Analytical structures and stability analysis of three-dimensional fuzzy controllers

We have revealed the analytical structures and stability analysis of the three-dimensional fuzzy controllers involving trapezoidal input fuzzy sets, singleton output fuzzy sets, Zadeh fuzzy AND triangular norm, Zadeh fuzzy OR triangular co-norm, Mamdani inference method and centroid defuzzification algorithm. This class of fuzzy controllers is a combination of a nonlinear PID controller with dynamic proportional gain, dynamic integral gain and dynamic derivative gain plus a piecewise constant term. Based on the mathematical structures, the bounded-input bounded-output (BIBO) stability conditions for fuzzy control systems have been obtained by the well-known small gain theorem. A computer simulation is provided to illustrate that the new fuzzy controller is effective and superior to the conventional PID controller.