Fuzzy control theory: A nonlinear case

Abstract

Sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs; fuzzy control rules; type of fuzzy logic used for evaluating the fuzzy control rules; and the defuzzification algorithm used for the controller output. We analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms. We prove theoretically that such a fuzzy controller, the smallest possible, with two inputs (error and rate change of error) and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional-integral (PI) controller with proportional-gain and integral-gain changing with error and rate change of error about a setpoint. Furthermore, this fuzzy controller is precisely equivalent to a conventional linear PI controller if a linear defuzzification algorithm is employed. Computer simulation showed that the performance of the fuzzy controller was almost the same as that of the PI controller when first-order and second-order linear processes were used. Furthermore, the fuzzy controller was significantly better when a first-order with a time delay model was used. More importantly, the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled, but the PI controller was unstable.