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Investigation of Fuzzy Rule Interpolation Techniques and the Universal Approximation Property of Fuzzy Controllers

By Domonkos Tikk


Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy If-then rules. Later on, the first approximating model can be tuned to obtain appropriate result. However, an essential problem of these algorithms is that their time complexity grows exponentially with the number of input variables. Fuzzy rule interpolation methods are one of the technique developed to reduce the complexity of fuzzy reasoning approaches. Purposes of this work are the following: -- Modification of the first published Kóczy--Hirota (KH) interpolation method to alleviate the so-called abnormal conclusion while maintaining its advantageous complexity behaviour. -- Investigation of the mathematical stability of the KH-method. -- Examination of the universal approximation property of certain fuzzy controllers. A modification of the original KH approach was proposed, whose main idea is the following. The consequent fuzzy sets are transformed by a proper coordinate transformation to such a space where the convexity of these consequents excludes abnormality of the conclusion. After the conclusion is calculated in this space, the inverse of the aforementioned transformation is used to obtain the corresponding conclusion in the original output space. The proposed method is closed for convex and normal fuzzy sets (Theorem 2.1). The new interpolation method was compared with the KH-approach one in several aspects. It was investigated how the proposed method differs form linear between characteristic points, and finally a comparison among the main interpolation techniques is given with respect to the relation of the observation's and conclusion's fuzziness. It was proven that the input-output function of the KH interpolation converges uniformly to the arbitrary approximated continuous function if the measurement points are uniformly distributed on the domain. A generalization of this theorem is also given for a wider class of interpolatory operators. It was pointed out that the stability of the well-known Shepard-interpolation (investigated extensively by approximation theorists) is can be derived from the one of the KH interpolation. The third main statement characterizes a set of certain type fuzzy controllers with bounded number of rules concerning the universal approximation property. As a generalization of Moser's result, it was shown that this property does not hold for the set of T-controllers (which includes Sugeno, Takagi--Sugeno, Takagi--Sugeno--Kang inference methods) if the number of rules is prerestricted, although that is a considerable practical limitation. It contradicts to those statements which state that fuzzy controllers are universal approximators, i.e., they lie dense in the space of continuous functions

Year: 1999
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