ELSEVIER Fuzzy Sets and Systems 106 (1999) 49-59 sets and systems Fuzzy Lyapunov-based approach to the design of fuzzy controllers Michael Margaliot, Gideon Langholz* Department of Electrical Engineering - Systems, Tel Aviv University, Tel Aviv 69978, Israel Received July 1998 Abstract In this paper we extend the classical Lyapunov synthesis method to the domain of computing with words. This new approach is used to design fuzzy controllers. Assuming minimal knowledge about the plant to be controlled, the proposed method enables us to systematically derive the fuzzy rules that constitute the rule base of the controller. We demonstrate the approach by designing Mamdani-type and Takagi-Sugeno-Kang-type fuzzy controllers for two well-known plants. @ 1999 Elsevier Science B.V. All rights reserved. Keywords: Lyapunov functions; Fuzzy control; Computing with words 1. Introduction The most difficult aspect in the design of fuzzy controllers is the construction of the rule base [2]. The process of extracting the knowledge of a human operator, in the form of fuzzy control rules, is by no means trivial, nor is the process of deriving the rules based on heuristics and a good understanding of the plant and control theory [7, 9]. We present a new method, based on extending the classical Lyapunov synthesis method to the design of fuzzy controllers. We show that it enables us to systematically derive the fuzzy rule base. Basically, we follow the classical Lyapunov synthesis method by constructing a Lyapunov function candidate V and then determining the conditions required to indeed make it a Lyapunov function of the closed-loop system. It turns out that, because we assume fuzzy knowledge about the plant to be controlled, the derived conditions can be stated as fuzzy if-then rules. We demonstrate our approach by applying it to the design of Mamdani-type stabilizing and tracking con- trollers for the inverted pendulum system, and by using it to design a Takagi-Sugeno-Kang-type controller for regulating the water level in a water tank. It should be noted that controllers for these plants were already synthesized by others (see, for example [1, 8]) using heuristics to derive the rule bases. In addition, properties such as stability and robustness have also been demonstrated but only through simulations. In contrast, we use the same examples in this paper to illustrate that our method can be used to derive the rule bases analytically (rather than heuristically). We also believe that our approach might lead to an analytical * Corresponding author. Tel.: +9723 6408764; fax: +972 3 6407095; e-mail: langholz@eng.tau.ac.il. 0165-0114/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00356-X