Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

Abstract

This paper deals with the simplest Takagi-Sugeno fuzzy two-term controllers. The analytical structures of the simplest fuzzy controllers were developed using a modified rule base. The number of tuning parameters of the controllers is reduced by introducing a novel rule base consisting of two rules. These controllers are termed as 'the simplest' since minimal (two) number of fuzzy sets is used for fuzzification. Algebraic product (AP)/minimum (min) triangular norm, bounded sum (BS)/maximum (max) triangular co-norm, different universes of discourses (UoDs) of inputs, and centre of gravity (CoG) defuzzification method are chosen to derive the mathematical models of the fuzzy controllers. The simplest fuzzy controller with modified rule base is equivalent to a (nonlinear) variable gain/structure PI/PD controller. The BIBO stability of the closed loop control system is investigated using the small gain theorem. The gain of the controller either varies or remains constant in different regions of the input plane. The gain variations and the computational burden of the fuzzy controllers are also studied. Two examples of nonlinear dynamical systems are considered to validate the developed models of the fuzzy controllers.