Theoretical analysis of a fuzzy-logic controller with unequally spaced triangular membership functions

Abstract The mathematical analysis concerning the explicit input/output relation of the fuzzy logic controller (FLC) is addressed in this article. Under the assumption of simple rule mapping and the use of arbitrarily located triangular fuzzy partitions as the membership functions, the inferred output of the FLC can be decomposed into two terms: the global multilevel relay and the local nonlinear compensator. The ultimate control behavior of this FLC and the equivalence to nonlinear proportional-integral (PI) controller, as the number of linguistic terms are approaching to infinity, are also investigated. The local stability criteria for PI-type FLCs are derived and the global stability properties for PD-type FLCs are established. This analysis is a generalization to previous related research works where the FLC is characterized by equally spaced triangular fuzzy partitions on the universal sets.