Alpha-cut representation used for defuzzification in rule-based systems

Abstract

Alpha-cut representation of fuzzy sets has been used as a basis for fuzzy numbers ranking in some applications but rarely used for defuzzification of rule-based systems or fuzzy controllers. Moreover, such alpha-cut defuzzification (called ACD here) is not yet formally linked to the membership function (MF) or to the common MF-based defuzzification methods, namely the centroid. The ACD can be considered as a generalisation of the similar algorithms in fuzzy numbers to any fuzzy set. A close-form formula for ACD is developed that involves both MF and its derivative, which shows that ACD reflects both static and dynamic aspects of a fuzzy set. Moreover, formal links between ACD and some MF-based defuzzification methods are shown. Through two groups of experiments, the utility of the new method is compared with centroid defuzzification. Particularly, we examined how the ACD significantly outperforms the centroid for noisy time-series prediction. Finally, the computation complexity of ACD is shown to be about the same as the centroid method, for convex fuzzy sets. Our tests suggest that ACD can be considered as a viable alternative defuzzification method for fuzzy system designers.