Analytical structure characterization and stability analysis for a class of Mamdani fuzzy controllers

We study how the components of a general class of Mamdani fuzzy controllers dictate the controller's input-output relationship. The controllers can use input fuzzy sets of any types, arbitrary fuzzy rules, arbitrary inference methods, either Zadeh or the product fuzzy logic AND operator, singleton output fuzzy sets, and the centroid defuzzifier. We theoretically prove that regardless of the choices for the other components, if and only if Zadeh fuzzy AND operator and piecewise linear (e.g., trapezoidal or triangular) input fuzzy sets are used, the fuzzy controllers become a peculiar class of nonlinear controllers. This necessary and sufficient condition becomes a sufficient condition if the product AND operator is employed instead. Taking advantage of this new structure knowledge, we have established a necessary and sufficient local stability condition for the fuzzy control systems. It can be used not only for the stability determination, but also for practically designing a fuzzy control system that is at least stable at the equilibrium point even when model of the controlled system is mathematically unknown.