Chaining of fuzzy IF-THEN rules in Mamdani-controllers

The problem of chaining of fuzzy IF-THEN rules has so far received a rather theoretic treatment in the literature on approximate reasoning. In particular, different types of composition operators, fuzzy implication operators, etc., have been identified such that the conclusion obtained via a chain of fuzzy rules coincides with the conclusion derived from the “abbreviated” version of the same chain. This “abbreviated” version is a single fuzzy rule which the rule-antecedent is the rule-antecedent of the first rule in the chain, and its rule-consequent is the rule-consequent of the last rule in the chain. However, in the case of more than one chain of rules and when the fuzzy sets defining the meaning of the rule-antecedents and rule-consequents from different chains overlap, then the above theoretical results do not hold in general. In the present paper we identify two major problems with the chaining of fuzzy rules in the case of more than one chain and overlapping rule-antecedents and rule-consequents that belong to different chains