What Procedure to Choose While Designing a Fuzzy Control? towards Mathematical Foundations of Fuzzy Control

Fuzzy control has been successfully applied in industrial systems. However, there is some caution in using it. The reason is that it is based on quite reasonable ideas, but each of these ideas can be implemented in several diierent ways, and depending on which of the implementations we choose we get diierent results. Some implementations lead to a high quality control, some of them not. And since are no theoretical methods for choosing the implementation, the basic way to choose it now is experimental. But if we choose a method that is good for several examples, there is no guarantee that it will work ne in all of them. Hence the caution. We are going to provide a theoretical basis for choosing the fuzzy control procedures. In order to choose a procedure that transforms a fuzzy knowledge into a control we need, rst, to choose a membership function for each of the fuzzy terms that the experts use, second, to choose operations of uncertainty values that corresponds to \and" and \or", and third, when we obtain a membership function for control, we must defuzzify it, that is, somehow generate a value of the control u that will be actually used. In the paper we describe a general approach that will help to make all these choices: namely, we prove that under reasonable assumptions membership functions should be linear or fractionally linear, defuzziication must be described by a centroid rule and describe all possible \and\ and \or\ operations. Thus we give a theoretical explanations of the existing semi-heuristic choices and formulate the basis for the further research on optimal fuzzy control. 1. BRIEF INTRODUCTION Why do we need mathematical foundations of fuzzy control? In order to design a fuzzy control, we must choose fuzzy variables, choose combination rules for uncertainty values and choose a defuzziication procedure. The eeciency of the resulting fuzzy control essentially depends on these choices. For example (see KFLL91] for details) diierent choices of combination rules can lead to relaxation times that diier twofold (and if we go for stability this is an essential increase). These choices are now made mainly on a semi-empirical basis: if a resulting system works, that's ne. This approach is acceptable for camcorders or dishwashers: even if something goes wrong with a picture for a moment or two, it is not a problem. However, this level of reliability is absolutely unacceptable for such …