Global Stability Analysis of Second-Order Fuzzy Systems
The design of fuzzy controllers (FCs) is an open problem in which the use of heuristics is one of the main design methodologies. In this way, available design methods do not give fully satisfactory results because they lack an analytical background which would allow a thorough analysis. Only analytical study will assure a good performance in all situations (or at least, in a large class of them). To introduce such analytical results, a concrete class of fuzzy control systems will be studied in this article. This class of systems has been chosen such that is at the same time simple in structure and general in the richness of the results obtained. This class is the one shown in Fig. 1 where there is a linear plant, represented by its transfer function G(s), and a FC with the associated nonlinearity u = Φ(e, ẏ) = Φ(-y, ẏ). The main objective of the analysis developed is to study an elementary system with a well known mathematical structure and of which all the behaviours it can exhibit can be fully analyzed. To that end a linear model for the plant has been adopted. All the nonlinearities in the system stand in the fuzzy controller. As will be seen, the qualitative analysis of this system allows to display a full variety of behaviours. For the sake of simplicity, in this article it will be assumed that (1) G(s) is second-order; (2) Φ(-y, -ẏ) = -Φ( y, ẏ); and (3) out of the region of normal operation, the controller saturates. This unavoidable saturation will have important consequences in the following.
KeywordsEquilibrium Point Hopf Bifurcation Operating Point Fuzzy Controller Global Stability
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