Abstract
A serious problem limiting the applicability of standard fuzzy controllers is the rule-explosion problem; that is, the number of rules increases exponentially with the number of input variables to the fuzzy controller. A way to deal with this “curse of dimensionality” is to use the hierarchical fuzzy systems. A hierarchical fuzzy system consists of a number of hierarchically connected low-dimensional fuzzy systems. It can be shown that the number of rules in the hierarchical fuzzy system increases linearly with the number of input variables. In this paper, we prove that the hierarchical fuzzy systems are universal approximators; that is, they can approximate any nonlinear function on a compact set to arbitrary accuracy. Our proof is constructive, that is, we first construct a hierarchical fuzzy system in a step-by-step manner, then prove that the constructed fuzzy system satisfies an error bound, and finally show that the error bound can be made arbitrarily small.
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