Robust 𝐻_∞ Filtering for General Nonlinear Stochastic State-Delayed Systems

Abstract

This paper studies the robust H_∞ 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.

1. Introduction

Robust 𝐻_∞ 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 on 𝐻_∞ 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; ℒ²_ℱ(ℛ₊,ℛ^𝑙): the space of nonanticipative stochastic processes 𝑦(𝑡) with respect to filtration ℱ_𝑡 satisfying ‖𝑦(𝑡)‖²_𝐿2∶=𝐸∫^∞₀‖𝑦(𝑡)‖²𝑑𝑡<∞; 𝐶^2,1(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; 𝑉_𝑡(𝑥,𝑡)∶=(𝜕𝑉(𝑥,𝑡))/𝜕𝑡; 𝑉_𝑥(𝑥,𝑡)∶=(𝜕𝑉(𝑥,𝑡)/𝜕𝑥_𝑖)_𝑛×1; 𝑉_𝑥𝑥(𝑥,𝑡)∶=(𝜕²𝑉(𝑥,𝑡)/𝜕𝑥_𝑖𝜕𝑥_𝑗)_𝑛×𝑛; 𝐶([−𝜏,0],ℛ^𝑛): a vector space of all continuous ℛ^𝑛-valued functions defined on [−𝜏,0].

2. Preliminaries

Consider the following nonlinear stochastic time delay system: (𝑡)=𝑓(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡)+(ℎ(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡(𝑡),𝑦(𝑡)(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡),𝑧(𝑡)(𝑥(𝑡)()),𝑥(𝑡)(𝑡)𝑑𝑥,𝑥𝑡−𝜏,𝑡+𝑔,𝑥𝑡−𝜏,𝑡𝑑𝑡,𝑥𝑡−𝜏,𝑡+𝑠,𝑥𝑡−𝜏,𝑡))𝑑𝑊=𝑙,𝑥𝑡−𝜏,𝑡+𝑘,𝑥𝑡−𝜏,𝑡=𝑚,𝑥𝑡−𝜏,𝑡=𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛, (2.1) where 𝑥(𝑡)∈ℛ^𝑛 is called the system state, 𝑦(𝑡)∈ℛ^𝑟 is the measurement, 𝑊(⋅) is a standard one-dimensional Wiener process defined on a complete filtered space (Ω,ℱ,{ℱ_𝑡}_𝑡∈ℛ+,𝒫) with a filtration {ℱ_𝑡}_𝑡∈ℛ+ satisfying usual conditions, 𝑧(𝑡)∈ℛ^𝑚 is the state combination to be estimated, 𝑣∈ℒ²_ℱ(ℛ₊,ℛ^𝑛𝑣) stands for the exogenous disturbance signal, which is a square integrable, ℱ_𝑡-adapted stochastic process, and 𝐶^𝑏_ℱ0([−𝜏,0];ℛ^𝑛) denotes all ℱ₀-measurable bounded 𝐶([−𝜏,0],𝑅^𝑛)-valued random variable 𝜉(𝑠) with 𝑠∈[−𝜏,0]. We assume that 𝑓,ℎ∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ^𝑛, 𝑔,𝑠∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ^{𝑛×𝑛𝑣}, 𝑙∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ^𝑟, 𝑘∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ^{𝑟×𝑛𝑣}, and 𝑚∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ^𝑛𝑧 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 stochastic 𝐻_∞ 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 (𝑡)(𝑥(𝑡)())(𝑥(𝑡)())(𝑡),𝑥(𝑡)(𝑡)𝑑𝑥=𝑓,𝑥𝑡−𝜏,𝑡𝑑𝑡+ℎ,𝑥𝑡−𝜏,𝑡𝑑𝑊=𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛, (2.2) is called exponentially mean square stable if there are positive constants 𝐴 and 𝛼 such that 𝐸‖𝑥(𝑡)‖²≤𝐴𝜙²𝑒^−𝛼𝑡, (2.3) where ‖𝜙‖²=𝐸max_{−𝜏≤𝑡≤0}‖𝜙(𝑡)‖².
Associated with (2.1) and 𝑉∶ℛ^𝑛×ℛ₊↦ℛ₊, we define an operator ℒ₁𝑉∶ℛ^𝑛×ℛ^𝑛×ℛ₊↦ℛ as follows: ℒ₁𝑉𝑥,𝑦,𝑡=𝑉_𝑡(𝑥,𝑡)+𝑉^′_𝑥(𝑓𝑣(𝑡+1𝑥,𝑡)𝑥,𝑦,𝑡+𝑔𝑥,𝑦,𝑡)2ℎ𝑥,𝑦,𝑡+𝑠𝑥,𝑦,𝑡𝑣(𝑡)′𝑉_𝑥𝑥(𝑥,𝑡)ℎ𝑥,𝑦,𝑡+𝑠𝑥,𝑦,𝑡𝑣(𝑡). (2.4)

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: (𝑡)=𝑓(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡)+(ℎ(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡(𝑡)𝑧(𝑡)(𝑥(𝑡)())(𝑡)(𝑡)𝑑𝑥,𝑥𝑡−𝜏,𝑡+𝑔,𝑥𝑡−𝜏,𝑡𝑑𝑡,𝑥𝑡−𝜏,𝑡+𝑠,𝑥𝑡−𝜏,𝑡))𝑑𝑊=𝑚,𝑥𝑡−𝜏,𝑡,𝑥=𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛., (2.5) If there exists a positive definite Lyapunov function 𝑉(𝑥,𝑡)∈𝐶^2,1(ℛ^𝑛,ℛ₊) solving the following HJI: Γ𝑥,𝑦,𝑡∶=𝑉_𝑡(𝑥,𝑡)+𝑉^′_𝑥()𝑓+1𝑥,𝑡𝑥,𝑦,𝑡2𝑉^′_𝑥(𝑥,𝑡)𝑔𝑥,𝑦,𝑡+ℎ^′𝑥,𝑦,𝑡𝑉_𝑥𝑥()𝑠×𝛾𝑥,𝑡𝑥,𝑦,𝑡₂𝐼−𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥(𝑥,𝑡)𝑠𝑥,𝑦,𝑡⁻¹×𝑔^′𝑥,𝑦,𝑡𝑉_𝑥(𝑥,𝑡)+𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥()ℎ+1𝑥,𝑡𝑥,𝑦,𝑡2‖𝑧(𝑡)‖²+12ℎ^′𝑥,𝑦,𝑡𝑉_𝑥𝑥()ℎ𝛾𝑥,𝑡𝑥,𝑦,𝑡<0₂𝐼−𝑠′𝑥,𝑦,𝑡𝑉_𝑥𝑥(𝑥,𝑡)𝑠𝑥,𝑦,𝑡>0,∀𝑥,𝑦,𝑡∈ℛ^𝑛×ℛ^𝑛×ℛ₊,𝑉(0,0)=0 (2.6) for some 𝛾>0, then the following inequality: ‖𝑧(𝑡)‖²_𝐿2≤𝛾²‖𝑣(𝑡)‖²_𝐿2,∀𝑣∈ℒ²_ℱℛ⁺,ℛ^𝑛𝑣,𝑣≠0, (2.7) holds with initial state 𝑥(𝑠)=0, a.s., for all 𝑠∈[−𝜏,0].

Proof. See Appendix A.

Lemma 2.3. Consider the unforced system (𝑡)(𝑥(𝑡)())(𝑥(𝑡)())(𝑡),𝑥(𝑡)(𝑡)𝑑𝑥=𝑓,𝑥𝑡−𝜏,𝑡𝑑𝑡+ℎ,𝑥𝑡−𝜏,𝑡𝑑𝑊=𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛. (2.8) If there exists a positive definite Lyapunov function 𝑉(𝑥,𝑡)∈𝐶^2,1(𝑅^𝑛,[−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)∈ℛ^𝑛×[−𝜏,∞),(ii)ℒ₁𝑉(𝑥,𝑦,𝑡)|_𝑣=0≤−𝑐₃‖𝑥‖²+𝑐₄‖𝑦‖², for all 𝑡>0,
then 𝐸‖𝑥(𝑡)‖²≤𝑐₄𝑐₂/𝑐₁𝜏+𝑐₂𝑐₁𝜙²𝑒^{−(𝑐3/𝑐2)𝑡}𝑐,0≤𝑡≤𝜏,₄𝑐₂/𝑐₁𝜏+𝑐₂𝑐₁‖𝜙‖²𝑒^{−((𝑐3/𝑐2)−(𝑐4/𝑐1))𝑡},𝑡>𝜏, (2.9) that is, (2.8) is exponentially mean square stable.

Proof. See Appendix B.

In what follows, we construct the following filtering equation for the estimation of 𝑧(𝑡): (𝑡)=𝑓((𝑡)())𝐺((𝑡)())𝑦(𝑡)𝑧(𝑡)((𝑡)())(0)𝑑𝑥𝑥,𝑥𝑡−𝜏,𝑡𝑑𝑡+𝑥,𝑥𝑡−𝜏,𝑡𝑑𝑡=𝑚𝑥,𝑥𝑡−𝜏,𝑡,𝑥=0, (2.10) where 𝑓, 𝐺, and 𝑚 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: (𝑡)((𝑡)())((𝑡)())𝑦(𝑡)((𝑡)())𝑧(𝑡)((𝑡)())(0)𝑑𝑥=𝑓𝑥,𝑥𝑡−𝜏,𝑡𝑑𝑡+𝐺𝑥,𝑥𝑡−𝜏,𝑡−𝑙𝑥,𝑥𝑡−𝜏,𝑡𝑑𝑡,=𝑚𝑥,𝑥𝑡−𝜏,𝑡,𝑥=0. (2.11) Set 𝜂(𝑡)=[𝑥^′(𝑡)𝑥^′(𝑡)]^′, and let 𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡)=𝑚(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)−𝑚(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡) (2.12) denote the estimation error; then we get the following augmented system: 𝑑𝜂(𝑡)=𝑓_𝑒𝜂(𝑡)+𝑔_𝑒𝜂(𝑡)𝑣(𝑡)𝑑𝑡+ℎ_𝑒𝜂(𝑡)+𝑠_𝑒𝜂(𝑡)𝑣(𝑡)(𝑡),𝑧(𝑡)(𝑡)−𝑧(𝑡)(𝑥(𝑡)())((𝑡)()),𝜂(𝑡)=𝜙(𝑡)0(𝑡)𝑑𝑊=𝑧=𝑚,𝑥𝑡−𝜏,𝑡−𝑚𝑥,𝑥𝑡−𝜏,𝑡,𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛,∀𝑡∈[−𝜏,0], (2.13) where 𝑓_𝑒𝜂(𝑡)=𝑓(𝑥(𝑡)())𝑓((𝑡)())+𝐺((𝑡)())𝑙(𝑥(𝑡)()),𝑔,𝑥𝑡−𝜏,𝑡𝑥,𝑥𝑡−𝜏,𝑡𝑥,𝑥𝑡−𝜏,𝑡,𝑥𝑡−𝜏,𝑡_𝑒𝜂(𝑡)=𝑔(𝑥(𝑡)())𝐺((𝑡)())𝑘(𝑥(𝑡)()),ℎ,𝑥𝑡−𝜏,𝑡𝑥,𝑥𝑡−𝜏,𝑡,𝑥𝑡−𝜏,𝑡_𝑒𝜂(𝑡)=ℎ(𝑥(𝑡)())0,𝑥𝑡−𝜏,𝑡,𝑠_𝑒𝜂(𝑡)=𝑠(𝑥(𝑡)())0.,𝑥𝑡−𝜏,𝑡 (2.14) In Section 3, we let ℒ_𝜂 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 square 𝐻_∞ filtering). Find the matrices 𝑓, 𝐺, and 𝑚 in (2.10), such that(i)the equilibrium point 𝜂≡0 of the augmented system (2.13) is exponentially mean square stable in the case 𝑣=0,(ii)for a given disturbance attenuation level 𝛾>0, the following 𝐻_∞ performance holds for 𝑥(𝑡)≡0 on 𝑡∈[−𝜏,0]: ‖𝑧‖²_𝐿2≤𝛾²‖𝑣‖²_𝐿2,∀𝑣∈ℒ²_ℱℛ⁺,ℛ^𝑛𝑣,𝑣≠0. (2.15)

Definition 2.5 (asymptotic mean square 𝐻_∞ filtering). If in (i) of Definition 2.4 the equilibrium point 𝜂≡0 of the augmented system (2.13) is asymptotically mean square stable, that is, lim_𝑡→∞𝐸𝜂(𝑡)²=0 (2.16) 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 square 𝐻_∞ filter.

Theorem 3.1. Suppose that there exists a positive Lyapunov function 𝑉(𝜂,𝑡)=𝑉(𝑥,𝑥,𝑡)∈𝐶^2,1(ℛ^2𝑛×[−𝜏,∞)), 𝑐₁,𝑐₂,𝑐₃,𝑐₄>0 with 𝑐₁𝑐₃>𝑐₂𝑐₄, such that 𝑐₁‖𝑥‖²+‖𝑥‖²≤𝑉(𝑥,𝑥,𝑡)≤𝑐₂‖𝑥‖²+‖𝑥‖²,∀(𝑥,𝑥,𝑡)∈ℛ^2𝑛×[),−1−𝜏,∞2𝑚(𝑥,𝑦,𝑡)−𝑚(𝑥,𝑦,𝑡)²≤−𝑐₃‖𝑥‖²+‖𝑥‖²+𝑐₄𝑦²+𝑦²,∀𝑡>0. (3.1) For given disturbance attenuation level 𝛾>0, if 𝑉(𝜂,𝑡) solves the following HJI: Γ𝑥,𝑦,𝑥,𝑦∶=𝑉_𝑡+𝑉^′_𝑥𝑓𝑥,𝑦,𝑡+𝑉^′𝑓+𝐺𝑙+1𝑥𝑥,𝑦,𝑡𝑥,𝑦,𝑡𝑥,𝑦,𝑡2Θ^′𝑥,𝑥,𝑦,𝑦,𝑡𝛾²𝐼−𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥𝑠𝑥,𝑦,𝑡⁻¹Θ+1𝑥,𝑥,𝑦,𝑦,𝑡2𝑚𝑥,𝑦,𝑡−𝑚𝑥,𝑦,𝑡²+12ℎ^′𝑥,𝑦,𝑡𝑉_𝑥𝑥ℎ𝛾𝑥,𝑦,𝑡<0,₂𝐼−𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥𝑠𝑥,𝑦,𝑡>0,∀𝑥,𝑦,𝑥,𝑦,𝑡∈ℛ^𝑛×ℛ^𝑛×ℛ^𝑛×ℛ^𝑛×ℛ₊,𝑉(0,0)=0 (3.2) for some matrices 𝑓,𝐺, and 𝑚 of suitable dimensions, then the exponential mean square 𝐻_∞ filtering is obtained by (2.10), where Θ^′𝑥,𝑥,𝑦,𝑦,𝑡=𝑉^′_𝑥𝑔𝑥,𝑦,𝑡+𝑉^′𝑥𝐺𝑥,𝑦,𝑡𝑘𝑥,𝑦,𝑡+ℎ^′𝑥,𝑦,𝑡𝑉_𝑥𝑥𝑠𝑥,𝑦,𝑡. (3.3)

Proof of Theorem 3.1. In Lemma 2.2, we substitute 𝑉(𝑥,𝑥,𝑡), 𝑧=𝑚(𝑥,𝑦,𝑡)−𝑚(𝑥,𝑦,𝑡), 𝑓_𝑒=𝑓𝑓+𝐺𝑙𝑥,𝑦,𝑡𝑥,𝑦,𝑡𝑥,𝑦,𝑡𝑥,𝑦,𝑡,𝑔_𝑒=𝑔𝐺𝑘,ℎ𝑥,𝑦,𝑡𝑥,𝑦,𝑡𝑥,𝑦,𝑡_𝑒=ℎ0𝑥,𝑦,𝑡,𝑠_𝑒=𝑠0,𝑥,𝑦,𝑡 (3.4) 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 ℒ_𝜂^𝑣=0𝑉(𝑥,𝑥,𝑡)∶=𝑉_𝑡+𝑉^′_𝜂𝑓_𝑒+12ℎ^′_𝑒𝑉_𝜂𝜂ℎ_𝑒. (3.5) By (3.2), ℒ_𝜂^𝑣=0𝑉()1𝑥,𝑥,𝑡<−2𝑚(𝑥,𝑦,𝑡)−𝑚(𝑥,𝑦,𝑡)²−12Θ^′𝑥,𝑥,𝑦,𝑦,𝑡𝛾²𝐼−𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥𝑠𝑥,𝑦,𝑡⁻¹Θ1𝑥,𝑥,𝑦,𝑦,𝑡≤−2𝑚(𝑥,𝑦,𝑡)−𝑚(𝑥,𝑦,𝑡)²≤−𝑐₃‖𝑥‖²+‖𝑥‖²+𝑐₄𝑦²+𝑦². (3.6) 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)𝑉_𝑥𝑥𝑠(𝑥,𝑦,𝑡)>0 holds automatically, and HJI (3.2) becomes an unconstrained one.

The following theorem is about asymptotic mean square 𝐻_∞ filter, which is weaker than the exponential mean square 𝐻_∞ filter.

Theorem 3.2. Assume that 𝑉(𝜂,𝑡)∈𝐶^2,1(ℛ^2𝑛,ℛ₊) has an infinitesimal upper limit, that is, lim_{‖𝜂‖→∞}inf_𝑡>0𝑉𝜂,𝑡=∞. (3.7) 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), ℒ_𝜂^𝑣=0𝑉(𝑥,𝑥,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 𝐸𝑉𝜂(𝑡),𝑡=𝐸𝑉𝜂(0),0+𝐸^𝑡₀ℒ_𝜂𝑉𝜂(𝑠),𝑠∣_𝑣=0𝑑𝑠+𝐸^𝑡₀ℎ^′_𝑒𝜂(𝑠),𝑠𝑉_𝜂𝜂(𝑠)(𝑠)𝜂(0),𝑠𝑑𝑊=𝐸𝑉,0+𝐸^𝑡₀ℒ_𝜂𝑉𝜂(𝑠),𝑠∣_𝑣=0𝜂(0)−1𝑑𝑠≤𝐸𝑉,02𝐸^𝑡₀‖𝑚(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)−𝑚(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)‖²𝜂(0)𝑑𝑠≤𝐸𝑉,0<∞. (3.8) Set ℱ_𝑡=ℱ_𝑡∪𝜎(𝑦(𝑠),0≤𝑠≤𝑡); then (3.8) yields 𝐸𝑉𝜂(𝑡),𝑡∣ℱ_𝑠≤𝑉𝜂(𝑠),𝑠a.s., (3.9) which says that {𝑉(𝜂(𝑡),𝑡),ℱ_𝑡,0≤𝑠≤𝑡} is a nonnegative supermartingale with respect to {ℱ_𝑡}_𝑡≥0. By Doob’s convergence theorem [18] and the fact that lim_𝑡→∞𝜂(𝑡)=0 a.s., it immediately yields 𝑉(𝜂(∞),∞)=lim_𝑡→∞𝑉(𝜂(𝑡),𝑡)=0a.s. Moreover, lim_𝑡→∞𝐸𝑉(𝜂(𝑡),𝑡)=𝐸𝑉(𝜂(∞),∞)=𝐸𝑉(0,∞)=0. Because 𝑉(𝜂,𝑡)≥𝑐‖𝜂‖² for some 𝑐>0, it follows that lim_𝑡→∞𝐸‖𝜂(𝑡)‖²=0. This theorem is proved.

As one application of Theorem 3.2, we concentrate our attention on linear stochastic time delay 𝐻_∞ filtering design. Consider the following linear time-invariant stochastic time delay system: 𝑑𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜏)+𝐵𝑣(𝑡)𝑑𝑡+𝐶₀𝑥(𝑡)+𝐶₁𝑥()(𝑡)(𝑡),𝑦(𝑡)𝑡−𝜏+𝐷𝑣𝑑𝑊=𝑙₀𝑥(𝑡)+𝑙₁𝑥()(𝑡),𝑧(𝑡)𝑡−𝜏+𝐾𝑣=𝑚₀𝑥(𝑡)+𝑚₁𝑥(),𝑥(𝑡)(𝑡)𝑡−𝜏=𝜙∈𝐶^𝑏_ℱ0[−𝜏,0];ℛ^𝑛, (3.10) where, in (3.10), all coefficient matrices are assumed to be constant. Consider the following Luenberger-type filtering equation: 𝑑𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜏)𝑑𝑡+𝐺𝑦(𝑡)−𝑙₀𝑥(𝑡)−𝑙₁()𝑧(𝑡)𝑥𝑡−𝜏𝑑𝑡,=𝑚₀𝑥(𝑡)+𝑚₁𝑥(𝑡−𝜏),𝑥(0)=0, (3.11) with 𝐺 a constant matrix to be determined later. In this case, 𝑓(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜏)−𝐺𝑙₀𝑥(𝑡)+𝑙₁𝑥(𝑡−𝜏),𝐺=𝐺. (3.12) Set 𝑉(𝑥,𝑥,𝑡)=𝑥^′(𝑡)𝑃𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^′(𝜃)𝑃₁𝑥(𝜃)𝑑𝜃+𝑥^′(𝑡)𝑄𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^′(𝜃)𝑄₁𝑥(𝜃)𝑑𝜃, (3.13) 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 𝑉_𝑡=𝑥′(𝑡)𝑃₁𝑥(𝑡)−𝑥^′(𝑡−𝜏)𝑃₁𝑥(𝑡−𝜏)+𝑥′(𝑡)𝑄₁𝑥(𝑡)−𝑥′(𝑡−𝜏)𝑄₁(),𝑉𝑥𝑡−𝜏_′𝑥𝑓𝑥,𝑦,𝑡=𝑥^′𝑦^′𝑥^′𝑦^′𝑃𝐴₀+𝐴^′₀𝐴𝑃⋆00_′1𝑥𝑦,𝑉𝑃00000000000𝑥_′𝑦𝑥𝐺𝑙𝑥,𝑦,𝑡=𝑥^′𝑦^′𝑥^′𝑦^′00⋆000⋆0𝑄𝐺𝑙₀𝑄𝐺𝑙₁𝑥𝑦,𝑉000000𝑥_′𝑦𝑥𝑓𝑥,𝑦,𝑡=𝑥^′𝑦^′𝑥^′𝑦^′𝐴0000000000𝑄₀−𝐺𝑙₀+𝐴₀−𝑙^′₀𝐴𝐺′𝑄⋆00₁−𝐺𝑙₁𝑥𝑦,1′𝑄0𝑥𝑦2ℎ^′𝑥,𝑦,𝑡𝑉^′_𝑥𝑥ℎ=𝐶𝑥,𝑦,𝑡𝑥′𝑦′𝑥′𝑦′^′0𝑃𝐶₀𝐶⋆00_′1𝑃𝐶₀𝐶^′₁𝑃𝐶₁𝑥𝑦,10000000000𝑥𝑦2𝑚(𝑥,𝑦,𝑡)−𝑚(𝑥,𝑦,𝑡)²=𝑥^′𝑦^′𝑥^′𝑦^′12𝑚^′₀𝑚₀𝑚⋆⋆⋆_′1𝑚₀𝑚^′₁𝑚₁⋆⋆−𝑚^′₀𝑚₀−𝑚^′₀𝑚₁𝑚^′₀𝑚₀⋆−𝑚^′₁𝑚₀−𝑚^′₁𝑚₁𝑚^′₁𝑚₀𝑚^′₁𝑚₁𝑥𝑦,1𝑥𝑦2Θ^′𝑥,𝑥,𝑦,𝑦,𝑡𝛾²𝐼−𝑠^′𝑥,𝑦,𝑡𝑉_𝑥𝑥𝑠𝑥,𝑦,𝑡⁻¹Θ=𝐶𝑥,𝑥,𝑦,𝑦,𝑡𝑥′𝑦′𝑥′𝑦′^′0𝐶𝑃𝐷+2𝑃𝐵_′10𝑃𝐷12𝑄𝐺𝐾2𝛾²𝐼−2𝐷^′𝑃𝐷⁻¹𝐶^′₀𝐶𝑃𝐷+2𝑃𝐵_′10𝑃𝐷2𝑄𝐺𝐾^′𝑥𝑦,𝑥𝑦 (3.14) where ⋆ is derived by symmetry. Hence, HJI (3.2) is equivalent to𝐴₁₁𝐴⋆⋆⋆₂₁𝐴₂₂⋆⋆𝑄𝐺𝑙₀−12𝑚^′₀𝑚₀𝑄𝐺𝑙₁−12𝑚^′₀𝑚₁𝐴₃₃⋆−12𝑚^′₁𝑚₀−12𝑚^′₁𝑚₁𝐴₁−𝐺𝑙₁^′1𝑄+2𝑚^′₁𝑚₀12𝑚^′₁𝑚₁−𝑄₁+𝐶^′₀𝐶𝑃𝐷+2𝑃𝐵_′10𝑃𝐷12𝑄𝐺𝐾2𝛾²𝐼−2𝐷^′𝑃𝐷⁻¹𝐶^′₀𝐶𝑃𝐷+2𝑃𝐵_′10𝑃𝐷2𝑄𝐺𝐾^′𝛾<0,₂𝐼−2𝐷′𝑃𝐷>0 (3.15) with 𝐴₁₁=𝑃𝐴₀+𝐴^′₀𝑃+𝐶^′₀𝑃𝐶₀+𝑃₁+12𝑚^′₀𝑚₀,𝐴₂₁=𝐴^′₁𝑃+𝐶^′₁𝑃𝐶₀+12𝑚^′₁𝑚₀,𝐴₂₂=−𝑃₁+𝐶^′₁𝑃𝐶₁+12𝑚^′₁𝑚₁,𝐴₃₃=𝑄𝐴₀−𝐺𝑙₀+𝐴₀−𝐺𝑙₀′𝑄+𝑄₁+12𝑚^′₀𝑚₀. (3.16) By Schur’s complement, (3.15) are equivalent to 𝐴₁₁𝐴⋆⋆⋆⋆₂₁𝐴₂₂𝐺⋆⋆⋆₁𝑙₀−12𝑚^′₀𝑚₀𝐺₁𝑙₁−12𝑚^′₀𝑚₁𝐴₃₃−1⋆⋆2𝑚^′₁𝑚₀−12𝑚^′₁𝑚₁𝐴₁𝑄−𝑙^′₁𝐺^′₁+12𝑚^′₁𝑚₀12𝑚^′₁𝑚₁−𝑄₁02𝐵^′𝑃+𝐷^′𝑃𝐶₀𝐷^′𝑃𝐶₁2𝐾^′𝐺^′₁0−2𝛾²𝐼+4𝐷′𝑃𝐷<0 (3.17) 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 square 𝐻_∞ filtering). Suppose that a stochastic signal 𝑧 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: 𝑑𝑥(𝑡)=−10𝑥(𝑡)−𝑥(𝑡)𝑥²(()𝑣(𝑡(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑡−𝜏)+𝑥𝑡−𝜏)𝑑𝑡+𝑥𝑑𝑊=𝜙∈𝐶^𝑏_ℱ0]),𝑦(𝑡)([−𝜏,0;ℛ=−252𝑥(𝑡)(𝑡)𝑥()(𝑡),𝑧(𝑡)(𝑡).−2𝑥𝑡−𝜏+𝑣=5𝑥 (4.1) For given disturbance attenuation level 𝛾=1, according to Theorem 3.1, in order to determine the filtering parameters 𝑓,𝐺, and 𝑚, we must solve HJI (3.2). Set 𝑉(𝑥,𝑥)=𝑥²+𝑥², 𝑚=−5𝑥; then (3.1) hold obviously. In addition, we can easily test that Γ(𝑥,𝑦,𝑥,𝑦)=−6.5𝑥²−13.5𝑥²<0 when we take 𝑓=−14𝑥, 𝐺=1, 𝑚=5𝑥. So the exponential mean square 𝐻_∞ filter is given as 𝑑𝑥(𝑡)=−14𝑥(𝑡)𝑑𝑡+𝑦(𝑡)𝑑𝑡,𝑧(𝑡)=−5𝑥(𝑡). (4.2) Because there may be more than one triple (𝑓,𝐺,𝑚) solving HJI (3.2), 𝐻_∞ filtering is in general not unique. The simulation result can be seen in Figures 1(a) and 1(b).

(a) The trajectories of 𝑥 and 𝑥

(b) The trajectories of 𝑧 and 𝑧

Figure 1 

Simulation results for Example 4.1.

Example 4.2 (linear mean square 𝐻_∞ filtering). In (3.10), we take the power of 𝑣 to be 0.01, and 𝐴₀=−2.6−0.20.4−1.8,𝐴₁=,𝐶−1.80.2−0.7−0.9,𝐵=0.70.94₀=−0.800−0.9,𝐶₁=,𝑙−0.30.40.21−1.05,𝐷=0.20.3₀=1.30.8,𝑙₁=𝑚1.23,𝐾=0.5,₀=−0.110.3,𝑚₁=0.280.63. (4.3) Obviously, substituting the above data into (3.17) with 𝛾=2 and solving LMI (3.17), we have 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=𝐺3.64870.13330.13333.6199>0,₁=−0.07720.0235,𝐺=𝑄⁻¹𝐺₁=.−0.07770.0194 (4.4) The simulation result can be found in Figures 2(a) and 2(b).

(a) The trajectories of 𝑥 and 𝑥

(b) The trajectories of 𝑧 and 𝑧

Figure 2 

Simulation results for Example 4.2.

5. Conclusions

This paper presents an approach to the design of 𝐻_∞ 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].

Appendices

A. Proof of Lemma 2.2

As done in [9], applying the completing squares technique and considering (2.6), it is easy to obtain ℒ₁𝑉≤1𝑥,𝑦,𝑡2𝛾²𝑣^′(𝑡)𝑣(𝑡)−12𝑧^′(𝑡)𝑧(𝑡). (A.1) In addition, by Itô’s formula, for any 𝑇>0, we have 𝐸𝑉(𝑥(𝑇),𝑇)=𝐸𝑉(𝑥(0),0)+𝐸^𝑇₀(𝑥(𝑠))(𝑥(0))𝑑𝑉,𝑠=𝐸𝑉,0+𝐸^𝑇₀(𝑥(𝑡))(𝑥(0))ℒ𝑉,𝑡𝑑𝑡=𝐸𝑉,0+𝐸^𝑇₀ℒ₁𝑉(𝑥(𝑡)((𝑥(0))+1,𝑥𝑡−𝜏,𝑡))𝑑𝑡≤𝐸𝑉,02𝐸^𝑇₀𝛾²‖𝑣(𝑡)‖²−‖𝑧(𝑡)‖²𝑑𝑡, (A.2) where, in (A.2), ℒ is the so-called infinitesimal operator of (2.5), which is defined by ℒ𝑉(𝑥(𝑡),𝑡)=𝑉_𝑡(𝑥(𝑡),𝑡)+𝑉^′_𝑥(𝑥(𝑡)𝑓(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡+1,𝑡),𝑥𝑡−𝜏,𝑡+𝑔,𝑥𝑡−𝜏,𝑡)2[ℎ(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)+𝑠(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)𝑣(𝑡)]^′𝑉_𝑥𝑥(𝑥(𝑡))⋅[ℎ(𝑥(𝑡)())(𝑥(𝑡)())𝑣(𝑡.,𝑡,𝑥𝑡−𝜏,𝑡+𝑠,𝑥𝑡−𝜏,𝑡)] (A.3) In view of 𝑉 being positive and 𝑉(0,0)=0, it follows that for the zero initial condition 𝑥(𝑠)≡0, forall𝑠∈[−𝜏,0], 𝐸^𝑇₀‖𝑧(𝑡)‖²𝑑𝑡≤𝐸^𝑇₀‖𝑣(𝑡)‖²𝑑𝑡, (A.4) which proves Lemma 2.2.

B. Proof of Lemma 2.3

By (A.2), we know that, for any 𝑡>0, 𝐸𝑉(𝑥(𝑡),𝑡)−𝐸𝑉(𝑥(0),0)=^𝑡₀𝐸ℒ₁𝑉(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)∣_𝑣=0𝑑𝑠. (B.1) By given conditions (i) and (ii), (B.1) yields 𝐸𝑉(𝑥(𝑡),𝑡)−𝐸𝑉(𝑥(0),0)≤−𝑐₃^𝑡₀𝐸‖𝑥(𝑠)‖²𝑑𝑠+𝑐₄^𝑡₀𝐸‖𝑥(𝑠−𝜏)‖²𝑐𝑑𝑠≤−³𝑐₂^𝑡₀(𝑥(𝑠))𝑐𝐸𝑉,𝑠𝑑𝑠+⁴𝑐₁^𝑡₀𝐸𝑉(𝑥(𝑠−𝜏),𝑠−𝜏)𝑑𝑠. (B.2) When 0≤𝑡≤𝜏, we have (𝑥(𝑡))≤𝑐𝐸𝑉,𝑡⁴𝑐₂𝑐₁𝜏+𝑐₂‖𝜙‖²−𝑐₃𝑐₂^𝑡₀𝐸𝑉(𝑥(𝑠),𝑠)𝑑𝑠. (B.3) Applying Gronwall’s inequality, it follows that 𝐸𝑉(𝑥(𝑡),𝑡)≤((𝑐₄𝑐₂/𝑐₁)𝜏+𝑐₂)‖𝜙‖²𝑒^{−(𝑐3/𝑐2)𝑡}. Again, using condition (i), 𝐸‖𝑥(𝑡)‖²≤𝑐₄𝑐₂/𝑐₁𝜏+𝑐₂𝑐₁𝜙²𝑒^{−(𝑐3/𝑐2)𝑡}. (B.4) When 𝑡>𝜏>0, letting 𝜇=𝑠−𝜏, (B.2) yields 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑐₂𝜙²−𝑐₃𝑐₂^𝑡₀(𝑥(𝑠))𝑐𝐸𝑉,𝑠𝑑𝑠+⁴𝑐₁^𝑡−𝜏_−𝜏𝐸𝑉𝑥𝜇,𝜇𝑑𝑡𝜇≤𝑐₂𝜙²−𝑐₃𝑐₂^𝑡₀(𝑥(𝑠))+𝑐𝐸𝑉,𝑠𝑑𝑠₄𝑐₁⁰_−𝜏𝑥𝜇𝑐𝐸𝑉,𝜇𝑑𝜇+⁴𝑐₁^𝑡₀𝑥𝜇=𝑐𝐸𝑉,𝜇𝑑𝜇₄𝑐₂𝑐₁𝜏+𝑐₂𝜙²−𝑐₃𝑐₂−𝑐₄𝑐₁^𝑡₀𝐸𝑉(𝑥(𝑠),𝑠)𝑑𝑠. (B.5) Repeating the same procedure as above, we have 𝐸‖𝑥(𝑡)‖²≤𝑐₄𝑐₂/𝑐₁𝜏+𝑐₂𝑐₁𝜙²𝑒^{−((𝑐3/𝑐2)−(𝑐4/𝑐1))𝑡}. (B.6) Lemma 2.3 is hence proved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (60874032, 60804034, 61174078), Specialized Research Fund for the Doctoral Program of Higher Education (20103718110006) and Key Project of Natural Science Foundation of Shandong Province (ZR2009GZ001).