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

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.


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 L 2 -gain from the external disturbance to the estimation error is below a prescribed level γ > 0. Stochastic It ô 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 It ô-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 Mathematical Problems in Engineering 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; L 2 F R , R l : the space of nonanticipative stochastic processes y t with respect to filtration F t satisfying y t 2 L 2 : E ∞ 0 y t 2 dt < ∞; 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 : ∂ 2 V 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 .

Preliminaries
Consider the following nonlinear stochastic time delay system: and m : R n × R n × R → R n z 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 is called exponentially mean square stable if there are positive constants A and α such that where

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:
If there exists a positive definite Lyapunov function V x, t ∈ C 2,1 R n , R solving the following HJI: for some γ > 0, then the following inequality: holds with initial state x s 0, a.s., for all, s ∈ −τ, 0 .
Proof.See Appendix A.

2.8
If there exists a positive definite Lyapunov function 4 satisfying the following conditions: 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 : 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:

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 Mathematical Problems in Engineering 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 :

Main Results
Our first main result is about exponential mean square H ∞ filter.
Theorem 3.1.Suppose that there exists a positive Lyapunov function

3.1
For given disturbance attenuation level γ > 0, if V η, t solves the following HJI: Next, we show the augmented system 2.13 to be exponential mean square stable for v ≡ 0. Set

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 γ 2 I − s x, y, t V xx s 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.

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 v 0 η V x, x, t < 0, so 2.13 is globally asymptotically stable in probability 1 according to the result of 17 .
By It ô's formula and the property of stochastic integration, we have  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: where, in 3.10 , all coefficient matrices are assumed to be constant.Consider the following Luenberger-type filtering equation: with G a constant matrix to be determined later.In this case, where P > 0, P 1 > 0, Q > 0, and Q 1 > 0 are to be determined.Then by a series of computations, we have from HJI 3.2 that where is derived by symmetry.Hence, HJI 3.2 is equivalent to

3.16
By Schur's complement, 3.15 are equivalent to with QG G 1 .Obviously, 3.17 is an LMI on P, P 1 , Q, Q 1 , G 1 .By Theorem 3.2, we immediately obtain the following corollary.

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:

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 x 2 x 2 , m −5 x; then 3.1 hold obviously.In addition, we can easily test that Γ x, y, x, y −6.5x 2 −13.5 x 2 < 0 when we take f −14 x, G 1, m 5 x.So the exponential mean square H ∞ filter is given as

4.4
The simulation result can be found in Figures 2 a and 2 b .

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

A.3
In view of V being positive and V 0, 0 0, it follows that for the zero initial condition x s ≡ 0, for all s ∈ −τ, 0 ,
∈ 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 ∈ L 2 F R , R n v stands for the exogenous disturbance signal, which is a square integrable, F t -adapted stochastic process, and 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, and 2.15 holds, then 2.10 is called an asymptotic mean square H ∞ filter.
Proof of Theorem 3.1.In Lemma 2.2, we substitute V x, x, t , z m x, y, t − m x, y, t , x G x, y, t k x, y, t h x, y, t V xx s x, y, t .3.3 Example 4.2 linear mean square H ∞ filtering .In 3.10 , we take the power of v to be 0.01, and 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 .
In addition, by It ô's formula, for any T > 0, we have L is the so-called infinitesimal operator of 2.5 , which is defined by x x t , t f x t , x t − τ , t g x t , x t − τ , t v t 1 2 h x t , x t − τ , t s x t , x t − τ , t v t V xx x t , t • h x t , x t − τ , t s x t , x t − τ , t v t .
B. Proof of Lemma 2.3