SLIDE: A simple adaptive defuzzification method

Abstract

We introduce a parameterized family of defuznfica- tion operators called the Semi LInear DEfuzzification (SLIDE) method. This method is based upon a simple transformation of the fuzzy output set of the controller. We suggest an algorithm for the learning of the parameter from a data set. In an attempt to simplify the parameter learning we suggest a modified version of the SLIDE method which results in a simple learning algorithm. The development of the learning algorithm is based upon the use of the Kalman filter N fuzzy logic control systems (l), (2) the defuzzification I step involves the selection of one value as the output of the controller. More specifically, starting with a fuzzy subset (possibility distribution) F over the output space X of the controller, the defuzzification step uses this fuzzy subset to select a representative element x*. The two most often used methods of defuzzification found in the literature are the center of area (COA) and mean of maxima (MOM) (3)-(6) methods. We recall that the MOM method takes as its defuzzified value the mean of the elements that attain the maximum membership grade in F. The COA method takes as its defuzzified value d = Ci(ui * xi), where ui = F(xi)/Cj F(xj). In (7) and (8) we formulated a general defuzzification method via BAsic Defuzzification Distribution (BADD) transformations. The main idea of this BADD approach is to transform the possibility distribution F into a probability distribution P and then the defuzzified value is the expected value of the probability distribution. The process of transforming F into P was seen as a two-step process. The first step was to transform F into a new possibility distribution, E, and then to normalize E to obtain P. The step of obtaining E from F involved the use of a BADD transformation in which E(xi) = (F(z;))~, where 6 E (O,co). It was shown in (7) that the generalized BADD method implies the COA and MOM methods as special cases. In particular the COA method is obtained for a = 1 and the MOM is obtained for 6 = 00. When S = 0 we get a defuzzified value that is the unweighted mean of the output space, d = Xi xi. The BADD transformation has the advantage of being based on a single parameter, which allows us the potential of adaptively learning the best method of defuzzification for a given controller. While the family of all defuzzified values that can be obtained by the generalized BADD defuzzification method was parameterized, an undesirable property of the method was the nonlinear