Foundations of Generic Optimization pp 433-456 | Cite as

# An Asymptotic Consistency Criterion for Optimizing Defuzzification in Fuzzy Control

## Abstract

In [6], we already pointed out that in a fuzzy control process, the choice of a good defuzzification method is quintessential. Throughout the literature, various defuzzification methods have been proposed, classified according to the properties they fulfil, such as continuity, scale invariance, core consistenty and so forth. In [6] we added a new criterion, by demanding that the defuzzification of the fuzzy image of a basic function, such as the identity, should still yield the identity, and we immediately found that this is almost never the case. However, the numerical deviation of this result can be established as a measure of fitness for the fuzzy controller in the particular problem. Moreover, given a parametric family of such defuzzification operators, such as D.P. Filev and R.R. Yager’s BADD-defuzzification ([3]), we were able to optimize the problem with respect to the arbitrary parameter. In this chapter, we will weaken out Consistency Criterion posed in [6] to a version that only needs to hold in an asymptotic case, namely with an infinite refinement of the width of the fuzzy antecedent rules. We will show that what ensues is a nice numerical description of the fitness of certain (families of) fuzzy defuzzification operators.

## Keywords

fuzzy control defuzzification consistency antecedent rule base## Preview

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