A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions

TLDR

Research on fuzzy control systems has increasingly focused on linear matrix inequality (LMI)–based analysis and design. This study introduces a descriptor system framework that employs fuzzy Lyapunov functions for fuzzy control design, demonstrated through two illustrative examples. By exploiting descriptor system redundancy, the authors reduce the number of LMI constraints, propose new fuzzy controllers and Lyapunov functions, and link LMI feasibility to the switching speed of each subsystem. The first example shows that the proposed fuzzy Lyapunov–based LMIs are less conservative than those using a common Lyapunov function, while the second example demonstrates their advantage over a piecewise Lyapunov approach.

Abstract

There has been a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs). This paper presents a descriptor system approach to fuzzy control system design using fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. To obtain relaxed LMI conditions, new types of fuzzy controller and fuzzy Lyapunov function are proposed. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystem (to be exact, to the lower bounds of time derivatives of membership functions). To illustrate the validity and applicability of the proposed approach, two design examples are provided. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach.