Fuzzy systems and approximation

Abstract

The basic motivation of using fuzzy rule-based systems especially for control purposes is to deduce simple and fast approximations of the unknown or too complicated models. Fuzzy rule-based systems have become very popuar because of their transparency and easiness of tuning and modification. Recently, some results concerning the explicit functions implemented by realistic fuzzy controllers presented the class of functions that could be implemented in this way. Some parallel results, on the other hand, attempted to prove that the main advantage of using fuzzy systems was the suitability for approximation with arbitrary accuracy in their universality. The explicit formulas and some very recent theoretical results made it clear however that fuzzy systems were not really good approximators, as realistic fuzzy controllers could generate only very rough approximations of given transference functions. In connection with approximation the question can be asked, whether there is an optimal fineness/roughness of a fuzzy rule-base that controls a certain action with roughness gives minimal time complexity. As an example, a target tracking problem was chosen (“Cat and Mouse”, or “Hawk and Sparrow” problem) where the antagonistic criteria of minimizing inference time by the given rule-base and minimizing action time (search for the target, with given uncertainty provided by the rule model) were examined. Under certain assumptions the solution of this optimization problem leads to nontrivial rule-base sizes. These results have also practical applicability since if a fine enough model of the system is known it is always possible to generate a rougher version of the same, by applying the model transformation technique offered by rule interpolation with α-levels.