ON FUZZY MODELING OF NONLINEAR DYNAMICAL SYSTEMS

Modeling nonlinear dynamical systems using fuzzy systems has increasingly been recognized as a distinct and important system identification paradigm. It refers to a process whereby a dynamical system is modeled not in the form of conventional differential and difference equations, but in the form of a set of fuzzy rules and corresponding membership functions. In this thesis, we approach the fuzzy modeling problem in a rigorous, yet application oriented fashion. More specifically, we have two objectives to achieve in the present thesis. One is to establish a proper framework for formulating the fuzzy modeling problem in a way that allows the essential aspects of the problem to be examined with ease. The other is to develop efficient schemes that can build fuzzy models of various types for engineering applications. With a proper framework based on the key notions of fuzzy quantization and function approximation, three types of fuzzy modeling schemes are developed in this thesis. Fuzzy modeling with fixed model structure (uniform fuzzy quantization) is the simplest to start. The simplicity makes it easier and clearer to rigorously analyze its modeling accuracy in terms of modeling error estimates and convergence rate. It is also very illustrative to use this simple fuzzy model structure to design efficient algorithms for fuzzy rule updating. However, this scheme requires some in-depth a priori knowledge of the unknown system and there is also a potential problem of over-fitting. An adaptive fuzzy modeling scheme is then developed to overcome the difficulties. It is based on the idea of structure-adaptation, or adaptive fuzzy quantization which intends to adapt both the fuzzy rules and the fuzzy quantization at the same time. A theoretical analysis is carried out to show its better performance in modeling a general class of nonlinear smooth systems with local singularities, e.g. nonlinear systems in Besov spaces. However, the structure adaptation usually takes longer time to converge. We thus present yet another scheme to solve the fuzzy structure determination problem with faster convergence. This scheme has to do with the notion of fuzzy modeling at different scales and its essential parts are multi-scale fuzzy quatization and residual based coarse-to-fine modeling technique. While the emphasis of the thesis is on the theoretical development, extensive simulation analysis is also used to compare these three fuzzy modeling schemes along with detailed discussions. We also identify the complementarity and equivalence relationships between fuzzy systems and wavelets in this thesis. The complementarity relationship states that wavelets are complement to fuzzy systems under some minor restrictions. Thus we can take fuzzy systems as the initial approximation of the unknown function awl take wavelets as the finer modification. With the proper definition of multi-scale fuzzy systems, the equivalence relationship indicates that they are functionally equivalent. These relationships further imply that advances in each field can be applied to the other directly.

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• YU YI