Fuzzy Entropy And Its Applications To The Optimization Of Fuzzy Systems

It is a very urgent and vital task to build a set of new, effective, reliable and more solidly theory-based approaches to analyze, design and optimize a fuzzy system in intelligent control and intelligent information processing. So far, existing related research results, however, are far from perfectness in that no adequate attention is paid in better studying and effectively using the information implied in a fuzzy system itself. Based on the order relation, this dissertation systematically studies the information representation theories of a fuzzy system from three different levels: fuzzy set, fuzzy partition and fuzzy system, and proposes the maximum (fuzzy partition or fuzzy system) fuzzy entropy method for the learning, training and optimization of a fuzzy system. So a theoretical foundation and new directions for the research of intelligent control are provided. The main contents of this dissertation involve the following nine aspects.1. A new definition of fuzzy entropy of a fuzzy set is proposed by converting the order relation ≤F on [0, 1]~n to the order relation ≤ on [0, 1/2]~n. The proposed method can simplify the discussion of some properties of fuzzy entropy, especially, simplify the proof of the uniqueness theorem of fuzzy entropy compared with existing results.2. The influence on fuzzy entropy imposed by the change of shape and support interval of membership function is discussed detailedly, which paves the way of more effectively measuring the uncertainty information of fuzzy partitionand fuzzy system. Let X=[a, b], h(u) =4u(1-u) , e(A)= ∫_xh(A(x))dxand HeightA denotes the height of fuzzy set A, the following results are obtained:(1) The ratio of fuzzy entropies of any two triangular fuzzy numbers amounts to the ratio of support intervals' length of the two triangular fuzzy numbers;(2) If A, B are any two triangular fuzzy sets with same support interval length, then:① if0 ≤ HeightA < HeightB ≤ 3 / 4, then e(A) < e{B); ② 3/4 ≤ HeightA < HeightB ≤1, then e(A)>e(B);...