Fuzzy model-based asynchronous H_∞ filter design of discrete-time Markov jump systems

Abstract

This paper investigates the issue of asynchronous H_∞ filter design for a class of discrete-time Markov jump fuzzy systems with uncertainties. Two totally different but equivalent methods are adopted to derive sufficient conditions that ensure stochastic stability as well as a prescribed H_∞ performance of the filtering error system. To reduce conservatism of the filter design methods, the fuzzy-basis-dependent Lyapunov function is chosen to depend both on mode information of the plant and the filter, which guarantees all available information is fully utilized. Then the existence criterion for the asynchronous H_∞ filter to be designed is presented in the form of linear matrix inequalities. Finally, a simulation example is provided to demonstrate the potential and effectiveness of the proposed methods.

Introduction

In the control field, it is well known that the system states contain or reveal some essential information of dynamic systems, such as the voltage, the current, the position and the velocity in physical systems. However, state variables are not always measurable in control systems, in such case we have to estimate state variables for control purposes. Over the past decades, due to the great significance of state variables, state estimation or filtering problem has been one of the most important research topics and many effective techniques have been developed to deal with this problem, such as Kalman filtering [1], H_∞ filtering [2], $l_2-l_{\infty }$ filtering [3], [4], [5]. Among all these techniques, H_∞ filtering, which is also called energy-to-energy filtering, has been one of the most popular techniques. The essence of H_∞ filtering is to minimize the gain between energy of the filtering error and that of the external disturbance [6]. Since H_∞ filtering is even applicable for systems whose information of noise signals may be unknown, it has attracted considerable attention from researchers and a great deal of work related to H_∞ filtering has been established [7], [8], [9].

In general, most of the dynamic systems suffer some random variations in parameters or structures in practice [10]. These random variations are usually caused by sudden environmental disturbance, component failures and component repairs [11], unfortunately, all of these have major influences on the performance of dynamic systems. To overcome this shortcoming, Markov jump systems (MJSs) have been introduced to describe dynamic models in various fields, such as biomedicine, economics and communication networks [12]. In the past decade, a number of results focused on H_∞ filtering of Markov jump systems have been obtained. For example, H_∞ filter design of Markov jump systems with sensor saturation and randomly occurring nonlinearities has been studied in [13], H_∞ filtering for Markov jump systems with partly known transition probabilities has been reported in [14]. Resilient asynchronous H_∞ filter design of Markov jump neural networks with multiplicative noises and unideal measurements has been considered in [15] while authors of [16] have studied H_∞ filtering for Markov jump linear systems whose jumping mode information is not accessible.

It should be pointed out that most of the results aforementioned are concerned with linear Markov jump systems, however, we are more commonly faced with some nonlinearities in practical engineering problems, which increase the difficulties of the analysis and synthesis of control problems. Fortunately, the Takagi–Sugeno (T–S) fuzzy model bridges a gap between nonlinear systems and linear control due to its powerful approximation of any smooth nonlinear systems [17], [18], [19], [20]. Recently, T–S fuzzy model has been utilized successfully to deal with H_∞ filtering problem of nonlinear Markov jump systems, for instance, H_∞ filter for a class of nonlinear nonhomogeneous Markov jump fuzzy systems has been designed in [21], robust H_∞ filtering for uncertain nonlinear singularly perturbed Markov jump fuzzy systems has been considered in [22], H_∞ filter design for a class of nonlinear Markov jump neutral systems based on T–S fuzzy model has been studied [23]. Up to now, although so many results on H_∞ filter design for Markov jump fuzzy systems have been reported, to the best of the authors’ knowledge, the issue of asynchronous H_∞ filter design for discrete-time Markov jump fuzzy systems with uncertainties has not been fully addressed, this is the motivation of the present study.

In this paper, our main purpose is to solve the asynchronous H_∞ filtering problem for a class of discrete-time Markov jump systems with uncertainties based on T–S fuzzy model, more specifically, we aim to design a filter which works asynchronously with the plant. Two different methods are proposed to deal with the filtering problem, which turn out to be equivalent. Sufficient conditions are derived to ensure the resultant filtering error system is stochastically stable with a prescribed H_∞ performance level. The fuzzy-basis-dependent Lyapunov function is chosen to depend both on mode information of the plant and the filter, which is obviously less conservative than the common Lyapunov function approach due to all available information is fully utilized. To demonstrate the correctness and effectiveness of the proposed methods, a simulation example is provided.

The rest portion of this paper is organized as follows. In Section 2 the problem to be addressed is formulated while in Section 3 stochastic stability with a given H_∞ performance of the filtering error system is analyzed. In Section 4, an sufficient condition is derived to deal with the filter design problem and the corresponding filter parameters are given explicitly. Finally, a simulation example is presented in Section 5 and the paper is concluded in Section 6.