Mode-Dependent Finite-Time H∞ Filtering for Stochastic Nonlinear Systems with Markovian Switching

This paper addresses the problem of finite-time H∞ filter design for a class of non-linear stochastic systems with Markovian switching. Based on stochastic differential equations theory, a mode-dependent finite-time H∞ filter is designed to ensure finite-time stochastic stablility (FTSS) of filtering error system and satisfies a prescribed H∞ performance level in some given finite-time intervals. Moreover, sufficient conditions are presented for the existence of a finite-time H∞ filter for the stochastic system under consideration by employing the linear matrix inequality technique. Finally, the explicit expression of the desired filter parameters is given.


Introduction
Since filtering plays an important role in control systems, signal processing and communication, there has been a rapidly growing interest in filter designing due to its advantages over the traditional Kalman filtering. In the past few years, many contributions on filtering for stochastic systems can be found in the literature [1]- [14], because it is an important research topic and has found many practical applications. In [1], a H ∞ filter was designed for nonlinear stochastic systems. H ∞ filtering problems for discrete-time nonlinear stochastic systems were addressed in [2]. Delay-dependent H ∞ filtering for discrete-time singular systems and fuzzy discrete-time systems were reported respectively in [3] [4] [5] [6]. In [7], a H ∞ filter was designed for discrete-time systems with stochastic in-How to cite this paper: Zhang, A.Q. As is well known, the previously mentioned literature was based on Lyapunov asymptotic stability which focuses on the steady-state behavior of plants over an infinite-time interval. But in many practical systems, it is only required that the system states remain within the given bounds. In these cases, the introduction of finite-time stability or short-time stability was needed, which has caused extensive attention [15]- [23]. The problem of finite-time stability and stabilization for a class of linear systems with time delay was addressed in [15]. In [16], the sufficient conditions were achieved for the finite-time stability of linear time-varying systems with jumps. The problem of robust finite-time stabilization for impulsive dynamical linear systems was investigated in [17]. In [18], fuzzy control method was adopted to solve finite-time stabilization of a class of stochastic system. A robust finite-time filter was established for singular discrete-time stochastic system in [19]. Finite-time H ∞ filtering was proposed respectively for T-S fuzzy systems, switched systems, nonlinear singular systems, Itô stochastic Markovian jump systems in [20] [21] [22] [23]. Motivated by the contributions mentioned above, we investigated the mode-dependent finite-time filtering problems for stochastic nonlinear systems, which could be used to detect generation of residuals for fault diagnosis problems. This paper will study the H ∞ filtering problem for a class of Markov Jump stochastic systems with Lipschitz nonlinearlity. The main purpose of this study is to construct a H ∞ filter such that the resulting filter error augmented system is FTSS. The sufficient condition for FTSS of the filter error system is obtained by constructing the Lyapunov-Krasovskii functional candidate combined with LMIs. We present an approach to design the desired FTSS filter. This paper is organized as follows. Some corresponding definitions and lemmas and the problem formulation are introduced in Section 2. In Section 3, we give a sufficient condition for FTSS of the mentioned filtering error system in terms of LMIs. Moreover, an approach of a finite-time H ∞ filter is presented.
Some conclusions are drawn in section 4.
We use n R to denote the n-dimensional Euclidean space. The notation X Y > (respectively, X Y ≥ ), where X and Y are real symmetric matrices, means that the matrix X Y − is positive definite (respectively, positive semi-definite). I and 0 denote the identity and zero matrices with appropriate dimensions.

Preliminaries
Consider a class of Itô stochastic nonlinear system with Markovian switching, which can be described as follows: The set N comprises the operation modes of the system. The transition probabilities for the process , , , , , , , We now consider the following filter for system (1) -(3): − then we can obtain the following filtering error system: We introduce the following definitions and lemmas, which will be useful in the succeeding discussion.
Lemma 1 (Gronwall inequality [25]): Let ( ) v t be a nonnegative function such that for some constants , 0 [26] [27]) Given a symmetric matrix 11 12 , the following three conditions are equivalent to each other: 2) 11 0 φ < , and Lemma 3 (Itô formula [28]) Let x (t) be an n-dimensional Itô process on 0 t ≥ with the stochastic differential where ( ) n f t R ∈ and ( ) n m g t R × ∈ , , V x t t is a real-valued Itô process with its stochastic differential where the weak infinitesimal operator

Main Results
Theorem 1: Suppose that the filter parameters , , such that the following LMIs hold and ( ) where Proof: Define the following stochastic Laypunov-Krasovskii functional candidate: By Itô formula, we have the weak infinitesimal operator of ( ) ( )

A. Q. Zhang Journal of Applied Mathematics and Physics
Applying (6) and the following well-known fact: it follows that Applying Schur complement, we have the following inequality by taking (17) into consideration: Multiplying the above inequality by e t α − and by Gronwall inequality (12), we obtain the following inequality Then, we have Therefore, it follows that condition (18) implies Proof: For the filtering error system (9) (10), consider the same stochastic Laypunov functional as in (19). Obviously, condition (33) implies that where Λ is given in (26). By theorem 1, conditions (17) and (18) guarantee that system (9) (10) is FTSS with respect to ( ) 1 2 , , , , c c T R d . Therefore, we only need to prove that (11) holds.
Noting that (27) and (34), we obtain Then using the similar proof as Theorem 1, condition (11) can be easily obtained.   In addition, the suitable parameters of the filter (7) (8) are given as follows:

Conclusion
In this paper, we deal with the finite-time H ∞ filter designing problem for a class of stochastic nonlinear systems with Markovian switching. The sufficient conditions for FTSS of the filtering error system have been presented and proved by the Lyapunov-Krasovski approach. The designed filter is provided to ensure the filtering error system FTSS and satisfies a prescribed H ∞ performance level in a given finite-time interval, which can be reduced to feasibility problems involving restricted linear matrix equalities.