Reliable mixed ${\mathcal{H}}_{\infty }$/passive control for T–S fuzzy delayed systems based on a semi-Markov jump model approach

Abstract

This paper investigates the problem of the reliable mixed ${\mathcal{H}}_{\infty }$/passive control for Takagi–Sugeno (T–S) fuzzy delayed systems based on a semi-Markov jump model (SMJM) approach. The focus is to design a fuzzy fault-tolerant controller such that the resulting closed-loop system is stochastically stable with a prescribed mixed ${\mathcal{H}}_{\infty }$/passive performance level even if the actuator failures appear. A semi-Markov process is employed to describe the encountered failures of the actuator. By applying the Lyapunov–Krasovskii method, in combination with some novel inequalities, some conditions on the performance analysis are established, where some negative quadratic terms are fully considered to reduce the conservatism. Based on the conditions, an explicit expression for the desired controller is given. Three numerical examples are presented to show the effectiveness and reduced conservatism of the proposed method.

Introduction

The study of Markov jump systems (MJSs) is inspired by many real world technical problems involving random abrupt variations and switches in many practical systems [1]. A Markov jump system that undergoes transitions from one mode (subsystem) to another, between a finite or denumerable number of possible modes. Its applications can be found in many practical systems, such as networked control systems [19], [30], fault-tolerant systems [33], [51], target tracking systems [18] and so on. It is therefore unsurprising that MJSs have received comprehensive research attention in recent years, see for example [14], [27], [40]. It should be pointed out that MJSs can be succeeded in modeling switching of practical subsystems. The key idea is based on an implicit assumption that the sojourn-time (the interval between two consecutive jumps) of each mode is subject to exponential distribution. Such an assumption, sometimes, is difficult to be satisfied in many practical applications, such as reliability analysis, DNA analysis, the bunch-train cavity interaction (BTCI) system and so on [2], [13], [23]. As a result, MJSs are no longer as the suitable mathematical models. In order to overcome this limitation, the concept of semi-Markov jump systems (SMJSs) has been introduced, in which the transition rates of semi-Markov process are sojourn-time-dependent, and then SMJSs have wider applications than the MJSs [23].

On another research direction, Takagi–Sugeno (T–S) fuzzy systems have been proven to be a powerful tool for controlling nonlinear systems owing to their universal approximation characteristics. The T–S fuzzy model approach combines the flexible fuzzy logic theory and fruitful linear system theory into a uniform framework to approximate a broad range of complex nonlinear systems. The advantages in using a small number of rules to model higher-order nonlinear systems based on T–S model were shown in [6], [9], [10], [32]. It is not surprising that various results on T–S fuzzy systems have been reported in terms of many kinds of methods, see [5], [9], [12], [31], [35], [41], [42], [47] and the references therein. As is known that time delays are an immanent property of many practical systems, such as biological systems, nuclear reactors, and chemical processes [11]. They may give rise to instability or significantly deteriorate performances for the corresponding closed-loop systems [11], [36]. Recently, the research on T–S fuzzy systems with time delays has also achieved substantial attention and a great many of results have been established [21]. To name just a few, several reliable control approaches have been proposed, see [17], [39]. In addition, the appearance of actuator failures should not be ignored due to the fact that the reliability and security demand of modern control systems is increasing, especially for achieving more industrial oriented applications [43], [44], [45], [46], [52]. However, how to design a fault-tolerant controller has not been fully studied in the above-mentioned papers, especially on the basis of the semi-Markov jump model (SMJM) approach, which limits their applications.

It is known that the ${\mathcal{H}}_{\infty }$ control theory as an important part has been thoroughly embedded in modern control theory during the last decades [28], [50]. The purpose is to design a controller which minimizes the ${\mathcal{H}}_{\infty }$ norm of some related systems [48]. Besides, it is recognized that the passivity theory plays a key role in the analysis and design of linear and nonlinear systems [16], [25], [26]. The application of passivity puts forward a new method to study the stability, and the storage function of passive systems can be utilized as a Lyapunov function candidate for complex systems. Therefore, the method of passive analysis can be taken advantage in the study of many systems [49]. Owing to the importance of ${\mathcal{H}}_{\infty }$ control theory and passivity theory, an interesting question appears here: how to address the mixed ${\mathcal{H}}_{\infty }$/passive problem of T–S fuzzy delayed systems? Furthermore, when the appearance of actuator failures is taken into account, how to use the SMJM approach to solve the fault-tolerant controller design problem? These questions have been seldom researched, which motivate our present work.

In this paper we deal with the reliable mixed ${\mathcal{H}}_{\infty }$/passive control problem for T–S fuzzy delayed systems based on a SMJM approach. The main contributions of this paper are three-fold: 1. The SMJM approach is used for the first time to solve the fault-tolerant controller design problem for T–S fuzzy systems; 2. A simple actual mixed ${\mathcal{H}}_{\infty }$/passive performance index is proposed in dealing with the control problem for T–S fuzzy delayed systems. Based on such an index, the control problem under consideration here is more general. It includes the ${\mathcal{H}}_{\infty }$ control or passive control problem as a special case by optimizing the weighting parameters; 3. Different from the existing methods in [22], we adopt some novel inequalities and an identical equation such that some negative quadratic terms (as stated in Remark 3) are reserved when a mode-dependent Lyapunov functional is chosen. As a consequence, the results could be less conservative. The effectiveness and reduced conservatism of the proposed method are demonstrated by three numerical examples.

The rest of this paper is organized as follows. The addressed problem is formulated in Section 2. Our main results are presented in Section 3, where some new mixed ${\mathcal{H}}_{\infty }$/passive performance conditions and the method to calculate the parameters of the controller are given. Section 4 provides three examples to demonstrate the effectiveness of the proposed method. Finally, we conclude the paper in Section 5.

Notation

The following notations will be used throughout the paper: ${\mathbb{R}}^n$ and ${\mathbb{R}}^{m\times n}$ denote the n-dimensional Euclidean space and the set of all $m\times n$ real matrices, respectively. $S>0$ means that matrix S is real symmetric and positive definite. The superscript “T” stands for the transpose. $|\cdot |$ denotes the Euclidean norm of a vector and its induced norm of a matrix. $\mathcal{E}\{\cdot \}$ denotes the expectation operator with respect to some probability measure $\mathcal{P}$. The symbol “⁎” is used to represent a matrix which can be inferred by symmetry. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.