Towards a logic for a fuzzy logic controller

Abstract

Fuzzy control is based on fuzzy logic [3], and the core of a fuzzy logic controller (FLC) is a set of fuzzy conditional statements related by the dual notions of fuzzy implication and the so-called compositional rule of inference[2]. However, despite of the use of the logic-related concepts of fuzzy conditional statements, fuzzy implication, and the compositional rule of inference, all of these are used in a mixed declarative/procedural manner. That is, there is no clear distinction between the part of a FLC (the fuzzy conditional statements) and the way this declarative knowledge is used in the inference component of a FLC (the compositional rule of inference). To be more specific, the compositionM rule of inference is defined directly on the extension (meaning) of a fuzzy conditional statement, rather than on its symbolic counterpart. Furthermore, the extension of a fuzzy conditional statement is constructed by applying a fuzzy implication operator on the extensions of its antecedent and consequent. Thus, a fuzzy implication is not any longer defined in terms of a function of the truth-values of the symbolic counterparts of the antecedent and the consequent. In this context, the declarative part of a FLC is nothing else but a collection of propositions (fuzzy conditional statements) represented via their extensions and thus yielding a set of equations. Then the inference amounts to solving such a system of equations. The clear consequence of all this is the complete lack of any syntactic representation of the declarative part of a FLC and hence, no underlying semantic characterization.