Necessary conditions on minimal system configuration for general MISO Mamdani fuzzy systems as universal approximators

Recent studies have shown that bath Mamdani-type and Takagi-Sugeno-type fuz zy systems are universal approximators in that they can uniformly approxima te continuous functions defined on compact domains with arbitrarily high ap proximation accuracy. In this paper, we investigate necessary conditions fo r general multiple-input single-output (MISO) Mamdani fuzzy systems as univ ersal approximators with as minimal system configuration as possible. The g eneral MISO fuzzy systems employ almost arbitrary continuous input fuzzy se ts, arbitrary singleton output fuzzy sets, arbitrary fuzzy rules, product f uzzy logic AND, and the generalized defuzzifier containing the popular cent roid defuzzifier as a special case. Our necessary conditions are developed under the practically sensible assumption that only a finite set of extrema of the multivariate continuous function to be approximated is available. W e have first revealed a decomposition property of the general fuzzy systems : A tau -input fuzzy system can always be decomposed to the sum of tau simp ler fuzzy systems where the first system has only one input variable, the s econd one two input variables, and the last one tau input variables. Utiliz ing this property, we have derived some necessary conditions for the fuzzy systems to be universal approximators with minimal system configuration. Th e conditions expose the strength as well as limitation of the fuzzy approxi mation: 1) only a small number of fuzzy rules may be needed to uniformly ap proximate multivariate continuous functions that have a complicated formula tion but a relatively small number of extrema; and 2) the number of fuzzy r ules must be large in order to approximate highly oscillatory continuous fu nctions, A numerical example is given to demonstrate our new results,