Fuzzy controllers: synthesis and equivalences

It has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy. In particular, any given linear control can be achieved with a fuzzy controller for a given accuracy. The aim of this paper is to show how to automatically build this fuzzy controller. The proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller. The main idea is to equate the output of the fuzzy controller with the output of the PI controller at some particular input values, called modal values. The rule base and the distribution of the membership functions can thus be deduced. The analytic expression of the output of the generated fuzzy controller is then established. For Sugeno-type fuzzy controllers, precise equivalence is directly obtained. For Mamdani-type fuzzy controllers, the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values. The usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed