Limit equation of fuzzy relation and its application

Mamdani proposed the first fuzzy controller, which was described by fuzzy linguistic rules and fuzzy relations. The stability of this type fuzzy controller is closely related to fuzzy relations. In this paper, a limit equation is first deduced from the stability conditions of generalized fuzzy control systems. Next, the structure of the solutions to the limit equation and the existence of minimal solution are discussed. Minimal solutions can be obtained by a path analysis based method. The minimal solutions of the limit equation produce elementary stable matrix forms, which are useful to analyze the stability of closed loop fuzzy systems. An example from the level control of water tank illustrates the proposed method.