A unified parameterized formulation of reasoning in fuzzy modeling and control

Abstract

A unified parameterized formulation for the reasoning process in fuzzy modeling and control is developed in this paper. First, by selecting a suitable parameterized family of triangular functions and extending to the n-ary operation, the parameterized form of Mamdani's approximation and formal logical reasoning approaches is introduced. Next, by unifying the inference mechanism for both approaches, a unified parameterized reasoning function is developed. It is also proved that for crisp input variables, the two methods of inference from a set of rules, i.e., first-aggregate-then-infer (FATI) and first-infer-then-aggregate (FITA), generate identical fuzzy outputs. The proposed reasoning formulation introduces four reasoning parameters. Depending on these parameters, the reasoning operation varies continuously among the extreme cases in each step of the inference. In order to reduce the computational effort, a fast algorithm for computing the parameterized family of triangular functions is suggested. Further, a simplified parameterized reasoning formulation is also suggested in which the defuzzified output can be calculated directly from the individual consequent fuzzy sets. This simplified formulation is comparable with Sugeno's and Yager's heuristic simplified reasoning functions. Some examples demonstrate the validity of the results.