On the Crisp-type Fuzzy Controller: Behaviour Analysis and Improvement

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Many fuzzy controller structures based on various inference methods have been presented [1–14]. Among them the most widely used methods in practice are: the Mamdani method proposed by Mamdani and his associates [2] who adopted the min-max compositional rule of inference based on an interpretation of a control rule as a conjunction of the antecedent and consequent; and the product-sum method proposed by Mizumoto [10–12] who suggested introducing the product and arithmetic mean aggregation operators to replace the logical AND (minimum) and OR (maximum) calculations in the min-max compositional rule of inference. The product-sum method greatly simplifies the algorithm of the fuzzy controller. In the algorithm of a fuzzy controller, the defuzzification calculation is also a complicated and time-consuming task. Using different defuzzification methods would not give a distinct improvement of performance. Hans and Christoph [22] have given a review on the defuzzyfication. Takagi and Sugeno proposed a crisp-type model in which the consequent parts of the fuzzy control rules are crisp functional representation or crisp real numbers in the simplified case instead of fuzzy sets [13, 14]. With this model of crisp real number output, the fuzzy set of the inference consequence will be a discrete fuzzy set with a finite number of points, this can greatly simplify the calculation of the defuzzification algorithm. The product-sum inference method and the crisp output model are often applied in a mixed manner. The mixed product-sum crisp model has a fine performance and the simplest algorithm that is very easy to be implemented in hardware system and converted into a fuzzy neural network model.