)+A1x̂(t-τ)dt+G(y(t)-l0x̂(t)-l1x̂(t-τ))dt,ẑ(t)=m0x̂(t)+m1x̂(t-τ),x̂(0)=0,
with G a constant matrix to be determined later. In this case, (3.12)f̂(x̂(t),x̂(t-τ),t)=A0x̂(t)+A1x̂(t-τ)-G(l0x̂(t)+l1x̂(t-τ)),Ĝ=G.
Set
(3.13)V(x,x̂,t)=x′(t)Px(t)+∫t-τtx′(θ)P1x(θ)dθ+x̂′(t)Qx̂(t)+∫t-τtx̂′(θ)Q1x̂(θ)dθ,
where P>0, P1>0, Q>0, and Q1>0 are to be determined. Then by a series of computations, we have from HJI (3.2) that
(3.14)Vt=x′(t)P1x(t)-x′(t-τ)P1x(t-τ)+x̂′(t)Q1x̂(t)-x̂′(t-τ)Q1x̂(t-τ),Vx′f(x,y,t)=[x′y′x̂′ŷ′][PA0+A0′P⋆00A1′P00000000000][xyx̂ŷ],Vx̂′Gl(x,y,t)=[x′y′x̂′ŷ′][00⋆000⋆0QGl0QGl1000000][xyx̂ŷ],Vx̂′f̂(x̂,ŷ,t)=[x′y′x̂′ŷ′][0000000000Q(A0-Gl0)+(A0-l0′G′)Q⋆00(A1-Gl1)′Q0][xyx̂ŷ],12h′(x,y,t)Vxx′h(x,y,t)=[x′y′x̂′ŷ′][C0′PC0⋆00C1′PC0C1′PC10000000000][xyx̂ŷ],12‖m(x,y,t)-m(x̂,ŷ,t)‖2=[x′y′x̂′ŷ′]12[m0′m0⋆⋆⋆m1′m0m1′m1⋆⋆-m0′m0-m0′m1m0′m0⋆-m1′m0-m1′m1m1′m0m1′m1][xyx̂ŷ],12Θ′(x,x̂,y,ŷ,t)(γ2I-s′(x,y,t)Vxxs(x,y,t))-1Θ(x,x̂,y,ŷ,t)=[x′y′x̂′ŷ′][C0′PD+2PBC1′PD2QGK0]12[γ2I-2D′PD]-1[C0′PD+2PBC1′PD2QGK0]′[xyx̂ŷ],
where ⋆ is derived by symmetry. Hence, HJI (3.2) is equivalent to(3.15)[A11⋆⋆⋆A21A22⋆⋆QGl0-12m0′m0QGl1-12m0′m1A33⋆-12m1′m0-12m1′m1(A1-Gl1)′Q+12m1′m012m1′m1-Q1]+[C0′PD+2PBC1′PD2QGK0]12[γ2I-2D′PD]-1[C0′PD+2PBC1′PD2QGK0]′<0,γ2I-2D′PD>0
with
(3.16)A11=PA0+A0′P+C0′PC0+P1+12m0′m0,A21=A1′P+C1′PC0+12m1′m0,A22=-P1+C1′PC1+12m1′m1,A33=Q(A0-Gl0)+(A0-Gl0)′Q+Q1+12m0′m0.
By Schur’s complement, (3.15) are equivalent to
(3.17)[A11⋆⋆⋆⋆A21A22⋆⋆⋆G1l0-12m0′m0G1l1-12m0′m1A33⋆⋆-12m1′m0-12m1′m1A1Q-l1′G1′+12m1′m012m1′m1-Q102B′P+D′PC0D′PC12K′G1′0-2γ2I+4D′PD]<0
with QG=G1. Obviously, (3.17) is an LMI on P,P1,Q,Q1,G1. By Theorem 3.2, we immediately obtain the following corollary.Corollary 3.3.
If (3.17) is feasible with solutions P>0,P1>0,Q>0,Q1>0, and G1, then (3.11) is an asymptotic mean square H∞ filter with the filtering gain G=Q-1G1.


## 4. Illustrative Examples

Below, we give two examples to illustrate the validity of our developed theory in the above section.Example 4.1 (one-dimensional exponential mean squareH∞ filtering).
Suppose that a stochastic signalz is generated by the following nonlinear stochastic system driven by a standard Wiener process and corrupted by a stochastic external disturbance v, where the power of v is 0.05. We construct an H∞ filter to estimate z from the measurement signal y:
(4.1)dx(t)=[(-10x(t)-x(t)x2(t-τ))+x(t-τ)v(t)]dt+x(t)dW(t)x(t)=ϕ(t)∈CF0b([-τ,0];R),y(t)=-252x(t)-2x(t)x(t-τ)+v(t),z(t)=5x(t).
For given disturbance attenuation level γ=1, according to Theorem 3.1, in order to determine the filtering parameters f̂,Ĝ, and m̂, we must solve HJI (3.2). Set V(x,x̂)=x2+x̂2, m̂=-5x̂; then (3.1) hold obviously. In addition, we can easily test that Γ(x,y,x̂,ŷ)=-6.5x2-13.5x̂2<0 when we take f̂=-14x̂, Ĝ=1, m̂=5x̂. So the exponential mean square H∞ filter is given as
(4.2)dx̂(t)=-14x̂(t)dt+y(t)dt,ẑ(t)=-5x̂(t).
Because there may be more than one triple (f̂,Ĝ,m̂) solving HJI (3.2), H∞ filtering is in general not unique. The simulation result can be seen in Figures 1(a) and 1(b).Simulation results for Example4.1.
(a)
The trajectories ofx and x̂(b)
The trajectories ofz and ẑExample 4.2 (linear mean squareH∞ filtering).
In (3.10), we take the power of v to be 0.01, and
(4.3)A0=[-2.6-0.20.4-1.8],A1=[-1.80.2-0.7-0.9],B=[0.70.94],C0=[-0.800-0.9],C1=[-0.30.40.21-1.05],D=[0.20.3],l0=[1.30.8],l1=[1.23],K=0.5,m0=[-0.110.3],m1=[0.280.63].
Obviously, substituting the above data into (3.17) with γ=2 and solving LMI (3.17), we have
(4.4)P=[1.6095-0.0293-0.02930.7909]>0,P1=[3.8622-0.5054-0.50541.6277]>0,Q=[1.00090.02750.02751.3260]>0,Q1=[3.64870.13330.13333.6199]>0,G1=[-0.07720.0235],G=Q-1G1=[-0.07770.0194].
The simulation result can be found in Figures 2(a) and 2(b).Simulation results for Example4.2.
(a)
The trajectories ofx and x̂(b)
The trajectories ofz and ẑ


## 5. Conclusions

This paper presents an approach to the design ofH∞ filtering for general nonlinear stochastic time delay systems via solving HJI (3.2). Although it is difficult to solve the general HJI (3.2), under some special cases such as linear time delay systems, HJI (3.2) reduces to LMIs, which can be easily solved. How to solve HJI (3.2) is a very valuable research topic, which deserves further study. In addition, in order to avoid solving HJI (3.2), a possible scheme is to adopt a fuzzy linearized method for the original system (2.1) as done in [19].



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*Source: 231352-2011-11-13.xml*