A unified parameterized formulation of reasoning in fuzzy modeling and control

A unified parameterized formulation for the reasoning process in fuzzy mode ling and control is developed in this paper. First, by selecting a suitable parameterized family of triangular functions and extending to the n-ary op eration, the parameterized form of Mamdani's approximation and formal logic al reasoning approaches is introduced. Next, by unifying the inference mech anism for both approaches, a unified parameterized reasoning function is de veloped. 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) a nd first-infer-then-aggregate (FITA), generate identical fuzzy outputs. The proposed reasoning formulation introduces four reasoning parameters. Depen ding on these parameters, the reasoning operation varies continuously among the extreme cases in each step of the inference. In order to reduce the co mputational 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 sim plified formulation is comparable with Sugeno's and Yager's heuristic simpl ified reasoning functions. Some examples demonstrate the validity of the re sults. (C) 1999 Elsevier Science B.V. All rights reserved.