Reliable guaranteed cost control for uncertain fuzzy neutral systems

Abstract

This paper focuses on the problem of robust reliable guaranteed cost control for a class of uncertain Takagi–Sugeno fuzzy neutral systems with linear fractional parametric uncertainties. The aim is to design a state feedback controller such that, for all admissible uncertainties as well as actuator failures occurring among the prespecified subset of actuators, the plant remains asymptotically stable and guarantees an adequate level of a quadratic cost index. Based on the Lyapunov–Krasovskii functional, the Barbalat lemma, the descriptor system approach and the free weighting matrix method, new delay-dependent sufficient conditions for solvability of this problem are presented in terms of LMIs. Based on that, the design problem of the optimal reliable guaranteed cost controller is formulated as a convex optimization problem. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.

Introduction

The Takagi–Sugeno (T–S) fuzzy model [1] has been proven to be a powerful tool for modeling complex nonlinear systems. It is well-known that, by means of the T–S fuzzy model, a nonlinear system can be represented as a weighted sum of some simple linear subsystems and then can be stabilized by a model-based fuzzy control. Therefore, it provides a good opportunity to employ well-established linear systems theory for theoretical analysis and design of the controllers for complex nonlinear systems. Over the past two decades, many issues related to stability analysis and control synthesis of T–S fuzzy systems have been reported, see, e.g. [2], [3], [4] and the references therein.

It is well known that, as a source of instability, time delay is often encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems [5]. The T–S fuzzy system with time delay was introduced in [6]; and during recent years, the study of the time-delay T–S fuzzy system has received much attention (see, e.g. [6], [7], [8]). Moreover, it is also well known that many practical delayed processes can be modeled as general neutral systems, which contain delays both in their states and in the derivatives of their states, such as circuit analysis, computer aided design, realtime simulation of mechanical systems, power systems, chemical process simulation, and optimal control [9]. So, recently, the stability and stabilization analysis of neutral systems have been extensively investigated, see, e.g. [10], [11], [12]. Quite recently, the T–S fuzzy time-delay system of neutral type was introduced in [13], where both the stabilization and $H_{\infty }$ control problems were studied by using the LMI approach. In [14], via the descriptor system approach, a generalized delay-dependent sufficient condition for time-delay fuzzy neutral systems to achieve $H_{\infty }$ disturbance attenuation was given. Utilizing the LKF and the LMI approach, robust $H_{\infty }$ control of uncertain T–S fuzzy neutral systems with both discrete and distributed delays was investigated in [15]. Based on the LKF, the descriptor system approach and the LMI approach, novel sufficient conditions for solvability of the non-fragile $H_{\infty }$ control problem for a class of uncertain fuzzy neutral systems were obtained in [16]. Very recently, by means of the LKF and the Barbalat lemma, the robust $H_{\infty }$ filtering problem for a class of uncertain T–S fuzzy neutral systems was investigated in [17].

On the other hand, when controlling a real plant, it is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance, such as disturbance attenuation, passivity and so on. Another approach to this problem is the guaranteed cost control approach introduced by [18]. This approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation is guaranteed to be less than this bound. Recently, there has been an increasing interest in the study of the guaranteed cost control problem for delayed fuzzy systems [19], [20], [21], [22]. Moreover, in actual implementation, the actuators may be subjected to failures. It is therefore of practical interest to design a control system which can tolerate some actuator failures. In the last few years, the problem of designing reliable control for fuzzy systems has attracted considerable attention [23], [24], [25], [26], [27], [28]. Furthermore, the reliable guaranteed cost control problem has been extensively investigated in recent years, see, e.g. [29], [30], [31], [32] and the references therein.

However, to the best of the authors’ knowledge, the reliable guaranteed cost control problem has not been addressed for T–S fuzzy neutral systems with time-varying delay and linear fractional parametric uncertainties, which motivates the present study. This paper aims to design a state feedback controller such that, for all admissible uncertainties as well as actuator failures, the plant remains asymptotically stable and guarantees an adequate level of a quadratic cost index. On the basis of the LKF, the Barbalat lemma, the descriptor system approach and the free weighting matrix method, new delay-dependent criterion for the reliable guaranteed cost controller design is derived in terms of LMIs. Furthermore, a method of selecting a optimal controller minimizing the upper bound of the guaranteed cost is presented.

The rest of this paper is organized as follows. The main problem is formulated in Section 2 and sufficient conditions for the solvability of the robust reliable guaranteed cost controller for the uncertain T–S fuzzy neutral systems are derived in Section 3. In Section 4, a simulation example is provided; and a concluding remark is given in Section 5.

Notations

$R^n$ and $R^{n\times m}$ denote, respectively, the $n$-dimensional Euclidean space and the set of all $n\times m$ real matrices. The notation $A>(\ge )B$ means that $A-B$ is positive (semi-positive) definite. $I\left(0\right)$ is the identity (zero) matrix with appropriate dimension. $A^{-1}$ denotes the inverse of matrix $A$ and $A^T$ the transpose. $\vec{A}$ represents the sum of $A$ and $A^T$. $\Vert •\Vert$ denotes the Euclidean norm in $R^n$. “*” denotes the elements below the main diagonal of a symmetric block matrix. $\mathrm{tr}\left(•\right)$ is the trace of matrix “•”. Let $C([-\tau ,0],R^n)$ be the family of continuous functions $\varphi$ from interval $[-\tau ,0]$ to $R^n$ with the norm ${\Vert \varphi \Vert }_{\tau }={sup}_{-\tau \le \theta \le 0}\Vert \varphi \left(\theta \right)\Vert$.