A multiple lyapunov function approach to stabilization of fuzzy control systems

TLDR

The paper studies stability and stabilization of Takagi‑Sugeno fuzzy systems using a multiple Lyapunov function. The authors propose a new parallel distributed compensation scheme that feeds back the time derivatives of premise membership functions to exploit this Lyapunov function. They define the fuzzy Lyapunov function by blending quadratic Lyapunov functions and derive stability conditions for open‑loop and closed‑loop systems, employing the new PDC scheme. The new PDC generalizes the ordinary PDC, and a design example demonstrates the effectiveness of the fuzzy Lyapunov function approach and the new stabilization method.

Abstract

This paper addresses stability analysis and stabilization for Takagi-Sugeno fuzzy systems via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. Based on the fuzzy Lyapunov function approach, we give stability conditions for open-loop fuzzy systems and stabilization conditions for closed-loop fuzzy systems. To take full advantage of a fuzzy Lyapunov function, we propose a new parallel distributed compensation (PDC) scheme that feedbacks the time derivatives of premise membership functions. The new PDC contains the ordinary PDC as a special case. A design example illustrates the utility of the fuzzy Lyapunov function approach and the new PDC stabilization method.