Abstract-In zyxwvutsrqpo this paper, we consider a fundamental theoretical question, Why does fuzzy control have such good performance for a wide variety of practical problems?. We try to answer this fundamental question by proving that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, and for each fixed type of membership function belonging to a wide class of membership functions, the fuzzy logic control systems using these two and any method of d e f d c a t i o n are capable of approximating any real continuous function on a compact set to arbitrary accuracy. On the other hand, this result can be viewed zyxwvuts as an existence theorem of an optimal fuzzy logic control system for a wide variety of problems. 1 zyxw I. INTRODUCTION IEEE TRANSACTIONS ON SYSTEMS, MAN, zyxwvutsr AND CYBERNETICS, VOL 25, NO. zyxwvutsrq 4, APRIL 1995 Fuzzy Logic Controllers Are Universal Approximators J. L. Castro 629 URING the past several years, fuzzy logic control (FLC) D has been successfully applied to a wide variety of practi- cal problems. Notable applications of that FLC systems include the control of warm water [7], robot [6], heat exchange [15], traffic junction [16], cement kiln [9], automobile speed [14], automotive engineering [25], model car parking and turning [19], [20], turning [17], power system and nuclear reactor [3], etc . .. It points out that fuzzy control has been effectively used in the context of complex ill-defined processes, specially those which can be controlled by a skilled human operator without the knowledge of their underlying dynamics. In this sense, neu- ral and adaptive fuzzy systems has been compared to classical control methods by B. Kosko in [8]. There, it is remarked that they are model-free estimators, i.e., they estimate a function without requiring a mathematical description of how the output functionally depends on the input; they learn from samples. However, some people criticize fuzzy control because its effectiveness has not been proved. That is, the very fundamen- tal theoretical question “Why does a fuzzy rule-based system have such good performance for a wide variety of practical problems?’ remains unanswered. There exist some qualitative explanations, e.g., “fuzzy rules utilize linguistic information”, “fuzzy inference simulates human thinking procedure”, “fuzzy rule systems capture the approximate and inexact nature of the real world,” etc., but mathematical proofs have not been obtained. A first approach to answer this fundamental question in a quantitative way was presented by Wang [ 181. He proved that a particular class of FLC systems are universal approximators, Manuscript received August 14, 1992; revised August 1, 1993 and June J. L. Castro is with the Department of Computer Science and Artificial IEEE Log Number 9406630. 3, 1994. Intelligence, Universidad de Granada, 1807 1 Granada, Spain. 0018-9472/95$04.00 zyxwvutsr 0 1995 IEEE a b C Fig. 1. Triangular membership function. i.e., they are capable of approximating any real continuous function on a compact set to arbitrary accuracy. This class is that with: 1) Gaussian membership functions, 2) Product fuzzy conjunction, 3) Product fuzzy implication, 4) Center of area defuzzification. Other approaches are due to Buckley [4], [5]. He has proved that a modification of Sugeno type fuzzy controllers are universal approximators. The modifications are: 1) The consequent part of the rules are polynomial func- tions, not only linear functions as in Sugeno type con- trollers, 2) The defuzzification is zyxw 6 = zyxw CXip(ni,m;), where Xi is the matching of the input value with the antecedent part of the rule Ri, while in the Sugeno controller it is Xi = X;/CXj. Although both results are very important, many real fuzzy logic controllers do not belong to these classes. The main reasons are that other membership functions are used, other inference mechanisms are applied or other type of rules are used. The most common membership functions are the triangular (see Fig. 1) or trapezoidal (see Fig. 2) functions. With respect the fuzzy inference, a wide variety of fuzzy implications are used: R-implications [21] and Mamdani implication [ 121 are the most common. Finally, in many fuzzy controllers the consequent part is not a polynomial function but a fuzzy proposition or a linear or constant function.