Formalization of Fuzzy Control in Possibility Theory via Rule Extraction

Possibility system has been recently recognized as a potential foundational theory for fuzzy set theory though the concept of possibility originated from the membership function of fuzzy sets. As an application of fuzzy set theory, fuzzy control has been widely used in engineering practices, where control laws are described by fuzzy if-then rules. This paper intends to formulate fuzzy control in the framework of possibility theory by regarding the fuzzification procedure of fuzzy control as extraction process of the fuzzy variable from crisp input, and the fuzzy if-then rules as extracted rules between fuzzy input/output. Typical procedures of fuzzy control, such as fuzzification, implication, aggregation, and defuzzification, are eventually formulated as a series of conditional possibilities operated by “max-product” operators. The reformulated fuzzy controller has a more general form, which encompasses the Mamdani fuzzy controller as a special case. Some fundamental concepts of possibility theory, such as randomness/fuzziness, possibility, and conditional possibility, are also discussed, which may be helpful for the correct understanding of possibility theory. Efforts have been made to bridge the two systems of possibility theory and fuzzy sets by the derivation of composition rule of fuzzy relations from conditional possibility. All results are derived for both normalized possibility and non-normalized possibility. This paper strengthened the role of possibility theory as a foundation for fuzzy sets, and as a complementary method to probability theory for handling information with fuzzy uncertainty.