A New Approach for Representing Control Surface by Fuzzy Rule Bases

We are interested in the variety of those fuzzy rule bases with crisp consequences which induce the same control surface when applying the so-called Sugeno inference. The goal is to find out fuzzy rule bases satisfying certain features we expect of a fuzzy system that is linguistically interpretable. For this purpose we discuss among others the followings criteria: 1.the maximal degree of inclusion of a fuzzy subset in another one of a fuzzy partition should be low; 2.the fuzzy subsets should be interpretable as fuzzy points with respect to an indistinguishability relation; 3.the number of fuzzy rules should be small. In order to cope with these criteria we start to discuss various concepts of a fuzzy partition proposed in literature. It is illustrated by examples that these concepts do not necessarily meet criterion 1. For that reason we introduce the concept of T-redundancy (with respect to a t-norm T), which turns out to be also related to the concept of a fuzzy point. It is shown that minimizing the T_L-redundancy (with respect to the Lukasiewicz t-norm T_L) leads to fuzzy partitions which meet criterion 2. (in a many-valued logical sense). Concerning criterion 3. we firstly demonstrate, that several universal approximation results for fuzzy controllers presented in literature do not longer hold, if the number of rules is bounded. In the sequel, we above all investigate fuzzy partitions consisting of a number of fuzzy subsets (minimal fuzzy partitions). I turns out, that there is a one-to-one correspondence between minimal fuzzy partitions and simplexes containing a bounded convex set G, which is determined by the Sugeno controller's surface.