We show in this paper that the set of functions, consisting of polytopic or TS models constructed from finite number of components, is nowhere dense in the approximation model space, if that is defined as a subset of continuous functions. This topological notion means that the given set of functions lies "almost discretely" in the space of approximated functions. As a consequence, by means of the mentioned models we cannot approximate in general continuous functions arbitrarily well, if the number of components are restricted. Thus, only functions satisfying certain conditions can be approximated by such models, or alternatively, we need unbounded number of components. The possible solutions are outlined in the paper.

Abstract: Following the wide spread usage of Fuzzy Systems, Rule Reduction has emerged as one of the most important areas of research in the field of Fuzzy Control. Many rule reduction methods have been proposed in the literature and can be broadly classified into Lossless or Lossy with respect to the inference, based on whether the outputs of the original and the reduced rule bases are identical or not. In a typical Multi-Input-Single-Output fuzzy system the number of rules far exceeds the number of fuzzy sets defined on the output domain. This suggests that the rule base can be partitioned into sets of rules, each set being mapped to a single consequent fuzzy set. In this paper, we investigate the conditions on the inference operators employed in a fuzzy system that enable “lossless ” merging of rules with identical consequents. After briefly surveying the many techniques that have been proposed towards reducing the number of rules, we propose a general framework for Inference in Fuzzy Systems and then propose some sufficiency conditions on this general framework that give us a class of Fuzzy Systems that allow lossless rule reduction of the type mentioned above. We then explore these conditions in the setting of Fuzzy Logic. We find that R- and S-implications play a very critical role. We give examples from the above class of Fuzzy Systems. In this study we apply the above technique only on rules whose antecedents and consequents are fuzzy sets.

The human liver--bile system is a complex, non-linear system. A soft-computing based universal approximator, the singleton based Product-Sum-Gravity inference is applied as a modeling technique. As the available information about the system is very limited, an additional technique, the partition refinement is used in the training process. Higher Order SVD was chosen for complexity reduction. These modeling techniques showed satisfactory results in the real life model.

The tensor product (TP) based models have been applied widely in approximation theory, and approximation techniques. Recently, a controller design framework working on dynamic systems has also been established based on TP model transformation combined with Linear Matrix Inequalities (LMI) within Parallel Distributed Compensation (PDC) framework. The effectiveness of the control design framework strongly depends on the approximation property of the TP model used. Therefore, the primary aim of this paper is to investigate the approximation capabilities of dynamic TP model. It is shown that the set of functions that can be approximated arbitrarily well by TP forms with bounded number of components lies no-where dense, i.e. "almost discretely" in the set of continuous functions. Consequently, this paper points out that in a class of control problems this drawback necessitates the application of trade-off techniques between accuracy and complexity of TP form. Such requirements are very difficult to consider in the analytical framework, but TP model transformation offers an easy way to deal with them.

In this paper we present a novel approach to datadriven fuzzy modeling which aims to create highly accurate but also easily comprehensible models. This goal is obtained by defining a flexible but expressive language automatically from the data. This language is then used to inductively learn fuzzy regression trees from the data. Finally, we present a detailed comparison study on the performance of the proposed method and an outlook to future developments. Keywords: Learning.