Delay-dependent fault detection for switched linear systems with time-varying delays—the average dwell time approach

Abstract

In this paper, the fault detection problem is investigated for a class of discrete-time switched linear systems with time-varying delays. The main purpose is to design a fault detection filter such that, for all unknown inputs, control inputs and time delays, the estimation error between the residual and fault is minimized in an exponential way. The fault detection problem is converted into an exponential H_∞ filtering problem. By using a newly constructed Lyapunov functional and the average dwell time scheme, a novel delay-dependent sufficient condition for the solvability of this problem is established in terms of linear matrix inequalities (LMIs). A numerical example is given to demonstrate the effectiveness of the developed theoretical results.

Introduction

Switched systems are hybrid dynamical (nonlinear) systems composed of subsystems with their own parameterizations subject to a rule orchestrating the switching law between the various subsystems [1]. The last decade has witnessed increasing research activities in the study of controllability, observability, stability, stabilization and filter designs of the switched systems. Among these research topics, stability analysis and stabilization for switched systems have attracted most of the attention. The readers may refer to the survey papers [2], [3] and the references therein. To date, many methods have been developed in the study of switched systems such as the multiple Lyapunov function approach [4], average dwell time method [5], [6], [7] and switched Lyapunov function approach [8]. Switched systems deserve the investigation for their practical importance. For example, switching among different controllers for a single process can be viewed as a switched system [9], and the discrete-time system with time-varying delays can also be viewed as a switched system [10]. It has been shown that this kind of switching control strategy provides an effective method to deal with highly complex systems. In addition, the study of the switched system provides a powerful tool for analysis of networked control systems (NCSs), e.g. NCSs subject to packet dropouts can be modeled as switched systems with both stable and unstable subsystems [11].

Time-delay phenomenon is very common in various practical systems. The presence of delay may lead to instability and poor performance of control systems [12], [13], [14], and may also make the analysis and synthesis problems for switched time-delay systems much more complicated. Switched time-delay systems have various applications in practical engineering systems. See, for example, power systems [15], power electronics [16] and networked control systems [17]. In the work [17], a discrete-time switched delay system model has been proposed to describe networked control systems with both network-induced delay and packet-dropout. Recently, some results have appeared on the stability analysis, controller and filter design for switched systems with time-delay. See e.g. [18], [19], [20], [21].

Fault detection and isolation (FDI), on the other hand, has been an active field of research over the past decades because of the increasing demand for higher performance, higher safety and reliability standards. Fruitful theoretical results and increasing applications in industrial practice, such as power plant, coal mills and robotic systems can be found in [22], [23], [24]. Furthermore, with the rapid development of communication networks, a great amount of efforts have recently been devoted to the problems of an FDI for networked control systems [25], [26]. For example, the authors in [25] have considered the fault detection problem for a class of discrete-time networked systems with multiple state delays and an unknown input. A new measurement model has been proposed to account for both the random measurement delays and the stochastic data missing (package dropout) phenomenon. As has been discussed before, the switched linear system is a powerful method to model the NCSs. Therefore, it is not only theoretically interesting but also practically important to study the FDI for the switched linear systems with delay or without delay. The pioneer of this topic has been published in [27], where the switched Lyapunov functional method is used for the FDI of discrete-time switched systems with state delays, and sufficient conditions for the solvability of the problem are established in terms of LMIs. However, it should be pointed out that only constant delays are considered. In addition, the obtained results are delay-independent. To the best of the authors’ knowledge, the delay-dependent fault detection for switched systems with time-varying delays has not yet been fully investigated, and it is our intention in this paper to shorten such a gap.

In this paper, the fault detection problem is investigated for a class of discrete-time switched linear systems with time-varying state delays. The fault detection problem addressed is firstly converted into an auxiliary H_∞ filtering problem. Then, the Lyapunov functional method and the average dwell time approach are proposed for the analysis and synthesis of the considered systems. A delay-dependent sufficient condition for the existence of the desired fault detection filters is derived and formulated in terms of LMIs. A numerical example is finally given to show the effectiveness of the proposed method.

Notations: the notation used throughout the paper is fairly standard. We use $W^T,W^{-1},\lambda (W),\text{Tr}(W)$ and ||W|| to denote, respectively, the transpose, the inverse, the eigenvalues, the trace and the induced norm of any square matrix W. We use W>0 to denote a positive-definite matrix W with λ_min(W) and λ_max(W) being the minimum and maximum eigenvalues of W and I to denote the identity matrix with an appropriate dimension. Let Rⁿ denote the n dimensional Euclidean space. R^m×n is the set of all m×n real matrices. The notation l₂[0,∞] refers to the space of square summable infinite vector sequences with the usual norm ||·||₂. The symbol ⁎ will be used in some matrix expressions to represent the symmetric terms.