Fuzzy controllers with conditionally firing rules

Mamdani controller was successfully used in many applications. One of its interpretations is that it uses a fuzzy relation as an approximation of the desirable input–output correspondence. We analyze mathematical properties of Mamdani controller and notice that it has lower computational complexity when compared to the residuum-based controller. However, we show that in standard situations, both these fuzzy controllers do not represent the rule base properly in the sense of finding a solution to the related system of fuzzy relational equations. First, we consider the premises and consequents as typical inputs and outputs and we want their correspondence to be kept. Second, we require that each normal input produces an output that bears nontrivial information. These two conditions appear to be almost contradictory for the previous controllers. We suggest a generalization of Mamdani controller which allows us to satisfy these requirements. Theory and experiments suggest that it performs better without any change of rule base and without a substantial increase of complexity.