Interval type-2 fuzzy control for nonlinear discrete-time systems with time-varying delays

Abstract

This paper is concerned with the problem of fault-tolerant control for discrete-time interval type-2 (IT2) fuzzy time delay system with actuator faults under imperfect premise matching. The time-varying delay and actuator fault are first taken into account for the IT2 fuzzy discrete-time systems. The novel IT2 state-feedback controller and the IT2 fuzzy model share different lower and upper membership functions. The fault-tolerant controller is designed to compensate for the effect of faults by stabilizing the closed-loop system under the actuator failures. Furthermore, the standard IT2 state-feedback controller is designed such that the closed-loop system is asymptotically stable and has an $H_{\infty }$ performance. The obtained conditions of the fault-tolerant controller and the standard IT2 controller can be expressed by the convex optimization problems. A numerical example is presented to show the effectiveness of the proposed results.

Introduction

It is well known that fuzzy sets have the capability to catch the complex system nonlinearities. On the strength of type-1 fuzzy sets [1], the fuzzy logic control systems have received considerable attention in [2], [3], [4], [5], [6], [7], [8]. In [4], by designing a decentralized adaptive fuzzy output feedback approach, a tracking control scheme for a class of large-scale nonlinear systems were designed. Considerable attention has been paid to type-1 Takagi–Sugeno (T–S) fuzzy system [5], [9], [10], [11] since it is a popular approach to represent nonlinear systems. Therefore, some results on stability, controller and filter design for T–S fuzzy systems were reported in [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. The delay widely exists in practical systems [22], [23], [24], [25], such as population dynamics, steel rolling process, chemical process, biological systems and economical systems. Recently, the stabilization problems of T–S fuzzy system with time-varying delays were illustrated in [12]. In [15], a new approach was proposed to the stability analysis and stabilization of discrete-time T–S fuzzy time-varying delay systems. On the other hand, in the actual circumstances, actuator failures may be encountered in many practical systems. Actuators play an important role in transforming the controller output to the plant and therefore it is significant to preserve the closed-loop control system performance under the circumstance of actuator failures [26], [27], [28], [29], [30]. The authors in [27] considered the problem of fault-tolerant control for T–S fuzzy discrete-time systems with actuator delay and infinite-distributed delay. However, it should be noted that the above stability analysis and control synthesis results are obtained in the frame of type-1 fuzzy set. One knows that the control problem for type-1 T–S fuzzy models can not be addressed if the membership functions contain uncertainties. Therefore, the advantages of stability analysis and controller synthesis by applying the parallel distributed compensation (PDC) design method will be vanished.

Recently, based on type-2 fuzzy set, an IT2 fuzzy logic model was proposed in [31] and utilized to deal with the nonlinear plants subject to parameter uncertainties which is superior than the type-1 T–S fuzzy uncertain systems. The main advantage of the IT2 fuzzy logic systems [31], [32], [33], [34] is that they are good in catching uncertainties by the lower membership function and upper membership function. More recently, many applications of the IT2 fuzzy logic systems have been given in [31], [35], [36], [37], [38], [39], [40]. Besides, the authors in [41] constructed the IT2 T–S fuzzy systems and designed an IT2 controller. It has been shown that the IT2 fuzzy state-feedback controller can obtain less conservative results than the usual type-1 PDC one. After the work [41], the authors in [42] designed a novel IT2 fuzzy controller to reduce the conservativeness. However, no work has been reported in the existing literature on the IT2 fuzzy systems with time delay and actuator failure, which motivates this study. Furthermore, there are few results on the IT2 fuzzy control design for discrete-time systems in the literature. Therefore, inspired by [41], [42], this paper will make a new attempt to model the discrete-time IT2 T–S fuzzy systems and the design of IT2 fuzzy controller for discrete-time IT2 T–S fuzzy systems with time-varying delay and actuator faults. By developing some new techniques, a new type fault-tolerant controller is designed to guarantee that the closed-loop system is asymptotically stable under the actuator failures. The existence condition of the fault-tolerant controller can be expressed by a convex optimization problem. The main contributions of this paper can be summarized as follows: (1) The nonlinear systems subject to parameter uncertainties are modeled by the IT2 T–S fuzzy model approach, in which the lower and upper membership functions are introduced to represent and capture the uncertainties. (2) The IT2 fuzzy systems and the IT2 controller do not need to share the same lower and upper membership, and rules number, which makes the controller design more flexible. (3) The time-varying delay and actuator fault are first taken into account for the IT2 fuzzy discrete-time systems. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.

The rest of this paper is structured as follows. Firstly, the IT2 T–S fuzzy model and controller with failures are provided in Section 2. An IT2 FMB state feedback controller is to be designed which can tolerate some actuator failures for the discrete-time IT2 fuzzy time delay system under imperfect premise matching in Section 3. A simulation example is used to show the effectiveness of the proposed results in Section 4. Finally, a conclusion is presented in Section 5.

Notation: The notation of the paper is fairly standard. The superscripts “T” represents the matrix transposition. “I” represents an identity matrix with appropriate dimension. “$0_{m\times n}$” represents m×n zero matrix. The notation $P>0(\ge 0)$ represents P is symmetric and positive definite (semi-definite). $\mathrm{diag}\left\{\dots \right\}$ represents a block diagonal matrix. ${\mathbb{R}}^n$ represents Euclidean space of n -dimension. “^⁎” is used as an ellipsis for the part that are induced by symmetry. If matrices are not explicitly stated, are assumed to be compatible dimensions.