By Kai Michels Dr., Frank Klawonn Professor Dr., Rudolf Kruse Professor Dr., Andreas Nürnberger Dr. (auth.)
ISBN-10: 3540317651
ISBN-13: 9783540317654
ISBN-10: 354031766X
ISBN-13: 9783540317661
The booklet presents a serious dialogue of fuzzy controllers from the point of view of classical keep watch over idea. distinct emphases are put on issues which are of significance for commercial functions, like (self-) tuning of fuzzy controllers, optimisation and balance research. The e-book is written as a textbook for graduate scholars in addition to a finished reference e-book approximately fuzzy keep watch over for researchers and alertness engineers. beginning with an in depth creation to fuzzy structures and keep watch over concept the reader is guided to updated study results.
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Additional resources for Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers
Example text
E. δ(x, y) ≤ 1 holds for all x, y ∈ X, so that similarity relations and pseudo-metrics (bounded by one) represent dual concepts. Later on, we other examples will provide a motivation to consider similarity relations with respect to other t-norms than the Lukasiewicz t-norm in order to characterize the vagueness or indistinguishability that is inherent in a fuzzy systems. 1 Fuzzy Sets and Extensional Hulls If we assume that a similarity relation characterises a certain indistinguishability, we should expect that elements that are (almost) indistinguishable should behave similar or have similar properties.
We generalize the concept of relations to fuzzy relations. Fuzzy relations are useful for representing and understanding fuzzy controllers that describe a vague connection between input and output values. Furthermore, we can establish an interpretation of fuzzy sets and membership degrees on the basis of special fuzzy relations called similarity relations. This interpretation plays a crucial role in the context of fuzzy controllers. 7. 1 Crisp Relations Before we introduce the definition of a fuzzy relation, we briefly review the fundamental concepts and mechanisms of crisp relations that are needed for understanding fuzzy relations.
20. 3 Cartesian Product and Cylindric Extension So far we have only extended mappings with one argument to fuzzy sets. In order to define operations like addition for fuzzy sets over real numbers, we need a concept how to apply a mapping f : X1 × . . × Xn → Y to a tuple (µ1 , . . , µn ) ∈ F(X1 ) × . . × F(Xn ) of fuzzy sets. Since we can interpret addition as a function in two arguments f : IR × IR → IR, (x1 , x2 ) → x1 + x2 , this concept will enable us to extend addition and other algebraic operations to fuzzy sets over real numbers.
Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers by Kai Michels Dr., Frank Klawonn Professor Dr., Rudolf Kruse Professor Dr., Andreas Nürnberger Dr. (auth.)
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