Size reduction by interpolation in fuzzy rule bases

Abstract

Fuzzy control is at present still the most important area of real applications for fuzzy theory. It is a generalized form of expert control using fuzzy sets in the definition of vague/linguistic predicates, modeling a system by If...then rules. In the classical approaches it is necessary that observations on the actual state of the system partly match (fire) one or several rules in the model (fired rules), and the conclusion is calculated by the evaluation of the degrees of matching and the fired rules. Interpolation helps reduce the complexity as it allows rule bases with gaps. Various interpolation approaches are shown. It is proposed that dense rule bases should be reduced so that only the minimal necessary number of rules remain still containing the essential information in the original base, and all other rules are replaced by the interpolation algorithm that however can recover them with a certain accuracy prescribed before reduction. The interpolation method used for demonstration is the Lagrange method supplying the best fitting minimal degree polynomial. The paper concentrates on the reduction technique that is rather independent from the style of the interpolation model, but cannot be given in the form of a tractable algorithm. An example is shown to illustrate possible results and difficulties with the method.

Improvement of Takagi‐Sugeno fuzzy model for the estimation of nonlinear functions

“…It was clearly shown, however, that the number of fuzzy rules increases as the approximation error tends to zero [15]. This expon en tial growth cannot be eliminated, so it becomes difficult to make use of the universal approximation property of TS fuzzy modelling for practical purposes.…”

“…In [15] the authors show that rule interpolation helps in reducing the identification complexity as it allows rule bases with gaps. It is suggested that only the minimal necessary number of rules remain which contain the essential information, and all other rules are replaced by the interpolation algorithm.…”

“…The approach proposed in [15] has some drawbacks such as subnormal conclusión for certain configuration of the involved fuzzy sets and does not always lead to interpretable fuzzy membership functions. Nevertheless, it is characterized by having the lowest computational complexity and has good approxiamtion aspects comparing with other fuzzy rule interpolation techniques.…”

“…In [31], a new technique for fuzzy rule interpolation was investigated. The aim is to combine the advantageous computational behavior of [15] and at the same time reduce the disadvantage of subnormality.…”

A General Framework for Ordering Fuzzy Sets

“…Similar questions arise in linguistic approximation [7,19] which may be considered as a kind of inverse procedure -finding a linguistic label for a given fuzzy set. A third application scenario is rule interpolation [15,16] which is concerned with obtaining conclusions for observations which are not covered by any antecedent in a fuzzy rule base. Orderings of fuzzy sets are able to provide criteria for determining between which rules the interpolation should take place [15].…”