How many rules are necessary to get a “good” fuzzy controller for a control problem?

This paper deal with a fundamental question in fuzzy control: How many rules are necessary to get a good fuzzy controller for my control problem? We shall give an upper bound answer to this question in the framework of fuzzy controllers with product conjunction, triangular or cubic membership functions and weighted sum of centroid defuzzification. In order to find this bound, under the hypothesis that the control function is continuous, and fixed the labels of the input variables of the controller we shall establish the existence of an optimal fuzzy controller for each control problem. Moreover, under the hypothesis that the control function is two or four times differentiable with continuity, we shall find an upper bound of the error of the optimal fuzzy controller