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Lipschitz continuity of fuzzy controller

Bartova D., Kukal J., Majerova D.

Archives of Control Sciences

2006

Vol. 16, no. 1

19-35

Any traditional fuzzy controller performs a sequence of three processes: fuzzification, control algorithm and defuzzification. It is useful when the contoller exhibits continuous behavior with constrained output and sensitivity. After the normalization of controller inputs and outputs into the interval [0;1], we designed the fuzzy controller to be Lipschitz continuous, which implies the constrained sensitivity of the controller. Łukasiewicz algebra enriched by ŁAsqrt was used for the realization of the proposed fuzzy controller. The realization of fuzzification and control algotithm is trivial. The only problem is in the defuzzification. Neither Mamdani nor Larsen approaches are continuous in general. Both MOM and COG technoques generate discontinuous output behaviour. That is why we developed a new defuzzification method based on Łukasiewicz algebra. Thus, the proposed technique of defuzzification is based on propositional logic and it helps to realize a class of Lipschitz continuous fuzzy controllers. the controllers were realized in the Matlab environment.