Generalized Fuzzy Lyapunov Stability Analysis of Discrete Type II/III TSK Systems Assem Sonbol M. Sami Fadali Electrical Engineering/260 Electrical Engineering/260 University of Nevada University of Nevada Reno, NV 89557 Reno, NV 89557 sonbol@unr.nevada.edu fadali@ieee.org Abstract--- We propose a new approach for the stability analysis of discrete Sugeno Types II and III fuzzy systems. The new approach uses arguments similar to those of traditional Lyapunov stability theory with positive and negative definite functions replaced by fuzzy positive definite and fuzzy negative definite functions, respectively. We introduce the concept of the equivalent fuzzy system for a cascade of two fuzzy systems. We use the cascade of a system and a fuzzy Lyapunov function candidate to derive new conditions for stability and asymptotic stability for Type II and Type III fuzzy systems. We apply our results to a numerical example. I. Introduction The stability analysis of fuzzy systems has been the subject of extensive research (see the review paper [5]. However, due to the nonlinear structure of fuzzy systems the development of a general approach is highly unlikely. For a systematic stability analysis, we start with a classification of fuzzy systems. Sugeno [5] classified fuzzy systems into three types. Type I, which was first introduced by Mamdani, uses fuzzy rules of the form [ ] [ ] n j N i A A x x H is y THEN is IF R j j T i n i i i T n i i i i i i i i n n n n n n , , 1 , , , 1 , , : 1 1 1 1 1 1 1 ... 1 ... ... ... ... K K L L = = = = A x A x (1) where j i j A and n i i H ... 1 are fuzzy sets. If we replace the consequents in (1) with [ ] n j N i h h j j T i i n i i i i n n n , , 1 , , , 1 , ... ... 1 ... 1 1 1 K K L = = = h (2) where n i i ... 1 h is a vector of singletons, we obtain Type II Takagi-Sugeno-Kang (TSK) fuzzy systems. Type II is a special case of Type III systems whose consequents are [ ] n j N i f f j j T i i n i i i i n n n , , 1 , , , 1 , ) ( ) ( ) ( ... ... 1 ... 1 1 1 K K L = = = x x x f (3) where ) ( ... 1 x f n i i are functions of x i , i = 1, …, n. While it is possible to transform one type of fuzzy system to another [11], these transformations do not allow the extension of sufficient stability tests for one type to others. In general, the stability analyses of Type II and Type III systems are significantly different [5]. Recently, the stability analysis of Type III systems has attracted considerable interest in the literature [1]-[10]. Most of these results require the existence of a common quadratic Lyapunov function [1]-[5]. Unfortunately, conditions for the existence of such functions are restrictive and difficult to establish [12]. For example, the search for a common Lyapunov function can be posed as a convex optimization problem in terms of linear matrix inequalities (LMIs) [9]. However, the LMI conditions for quadratic stability for fuzzy systems are often conservative. Moreover, the convex optimization problem often involves a large number of LMIs and a dramatically increasing computational load with the number of inputs [10]. Several authors were able to analyze the stability of fuzzy systems without a common Lyapunov function [6], [9], [10]. Lo and Chen [6] used Kharitonov theory to derive a sufficient condition for fuzzy controller stability. Unfortunately, Johansen and Slupphaug [7] showed by a counterexample that the conditions proposed in [6] are not sufficient. Dvorakova and Husek [8] also analyzed the results in [6] and showed that the computational procedure presented is not valid for fuzzy systems where the number of rules is greater than three. Johansson and Rantzer [9] presented a novel approach for stability analysis of fuzzy systems. The analysis was based on piecewise-continuous quadratic Lyapunov functions. The approach resulted in stability conditions that can be verified via convex optimization over LMIs. Feng and Harris [10] also used a piecewise-continuous quadratic Lyapunov functions. Their approach exploited the properties of the input membership functions to reduce the number of candidate Lyapunov functions and the associated LMIs. To date, there has been no theoretical study of the stability of Type I [5] and only two papers on Type II systems [5], [15]. Sugeno [5] gave stability conditions for both discrete-time and continuous time Type II systems. In [15], we introduced the concept of fuzzy positive definite and fuzzy negative definite systems. Then, we used them to derive Lyapunov like conditions for the stability analysis of discrete Type II systems. Here, we generalize our earlier results to allow more flexibility in the selection of the Lypunov function ( ) ) (k V x . Whereas our earlier results restricted the shape of the ( ) ) (k V x contour, our new results allow any piecewise linear closed contour. In addition, our earlier results are restricted to Type II systems while our new results are applicable to both Type II and Type III. We provide an example where no common Lyapunov function exists but where our method establishes the stability of the fuzzy system. The paper is organized as follows. Section II introduces basic definitions and concepts. In Section III, Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC WeA14.2 453