How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules Aj1(x1) & ... & Ajn(xn) → Bj(u) formulated in terms of a natural language, into an actual control strategy u(x⃗). Implication of uncertain statements is much more difficult to understand than “and”, “or”, and “not”. So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and”, “not”, and “or”. The first such translation was proposed by Mamdani in his pioneer paper on fuzzy control. According to this paper, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A11(x1) & ... & A1n(xn) & B1(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x⃗, we know what u(x⃗) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x⃗ may be different from the desired value u(x⃗). To overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x⃗). In this paper, we show that a natural modification of Mamdani’s approach can solve this problem without adding any ad hoc priorities.