The effect of membership functions in fuzzy systems on the stability region

The effect of the number of membership functions of the controller's input variables on the stability boundary of a fuzzy control system is studied. A nonlinear SISO plant in controllable canonical form is considered. The simulation was carried out, first for two membership functions, with different simulation points. It was noticed that inside the indefinite stability region there are some stable states. This means that the estimated stability boundaries were not optimal. It is shown analytically that the number of membership functions has no effect on the stability boundary of the system while the Lyapunov matrix P has an effect. This was justified by the simulation results.