Modeling and analysis of the simplest fuzzy PID controller of Takagi-Sugeno type with modified rule base

This paper deals with the simplest fuzzy PID controllers of the Takagi-Sugeno (TS) type. The term simplest is coined since minimal (two) number of fuzzy sets is used for fuzzification in these controllers. The mathematical models of these controllers are found using a modified rule base. The rule base consists of two rules which reduce the number of tuning parameters. Two classes of the simplest fuzzy PID controller are defined using algebraic product triangular norm and bounded sum/maximum triangular co-norm. It is shown that the fuzzy PID controller with modified TS rule base is equivalent to a nonlinear variable gain/ structure controller. The BIBO stability of the closed-loop control system is studied using the small gain theorem. The computational aspects of the controllers are investigated. The applicability of the simplest fuzzy PID controllers is demonstrated with the help of examples.