Further result on H∞ filter design for continuous-time Markovian jump systems with time-varying delay

Abstract

This paper investigates the problem of $H_{\infty }$ filtering for Markovian jump linear systems with time-varying delay. The aim of this problem is to design an $H_{\infty }$ filter that ensures stochastic stability of the filtering error system and a prescribed L₂-induced gain from the noise signals to the estimation error, for all admissible uncertainties. For solving the problem, we transform the system under consideration into an interconnection system. Based on the system transformation and the stochastic scaled small gain theorem, stochastic stability of the original system is examined via the stochastic stability version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time-varying delay and the stochastic scaled small gain theorem. The proposed $H_{\infty }$ filtering condition is demonstrated to be less conservative than most existing results. Moreover, the $H_{\infty }$ filter design condition is further presented via convex optimizations, whose effectiveness are also illustrated via numerical examples.

Introduction

Over the past decades, the $H_{\infty }$ filtering problem has drawn much attention due to its advantages over the traditional Kalman filtering [1], and many results have been obtained with respective to various filtering performance criteria, see [2], [3], [20], and references therein. The results of the problem for filter design are widely used for data processing in the fields of aerospace and detection. In the $H_{\infty }$ filtering setting, the noise sources are arbitrary signals with bounded energy or average power, and no exact statistics are required to be known [4], [5], [6], [7], [10], [21].

On the other hand, many practical systems are subject to random abrupt changes in their inputs, internal variables and other system parameters, which may result from abrupt phenomena such as random failures and repair of the components, changes in the interconnections of subsystems, sudden environmental changes, and modification of the operating point of a linearized model of a nonlinear system. Markovian jump linear system (MJLS) can represent these systems successfully and has drawn great attention during the past decades, and many significant results are presented and applied to many practical systems such as failure prone manufacturing systems, power systems and economics [11], [12], [18], [26], [27], [28].

It is well recognized that there always exist inherent delays and time delays in most of industrial processes, which are often the cause for instability and poor performance of the systems. In the past few years, lots of papers have appeared on this general topic of $H_{\infty }$ filtering problem for Markovian jump delayed systems. The problem of robust $H_{\infty }$ filtering is investigated in [17], [22], [23], [24], [25], while the reduced-order $H_{\infty }$ filter for Markovian jump systems with time delay is investigated in [16]. Furthermore, the paper in [19] presents the exponential $H_{\infty }$ filter design existence condition considering missing measurements, and the $H_{\infty }$ filter design for two-dimensional Markovian jump systems with time delay is studied in [30]. Although the problem of $H_{\infty }$ filter for Markovian jump systems has been widely investigated, there still exist some further problems, which need to be solved. For example, the filtering design results obtained so far are still quite conservative and leave much room for further improvement, which motivates our present study.

In this paper, our objective is to solve the problem of $H_{\infty }$ filtering for Markovian jump systems with time-varying delay from a perspective looking for feasible solutions to this problem with much reduced conservatism. The Markovian jump system with time-varying delay under consideration is transformed into an interconnected form by the transformation method. Second, based on the stochastic small gain theorem for time-delay systems with Markovian jump parameters, the problem of $H_{\infty }$ filtering is solved as a LMIs form. Finally, one illustrative example is exploited to demonstrate the merit of the developed $H_{\infty }$ filtering result. The result presented in this paper is less conservative than other references, and also this paper expands the input–output method to solve the problem of $H_{\infty }$ filtering for Markovian jump systems.

The remainder of the paper is organized as follows. The problems of switched system with time-varying delay are addressed in Section 2. In Section 3, based on the idea of input–output approach and the model transformation method, the new stability criterion is proved to guarantee interconnection system input–output stability via exponential scaled small gain theorem, which is further proved to guarantee interconnection system exponential stability. Meanwhile, the controller existence criterion and switching law criterion based on scaled small gain theorem are proposed. The illustrative examples are provided in Section 4, and finally we conclude the paper in Section 5.

Notations: ${\mathbb{R}}^n$ and ${\mathbb{R}}^{m\times n}$ represent the set of real n-vector and m×n matrices, respectively. ${\mathbf{G}}_1○{\mathbf{G}}_2$ represents the series connection of mapping ${\mathbf{G}}_1$ and ${\mathbf{G}}_2$. The superscript “T” stands for matrix transpose. The notation $P>0(\ge 0)$ means that matrix P is positive (semi)definite. I_n denotes an identity matrix with dimension n and $0_{m,n}$ denotes an m×n dimension zero matrix. We use “$⁎$” to denote the symmetric terms in a block matrix P, ${\{P\}}_i$ to represent the ith row of its explicitly expressed block structure, $\mathrm{sym}\left(P\right)$ to abbreviate $P+P^T$. For $x\left(t\right)$, $\parallel x{\parallel }_{E_2}\triangleq E\left\{\sqrt[2]{{\int }_0^{\infty }{x}^T\left(t\right)x\left(t\right)\mathit{dt}}\right\}$.