Optimal tuning of scaling factors and membership functions for mamdani type PID fuzzy controllers

This study addresses the problem of tuning Mamdani-type fuzzy Proportional-Integral-Derivative controllers by solving a constrained nonlinear optimisation problem. Two different schemes are discussed. One considers merely the scaling factors optimisation, while assuming the membership functions width unchanged. In second methodology both scaling factors and membership functions width adjusted. The cost function under minimisation is chosen as the closed loop performance index, with the system dynamics described by a nonlinear model. In addition, a set of constraints to the optimisation problem are taken into account, namely bounds on scaling factors, control actions and also on the system outputs. For the second approach, extra constraints on membership functions width are also included in problem formulation. Experimental results carried out on a Multi-Input Multi-Output benchmark system shows the out-performance of considering the optimising of both scaling factors and membership function widths.