$H_{\infty }$ descriptor fault detection filter design for T–S fuzzy discrete-time systems

Abstract

This paper deals with the problem of $H_{\infty }$ descriptor fault detection filter design for T–S fuzzy discrete-time systems. By using the descriptor system method, a $H_{\infty }$ fault detection filter is designed to guarantee that the residual system is admissible and satisfies the $H_{\infty }$ performance index when control inputs, actuator faults and unknown bounded disturbances are included in the systems. By means of Lyapunov functional approaches, a sufficient condition for the admissibility of the residual system is expressed in form of linear matrix inequalities. The desired detection filter is obtained. A numerical example is presented to illustrate the effectiveness of this developed method.

Introduction

For highly complicated nonlinear systems, an appealing and effective approach, that is, fuzzy logic control has been developed. Based on the existing fuzzy approaches, Takagi–Sugeno (T–S) fuzzy model which was described by a family of fuzzy IF–THEN rules was firstly introduced in [1]. It is one of the most significant modeling methods. The analysis and synthesis of the investigated nonlinear systems is simplified via the methods. In terms of the T–S fuzzy methodology, local dynamics describe different linear models in different state-space regions. Thus, on the basis of smoothly blending these local models together with membership functions, the overall fuzzy model of the nonlinear dynamics is obtained. Many literatures on T–S fuzzy systems have been reported in the literature [2], [3], [4], [5], [6] and the references therein. The problems of discrete-time T–S fuzzy systems have been considered, refer to [7], [8], [9], [10], [11], [12] and so on.

Descriptor system is a natural mathematical representation of many practical systems, which is also called singular system, generalized state-space system, implicit system, semi-state system or differential algebraic systems. It gives a description of the dynamic and the algebraic relationship between the chosen descriptor variables simultaneously. Descriptor system has a larger class of system than the normal linear system model, which has been applied extensively in electrical circuits, power systems, economics, and the other areas. In recent years, many papers on descriptor systems have been reported, for examples [13], [14], [15], [16], [17], [18], [19] and the references therein. The problem of fuzzy descriptor systems has been solved in [16], [17]. Discrete-time singular fuzzy systems [18], [19] have been investigated. Based on the descriptor system approach, in [20], [21], the problem of fault estimation for stochastic systems and uncertain systems has been dealt with, respectively. In the literature [22], [23], [24], the descriptor system approach has been used to solve the problem of fuzzy systems.

In the past few decades, there exist intensive researches on fault detection and isolation algorithm and their applications in a wide range of industrial processes. Many results on fault detection have been shown in the literature [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41] and the references therein. Main sources of false alarm are unknown inputs, uncertainties, faults, disturbances and so forth in many industrial systems. The model-based approach is that a residual signal is compared with a predefined threshold by designing a state observer or filter. When the residual evaluation function is larger than the threshold, an alarm signal appears. In [29], the fault detection filter for uncertain fuzzy systems based on the delta operator approach has been designed. Fault detection for networked control systems has been reported in [30], [31]. The problem of fault detection for discrete-time system [32], [33] has been solved. The fault detection filter or observer for singular system has been obtained in [34], [35], [36]. In finite-frequency domain, authors have investigated the fault detection for T–S fuzzy discrete systems [37]. For Takagi–Sugeno systems, a robust fault detection observer has been constructed by using the descriptor system approach in [38]. The problem of fault detection of fuzzy jump systems [39] has been considered. For Markovian jump systems with sensor saturations and randomly varying nonlinearities [40], the fault detection problem has been solved. The fault detection problem for discrete-time Markovian jump systems with incomplete knowledge of transition probabilities, randomly varying nonlinearities and sensor saturations has been addressed in [41]. The results could be further utilized for other related issues, such as the filtering problems in sensor networks [42].

To the best of the authors׳ knowledge, the problem of $H_{\infty }$ descriptor fault detection filter design for T–S fuzzy discrete-time systems has not yet been fully investigated. Therefore, it motivates us to solve the interesting and challenging problem in this paper. The advantages of this paper are shown as follows: (I) The problem of $H_{\infty }$ fault detection filter design for T–S fuzzy discrete-time systems is solved by using a descriptor system method. (II) For control inputs, actuator faults and unknown bounded disturbances in this system, a fault detection filter is designed via the descriptor system method such that the residual system is admissible in the $H_{\infty }$ sense. (III) In terms of Lyapunov functional approaches, a sufficient condition for the admissibility of the residual system is obtained by linear matrix inequalities. The desired detection filter is expressed. (IV) The estimates of states and disturbances are obtained via designing a fault detection filter, simultaneously. (V) A numerical example is presented to illustrate the effectiveness of this developed method.

Notation: I is the unit matrix of appropriate dimensions; ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space; the notation $X>Y(X\ge Y)$ means that the matrix $X-Y$ is positive definite $(X\ge Y$ is semi-positive definite, respectively); $\mathrm{deg}\left(B\right)$ and $\det \left(B\right)$ denote the degree and the determinant of the square matrix B, respectively; for matrix A, the inverse of A is $A^{-1}$ and A^T denotes the transpose of A; $\parallel ·\parallel$ denotes the Euclidean norm for vectors or the spectral norms of matrices; $\mathit{diag}\{M_1,M_2,\dots ,M_n\}$ stands for a block diagonal matrix; $\mathrm{Re}(·)$ notates the real part of the argument; $l_2[0,\infty )$ means the space of square summable infinite sequence. ⁎ denotes the symmetric terms in a symmetric matrix.