Analytical Structures of Typical Takagi - Sugeno Fuzzy PID Gain Scheduling Controllers

The analytical structure of a class of typical Takagi -Sugcno (TS) fuzzy controllers is revealed in this paper. The TS fuzzy controllers consist of three or more trapezoidal input fuzzy sets, Zadeh fuzzy logic AND operator, fuzzy rules with linear consequent, and the centriod de-fuzzifier. The TS fuzzy controllers are proved to be accurately nonlinear PID controllers with gains continuously changing with process output. The analytical expressions of the variable gains of the TS fuzzy controllers are derived and their mathematical characteristics including the bounds and geometrical shape of the gain variation are analyzed. The resulting explicit structures show that the TS fuzzy controllers are inherently nonlinear PID gain scheduling controllers with variable gains in different regions of input space.