The simplest fuzzy PID controllers: mathematical models and stability analysis

This paper reveals mathematical models for the
simplest fuzzy PID controllers which employ two fuzzy sets
for each of the three input variables and four fuzzy sets for the output variable. Mathematical models are derived via
left and right trapezoidal membership functions for each in-
put, singleton or triangular membership functions for output, algebraic product triangular norm, different combinations of triangular co-norms and inference methods, and center of sums (COS) defuzzification method. Properties of these structures are studied to examine their suitability for control application. For the structure which is suitable for control,bounded-input bounded-output (BIBO) stability proof is presented. An approach to design fuzzy PID controllers is given.Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controllers.