Finite-time boundedness of interval type-2 fuzzy systems with time delay and actuator faults

Abstract

This paper is concerned with the issue of finite-time boundedness of discrete-time uncertain interval type-2 fuzzy systems with time-varying delay and external disturbances via an observer-based reliable control strategy. According to the system output variable, a full-state observer that shares the same membership functions of the plant is constructed to estimate the unknown system states. In addition, a reliable controller subject to observer states and actuator faults is designed to formulate the closed-loop feedback control system, which does not share the same membership functions of the plant. Then, by constructing an appropriate Lyapunov–Krasovskii functional and using the finite-time stability theory, a new set of delay-dependent sufficient conditions guaranteeing the finite-time boundedness of the addressed system is established in the framework of linear matrix inequalities. Furthermore, the explicit expressions of gain matrices of the state observer and the reliable controller are given in terms of the established sufficient conditions. Finally, simulation results are presented to demonstrate the effectiveness of the obtained theoretical results.

Introduction

Takagi–Sugeno (T–S) fuzzy systems have attracted considerable amount of research attentions over the past three decades since they have been recognized as a most popular and successful approach to the systematic design and analysis of nonlinear dynamical systems [1], [2], [3]. The most important property of T–S fuzzy model is that over all dynamics of the smooth nonlinear systems can be described as a convex summation of relatively simple linear subsystems using membership functions. Due to this fact, many interesting results on nonlinear systems have been developed based on the T–S fuzzy model approach [4], [5], [6], [7], [8], [9], [10], [11], [12]. Though this approach effectively deals with nonlinearities, it does not capture the uncertainty effect completely. To overcome this difficulty, an interval type-2 (IT2) fuzzy model has been introduced in [13], which has the potential to provide better performance than the conventional fuzzy models in the presence of uncertainties. Inspired by this advantage, research communities have widely investigated the IT2 fuzzy systems and presented many important results in the recent literature [14], [15], [16], [17], [18], [19], [20]. To mention a few, in [14], the authors obtained a sufficient condition for the uniformly boundedness of IT2 fuzzy system by using the adaptive sliding mode control law. In the framework of IT2 fuzzy model, a robust controller has been designed in [18] to stabilize the inverted pendulum with uncertain parameters. In [20], the problem of robust stabilization of polynomial IT2 fuzzy system with different parametric uncertainties has been discussed.

It should be pointed out that most of the existing works on fuzzy systems including the aforementioned references are based on the assumption that all the system states are measurable. Nevertheless, this assumption may not be hold for many practical system, which makes difficulty to utilize the system states in the design of state feedback controller for stabilizing such practical systems. In particular, the premise variables in T–S fuzzy systems are often depending on the state variables. It is therefore important to estimate all the states of the systems, which could be possible by the system output variables. In recent decades, several important and interesting results about estimation of the immeasurable states of T–S fuzzy system by using observer-based approach have been proposed, which play a significant role in design of stabilization control laws for complicated and nonlinear practical models [21], [22], [23], [24], [25], [26], [27]. On the other hand, due to growing complexity of automated control systems, various faults, especially actuator and sensor faults, are likely to be encountered in control systems. Neglecting these faults may consequently cause unwanted, even catastrophic, actions for the closed-loop system. To overcome this difficulty, a large number of works on fault-tolerant control problems for nonlinear systems based on the T–S fuzzy model approach have been reported in the existing literature [28], [29], [30], [31], [32]. For instance, in [29], the issue of stochastic stability for a class of Markov jump nonlinear systems with actuator faults has been discussed via an adaptive sliding mode control approach. In [32], the reliable dissipative control problem for T–S fuzzy systems with Markov jumping parameters has been considered.

Most of the existing results related to stability problems concentrate on asymptotic stability, which is associated with the behavior of dynamical systems within a sufficiently long time interval. Nevertheless, in practice, it is more desirable that a dynamical system should possess the property that state trajectories of system converge to a stable equilibrium point in a desired finite-time interval rather than asymptotically. Based on this motivation, the concept of finite-time stability has been proposed [33], [34]. It is noted that systems with finite-time convergence usually demonstrate not only faster convergence rates, but also some other nice features, such as better disturbance rejection properties and better robustness against uncertainties. Therefore, the investigation of finite-time stability of dynamical systems has received increasing attention recently [35], [36], [37], [38], [39], [40]. More interestingly, the dissipative-based finite-time boundedness problem for a class of networked control systems has been analyzed through a cascade controller in [39], where for simplicity only norm bounded parameter uncertainty and time-varying transition probability matrix have been considered. In [40], the problem of robust finite-time mixed H_∞ and passive filter for Markovian jump T–S fuzzy systems with time delay and uncertainty has been addressed.

Motivated by the aforementioned discussions, in this paper, the issue of robust H_∞ finite-time boundedness is investigated for a class of IT2 fuzzy systems subject to time-varying delay, parameter uncertainties and actuator faults via an observer-based approach. The primary contributions of this study can be summarized as follows:

• An IT2 fuzzy controller is designed in the framework of observer state for a discrete-time IT2 fuzzy system with time-varying delay, external disturbance and actuator faults.

• The designed control has different membership functions than the ones in the system under consideration and thus, the proposed controller design is independent of the premise variables, membership functions and number of fuzzy rules of the system, which enhances the flexibility of the controller design.

• To utilize more information about system uncertainty, footprint of uncertainties is considered and to improve the reliability of the control performance, actuator faults are taken into the controller design. These facts make the proposed model stronger.

• Based on the Lyapunov direct method, sufficient conditions for finite-time boundedness of the addressed IT2 fuzzy system with a prescribed disturbance attenuation level are established in the form of linear matrix inequalities (LMIs) using some advanced inequality techniques and the boundary information on membership functions. Illustrative examples are also provided to show the potentiality of the proposed design technique.

The rest of contents of this paper are as follow. In Section 2, a class of IT2 T-S fuzzy systems subject to non-measurable states together with the formulation of fault-tolerant controller is presented. In Section 3, the problem of finite-time boundedness is discussed for the concerned fuzzy systems. Two numerical examples are illustrated the effectiveness of the proposed results in Section 4. Conclusion of this paper is finally provided in Section 5.