Fuzzy Control Revisited | Why Is It Working?

In the last decade fuzzy control (FC) has become a very popular control paradigm. Nowadays fuzzy controllers are very successful in a wide range of application domains and the market for such FC applications is still expanding. Nevertheless, the theoretical framework of FC is far beyond its practical success. While many convincing applications justify Mamdani's original FC design, from a theoretical point of view there is still a simple question to be answered: Why is FC working? And to answer this question is not only a task to satisfy theorists but also a matter of practice. That is, to give FC a sound theoretical background means to answer a very important related question, namely: When is FC working? In which contexts may we expect FC to work properly and how should we prepare the controller inputs and interpret the given results? In this paper we introduce a complete and sound framework to explain the most common FC mechanism, Mamdani's original approach. We present the idea of {distributions, which is a concept somewhat dual to possibility theory, and prove that regarding fuzzy sets as {distributions means to postulate FC in the sense of Mamdani. We point out that dealing with {distributions is based on dissonant bodies of evidence and therefore not the same as the use of possibility theory. Furthermore, with respect to incomplete or inconsistent knowledge both approaches show complementary characteristics, which is a very important observation from a practical point of view.