Mathematical Modelling and Analysis of the Simplest Fuzzy Two-Input Two-Output Two-Term Controller of Takagi−Sugeno Type

Abstract

Mathematical modelling of the simplest Takagi−Sugeno fuzzy Two-Input Two-Output (TITO) Proportional-Integral/Proportional-Derivative (PI/PD) controller is presented in this paper. Mathematical model of fuzzy PI/PD controller is proposed using Algebraic Product (AP) t-norm, Bounded Sum (BS) t-co-norm and Center of Gravity (CoG) defuzzifier. The inputs are fuzzified by fuzzy sets having <i>L</i> and <inline-formula id="M2"> <math id="mathml_M2" display="inline" overflow="scroll"><mi>Γ</mi></math></inline-formula> type membership functions. The rule base consists of five rules with different linear models in the consequent parts. Both static and dynamic coupling is taken into account while deriving the models. The model of the fuzzy PI/PD controller reveals that it is a variable gain/structure controller. Also, each output of the TITO fuzzy controller is the sum of two nonlinear PI or PD controllers with variable gains. The properties and the gain variations of the controllers are investigated.