Analytical structure of a two-input two-output fuzzy controller and its relation to PI and multilevel relay controllers

Abstract

This study investigates the analytical structure of a two-input two output fuzzy controller which employs triangular-shaped input fuzzy sets, trapezoidal-shaped output fuzzy sets, linear control rules, probabilistic AND fuzzy logic, Łukasiewicz OR fuzzy logic, Mamdani's minimum inference method and the center of gravity defuzzification algorithm. We analytically prove that the structure of the fuzzy controller is the sum of global four-dimensional multilevel relays and local nonlinear proportional-integral (PI) controllers with variable gains continuously changing with process outputs. The global multilevel relays play a major role in determining control action of the fuzzy controller while the local PI controllers locally fine-tune the control action of the relays. Properties of the fuzzy controller structure are analytically and quantitatively investigated. Moreover, it is proved that, as the number of control rules approaches ∞, each global four-dimensional multilevel relay becomes the sum of two global linear PI controllers while the local PI controllers disappear.