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.

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This paper deals with the approximation behaviour of soft computing techniques. First, we give a survey of the results of universal approximation theorems achieved so far in various soft computing areas, mainly in fuzzy control and neural networks. We point out that these techniques have common approximation behaviour in the sense that an arbitrary function of a certain set of functions (usually the set of continuous function, C) can be approximated with arbitrary accuracy e on a compact domain. The drawback of these results is that one needs unbounded numbers of ''building blocks'' (i.e. fuzzy sets or hidden neurons) to achieve the prescribed e accuracy. If the number of building blocks is restricted, it is proved for some fuzzy systems that the universal approximation property is lost, moreover, the set of controllers with bounded number of rules is nowhere dense in the set of continuous functions. Therefore it is reasonable to make a trade-off between accuracy and the number of the building blocks, by determining the functional relationship between them. We survey this topic by showing the results achieved so far, and its inherent limitations. We point out that approximation rates, or constructive proofs can only be given if some characteristic of smoothness is known about the approximated function.

In our previous papers, fuzzy model identification methods were discussed. The bacterial evolutionary algorithm for extracting fuzzy rule base from a training set was presented. The LevenbergMarquardt method was also proposed for determining membership functions in fuzzy systems. The combination of the evolutionary and the gradient-based learning techniques is usually called memetic algorithm. In this paper, a new kind of memetic algorithm, the bacterial memetic algorithm, is introduced for fuzzy rule extraction. The paper presents how the bacterial evolutionary algorithm can be improved with the Levenberg-Marquardt technique. C 2009 Wiley Periodicals, Inc.

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Hierarchical Fuzzy Signatures are generalizations of the Vector Valued Fuzzy Set concept introduced in the 1970s. A crucial question in the Fuzzy Signature context is what kinds of aggregations are applicable for combining data with partly different substructures. Our earlier work introduced the Weighted Relevance Aggregation method to enhance the accuracy of the final results of calculations based on Hierarchical Fuzzy Signature Structures. In this paper, we further generalise the weights and the aggregation into a new operator called Weighted Relevance Aggregation Operator (WRAO). WRAO enhances the adaptability of the fuzzy signature model to different applications and simplifies the learning of fuzzy signature models from data. We also show the methodology of learning these aggregation operators from data.

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