Finite-Time $\mathcal{H}_{\infty}$ Controller Synthesis of T–S Fuzzy Systems

This paper studies the finite-time H_∞ control problem of T-S fuzzy systems subject to external disturbances. A novel fuzzy control approach is developed on the basis of the control Lyapunov function technique and the finite-time Lyapunov theorem so that the closed-loop fuzzy system is finite-time stable with guaranteed H_∞ performance. It is shown that the proposed control approach does not require the existence of a Lyapunov function before design of the fuzzy controller in comparison with many existing control approaches to finite-time control of nonlinear systems via the technique of control Lyapunov functions, and thus, the finite-time controller and the corresponding Lyapunov function can be designed simultaneously. The procedures on how to obtain the fuzzy controller are provided in terms of linear matrix inequalities. Moreover, an upper bound on the time taken for state trajectories to arrive at their desired targets is estimated. The effectiveness of the proposed finite-time H_∞ control approach is finally illustrated via numerical simulations.

I. Introduction

Fuzzy logic control (FLC) has been adopted widely to handle complex nonlinear systems over the past few decades. FLC is based on the fuzzy set theory introduced in [1], and the first FLC algorithm implemented in practice was reported in [2] and [3]. Since then, increasing attention has been given to the development of FLC due to its simplicity and ease in implementation for complex nonlinear system control problems, and some valuable results on this topic such as [4]–[6] have been obtained. Even though conventional fuzzy control approaches have been well accepted by engineers, yet they often suffer from the lack of systematic tools because conventional fuzzy control approaches are generally model free. To address these problems, a variety of fuzzy control approaches on the basis of models were proposed, and T–S model-based fuzzy control approaches have proved to be particularly successful in dealing with complex nonlinear systems, see [7]–[16] for details.