Fuzzy control as interpolation on the basis of equality relations

In the model of fuzzy control the rules are considered to describe a partial specification of a crisp control function (or relation). The domains of the control variables are assumed to be endowed with equality relations reflecting the idea that values that are close together are not well distinguished. This indistinguishably transforms the crisp points or sets used in the control rules into fuzzy sets. Therefore, fuzziness is assumed to stem from indistinguishability. The formal framework for the description of indistinguishability by equality relations is provided. Within the model of fuzzy control described the min-max inference method can be derived and the use of triangular and trapezoidal membership functions can be justified. Although it is assumed that the equality relations on the domains of the control variables are known, it is not necessary to specify them explicitly, since they can be directly computed from the fuzzy sets appearing in the rules if these fuzzy sets satisfy certain restrictions, which are generally accepted in fuzzy control. The details are given. A criterion for coherent defuzzification strategies is provided