Compactness of Family of Fuzzy Sets in L2 Space with Application to Optimal Control
Takashi Mitsuishi, Yasunari Shidama · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2009
The optimization of nonlinear feedback fuzzy system using the product-sum-gravity method is described in this paper. The fuzzy control discussed here is the nonlinear feedback control in which the feedback laws are determined by IF-THEN type fuzzy production rules through product-sum-gravity method. To prove existence of optimal control, we applied compactness of a set of membership functions in L2 space and continuity of the approximate reasoning, and prepared some propositions concerning product-sum-gravity method. By considering fuzzy optimal control problems as problems of finding the minimum (maximum) value of the integral cost (benefit) function on an appropriate set of membership functions, the existence of fuzzy optimal control is shown.