Structure analysis of Mamdani fuzzy PID controllers with nonlinear input fuzzy sets

PID control is of particular importance because it presently controls the vast majority of industrial processes worldwide. In our previous papers and those by other researchers, explicit connections between PID control and various Mamdani fuzzy controllers, all of which use piecewise linear input fuzzy sets, have been established. In the present paper, we extend the study to cover fuzzy controllers that employ nonlinear fuzzy sets. The fuzzy controllers in this study consist of two fuzzy sets for each input variable, product or Zadeh AND operators, and the centroid defuzzifier. We have developed a new approach that establishes necessary and sufficient conditions on the input fuzzy sets in order for the fuzzy controllers to be nonlinear PI, PD, or PID controllers with variable gains. The treatment is general and covers two different and general types of nonlinear fuzzy sets.