Investigating approximate reasoning and fuzzy control by concepts of functional analysis

Abstract

We investigate the input-output behaviour of the inference machine of a fuzzy controller, which is interpreted as a functional operator operating on the set of all fuzzy sets defined on a given universe. The concepts of fixed point, eigen function, eigen value and linearity are suitably translated into the fuzzy approach presented. Classical fixed point theorems (Birkhoff, Zorn, Banach) are applied in order to investigate the behaviour of inference machines. The concept of linearity is studied for Mamdani interpretations.