Stochastic input-to-state stability and $H_{\infty }$ filtering for a class of stochastic nonlinear systems

Abstract

In this paper, an $H_{\infty }$ filtering problem is considered for a class of stochastic nonlinear systems with time-delay and exogenous disturbances. Sufficient conditions for stochastic input-to-state stability (SISS) of the stochastic nonlinear systems are developed via Lyapunov function method and linear matrix inequalities (LMIs) technique. The purpose of this work is to design a nonlinear filter to ensure both the SISS and a prescribed $H_{\infty }$ attenuation level for the corresponding filtering error system. By using the SISS results, a set of sufficient conditions for the existence of $H_{\infty }$ filter is given in the form of LMI. When the LMI is feasible, an explicit expression of a desired filter is presented. Finally, some numerical and practical examples are employed to demonstrate the effectiveness of the proposed approaches.

Introduction

It is well known that the term of input-to-state stability (ISS) plays an important role in stability analysis and controller design of deterministic nonlinear systems [1]. Lyapunov stability theory mainly devotes to analyzing systems without external perturbations. The states of the systems can return to their inherent equilibrium states for arbitrary initial point. However, Lyapunov stability theory is not suitable for analyzing and processing the state responses of the systems with perturbations when it is required to determine whether the responses are controlled in a pre-set region or not. The ISS implies that the unperturbed systems are asymptotically stable in the Lyapunov sense, and the responses of the systems with bounded inputs are bounded. The properties of ISS have been presented in [2], [3]. The ISS problem has been extensively investigated by many authors [4], [5], [6], [7], [8]. At the same time, the ISS has been investigated for different systems such as impulsive systems, switching systems, time-delay systems and stochastic systems etc. [9], [10], [11], [12], [13]. As the natural extension for ISS, integral input-to-state stability (iISS), input–output-to-state stability (IOSS) and stochastic input-to-state stability (SISS) have been introduced in [14], [15], [16]. The ISS properties of stochastic nonlinear systems were considered, and the terms of SISS and iISS in mean were introduced in [12]. The SISS of Lur’e distributed parameter control systems was addressed, and sufficient conditions for SISS in Hilbert spaces were established in terms of linear operator inequalities in [13]. The ISS idea has also been introduced by different approaches [17], [18]. In [17], the sufficient conditions for the ISS and iISS of nonlinear systems were obtained, which were based on an indefinite Lyapunov function rather than a positive definite one. The nonlinear matrix inequalities (NLMIs) approach was extended to ISS. Sufficient conditions for ISS of systems with time-delay were obtained and NLMIs were derived then for a class of systems with delayed state-feedback by using the S-procedure in [18].

State estimation has been found in many practical applications, which has been extensively investigated in the past decades. The objective of the state estimation is to estimate the unavailable state variables (or a linear combination of the state variables) for a given system. It has been recognized that one of the most celebrated estimation methods is Kalman filtering. Kalman filtering method requires that the given system is known in which disturbances are stationary Gaussian noises with known statistics. In practical applications, the statistical property of the disturbances may not be known exactly. So some approaches have been proposed to deal with the filtering problem such as $H_{\infty }$ filtering [19], [20], [21], [22], [34] and $L_2-L_{\infty }$ filtering [23], [24] etc. Compared with the Kalman filtering, one of the main advantages of $H_{\infty }$ filtering is that the statistical information for the external noises is not required. There are many approaches for designing $H_{\infty }$ filters which mainly based on the algebraic Riccati equations and linear matrix inequalities (LMIs) [25], [26]. In the past few years, stochastic systems have received much attention since stochastic modeling played an important role in many branches of science and engineering applications. It is well-known that time-delay is frequently encountered in various real systems [35], [36], [37], [38], [39], and it is often one of the sources of poor performance of systems. A considerable number of terms and results for stochastic time-delay systems have been presented. Hence, the $H_{\infty }$ filtering problem for the stochastic time-delay systems is an important research area that has attracted considerable interest [30], [31]. From time-delay point of view, there are two kinds of filters: delay-independent filter and delay-dependent filter. Generally speaking, the delay-independent filter is more conservative than the delay-dependent one. For structured uncertainty time-delay systems and stochastic Markovian jump time-delay systems, the $H_{\infty }$ filtering problem has been discussed in [4], [27], [28], [29].

As above discussed, there are many methods for solving filtering problem for linear systems. Compared with linear systems, the filtering problem of nonlinear systems has few reports although it is important in control design and signal processing applications [32], [33]. So far, to the best of the authors’ knowledge, the filtering problem for stochastic nonlinear systems has not been fully investigated and is still full of challenging.

In this paper, an $H_{\infty }$ filtering problem is studied for a class of stochastic nonlinear systems with time-delay and exogenous disturbances. The stochastic systems considered in this paper are It$\hat{o}$ type, in which the time-delay is constant and the nonlinear function satisfies a linear-type growth condition and local Lipschitz condition. By using Lyapunov function method, sufficient conditions of SISS for the stochastic time-delay nonlinear systems are given in terms of LMI, which is convenient for the parameters design. By using the sufficient conditions of SISS, an $H_{\infty }$ filter is designed such that the filtering error system is SISS and satisfies the prescribed $H_{\infty }$ performance level.

The rest of this paper is organized as follows. In Section 2, we provide some definitions that will be needed throughout this paper. In Section 3, the main contributions of this paper are stated and proved. Three examples are to be presented to demonstrate the effectiveness of the proposed approaches in Section 4. Finally, conclusions are given in Section 5.

This paper uses the following notations: $\mathbb{R}$ and ${\mathbb{R}}^{+}$ denote the set of real numbers and the set of nonnegative real numbers, respectively. ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space. $L_2[0\text{,}\infty )$ is the space of square-integrable vector functions over $[0\text{,}\infty )$. Let I denote the identity matrix. The expression $A<B$ means that the matrix $B-A$ is positive definite. ${\vartheta }^{i\text{,}k}$ denotes all the functions with ith continuously differentiable first component and kth continuously differentiable second component. $\epsilon (•)$ denotes the expectation. $‖x‖$ denotes the Euclidean norm defined by $‖x‖=\sqrt{{\sum }_{i=1}^n{x}_i^2}$ for $x={(x_1^T\text{,}{x}_2^T\text{,}\cdots \text{,}{x}_n^T)}^T\in {\mathbb{R}}^n$. ${\lambda }_{\max }(A)$ and ${\lambda }_{\min }(A)$ are used to denote the maximum and minimum eigenvalue of A respectively. $L_{\infty }$ denotes the set of all functions endowed with the essential supremum norm defined by $\left|u\right|=\sup \{‖u‖\text{,}t⩾0\text{,}u\in {\mathbb{R}}^q\}<\infty$.