Takagi-Sugeno fuzzy PID controllers: mathematical models and stability analysis with multiple fuzzy sets

This paper deals with nonlinear Takagi-Sugeno (TS) fuzzy PID controllers with multiple fuzzy sets. Two models of fuzzy PID controllers are proposed using algebraic product (AP) triangular norm, bounded sum (BS)/maximum (Max) triangular co-norm and centre of gravity (CoG) defuzzifier. The inputs are fuzzified by three or more fuzzy sets with trapezoidal/triangular type membership functions. A new rule base is proposed consisting of four rules which reduce the number of tunable parameters. The models of the fuzzy PID controllers reveal that they are (nonlinear) variable gain/structure controllers, i.e., the gains are a function of input variables and the structure of the controller changes in the input space. The variations of gain and the properties of the controllers are investigated. The bounded-input bounded-output (BIBO) stability of the closed loop system with one of the proposed models in the loop is studied. The applicability of the controllers is demonstrated with the help of two examples.