Modelling, stability analysis and computational aspects of nonlinear fuzzy PID controllers using Mamdani minimum inference

Abstract

This paper presents two new mathematical models of the simplest fuzzy PID controller which employ two fuzzy sets (negative and positive) on each of the three input variables (error, change in error and double change in error) and four fuzzy sets (−2, −1, +1, +2) on the output variable (incremental control). L-type, Γ-type and Π-type membership functions are considered in fuzzification process of input and output variables. Controller modelling is done via algebraic product AND operation, maximum/bounded sum OR operation, Mamdani minimum inference method, and centre of sums (CoS) defuzzification. The new models obtained in this manner turn out to be nonlinear, and their properties are studied. Since digital controllers are implemented on the digital processors, the computational and memory requirements of the fuzzy controllers and conventional (non-fuzzy) controller are compared. Stability analysis of closed loop systems containing the fuzzy controller models is done using the small gain theorem.