On Optimal Defuzzification and Learning Algorithms: Theory and Applications

Abstract

In this paper, the issue of optimal defuzzification which is advocated in the Optimality Principle of Defuzzification (Song and Leland (1996)) is addressed. It was shown that defuzzification can be treated as a mapping from a high dimensional space to the real line. When system performance indices are considered, the defuzzification mapping which optimizes the performance indices for the given fuzzy sets is known as the optimal defuzzification mapping. Thus, finding this optimal defuzzification mapping becomes the essence of defuzzification. The problem with this idea, however, is that the space formed by all possible continuous defuzzification mappings is so large to search that the only recourse is an approximation to the optimal defuzzification mapping. With this, learning algorithms can be devised to find the optimal parameters of defuzzifiers with fixed structures. The proposed method is rigorously examined and compared with some well-known defuzzification methods. To overcome the resultant enormous computational load problem with this algorithm, the concept of defuzzification filter is additionally proposed. An application of the method to the power system stabilization problem is presented.