Membership Function Independent Global Stability Condition for Takagi-Sugeno Fuzzy Systems

We provide a less conservative approach for the stability analysis of Takagi–Sugeno fuzzy systems via multiple Lyapunov function with an integral of the membership function. In many papers, fuzzy Lyapunov function and the stability of fuzzy systems have been discussed. Since they assumed the upper/lower bounds of the derivative of the membership function derived from fuzzy Lyapunov function, they basically give local stability conditions. In this paper, we employ a Lyapunov function constructed with an integral of the membership function, from which less conservative stability conditions have been obtained in the literature. Furthermore, by combining linear fractional transformation (LFT) method and S-procedure method, we obtain a global asymptotic stability condition of fuzzy systems. Finally, an example is shown to confirm the advantage of the proposed method.

I. Introduction

In nonlinear control theory, Takagi–Sugeno fuzzy model, that efficiently represents a nonlinear system, has been extensively used due to its properties. These days, they are found in a variety of applications [1], [2]. Hence, analysis of the stability for nonlinear systems is important. Lyapunov’s stability theorem have been used as a method for guaranteeing stability for Takagi–Sugeno fuzzy systems in the literature. A solution based on linear matrix inequality (LMI) has been applied, and the problem of constructing a Lyapunov function in fuzzy systems have been basically solved [3]. However, many papers still tackle how to handle the derivative of a membership function.