Conditions for general Mamdani fuzzy controllers to be nonlinear

Fuzzy controllers are best used as nonlinear controllers although they can be linear, piecewise linear or nonlinear. Currently, there exist no theoretical methods to determine whether a fuzzy controller is nonlinear. Because a fuzzy controller has many degrees of freedom in terms of its components selection (e.g., input fuzzy sets, output fuzzy sets, and fuzzy rules), linear controllers can be unconsciously and undesirably generated. In the present paper, we establish conditions under which nonlinearity of a general class of Mamdani fuzzy controllers can be determined. These fuzzy controllers can use input fuzzy sets of any types, arbitrary fuzzy rules, arbitrary singleton output fuzzy sets, arbitrary inference methods, Zadeh fuzzy logic AND operator, and the centroid defuzzifier. We prove that the fuzzy controllers using Zadeh AND operator are always nonlinear, regardless of choice of the other components. The general fuzzy controllers using the product AND operator are also always nonlinear except when all input fuzzy sets are triangular or trapezoidal and a couple of other conditions are satisfied. The exceptions lead to piecewise linear or linear controllers.