By Werner Peeters (auth.), Robert Lowen, Alain Verschoren (eds.)
This is a complete evaluate of the fundamentals of fuzzy regulate, which additionally brings jointly a few fresh examine ends up in tender computing, particularly fuzzy good judgment utilizing genetic algorithms and neural networks.
This publication bargains researchers not just an exceptional heritage but additionally a image of the present state-of-the-art during this field.
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Extra resources for Foundations of Generic Optimization: Volume 2: Applications of Fuzzy Control, Genetic Algorithms and Neural Networks
Sample text
We will distinguish between the cases where X has no structure at all, (X, ≤) is an ordered lattice, (X, T ) is a topological space and (X, T +, ·) is a topological vector space. m(x) Fig. 23 Fuzzy consequence after aggregation X 38 W. 1 Uniqueness Criterion (UC) For an arbitrary universe X, the defuzzification value should be unique, and therefore not dependent anymore of any stochastic process. Stated differently, the output of the defuzzification process should be unique for every choice of the fuzzy set µ ∈ F(X).
Therefore, in [85] and [86], we stated two new criteria that a defuzzifier may or may not satisfy. , αn } be an antecedent rule base that covers X. Furthermore, let µ ∈ F(X) be the fuzzy set resulting from aggregation and implication. A defuzzifier D will be called consistent if and only if for all x ∈ X, D(µ (x)) = x(= id(x)). One will rarely encounter a defuzzification operator that is consistent. Mostly, our goal is to find an upper bound for the supremum distance D◦µ − f ∞ ≤ l(n), where n is the number of defuzzifiers.
P. Filev in [160]. 9 Definition (SLIDE-defuzzification) For a universe X ⊆ R compact, for any α , β ∈ [0, 1], the SemiLineair Defuzzification DSLIDE (Figure 31) (see [160]) is a function that maps µ ∈ F(X) to (1 − β ) DSLIDE (µ , α , β ) = xµ (x)dx + (Γα (µ ))C (1 − β ) (Γα µ (x)dx + (µ ))C xµ (x)dx Γα ( µ ) µ (x)dx Γα ( µ ) Whereas the parameter α is again a measure of confidence in the system, the parameter β on the contrary is a parameter that denotes the degree of rejection of all points with membership µ (x) < α .