Modeling, stability analysis and computational aspects of nonlinear fuzzy PID controllers

This paper presents mathematical models of the simplest fuzzy PID controller of Mamdani type. This controller is called the "simplest" as it employs minimum number of fuzzy sets (two on each input variable and four on output variable) while satisfying the control rule base that contains four rules relating all six input fuzzy sets to all four output fuzzy sets. L - type, Gamma - type and Pi - type membership functions are considered in fuzzification process of input and output variables. Controller modeling is done via algebraic product AND operator-maximum OR operator-Larsen product inference method-Centre of Sums (CoS) defuzzification process combination. The new model obtained in this manner turns out to be nonlinear, and its properties are studied. Since digital controllers are implemented on the digital processors, the computational and memory requirements of the fuzzy controller and conventional (nonfuzzy) controller are compared. Stability analysis of closed loop systems containing the fuzzy controller models is done using the small gain theorem.