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Modeling of control functions by fuzzy controllers. (English) Zbl 0851.93038

Yager, Ronald R. (ed.) et al., Theoretical aspects of fuzzy control. Proceedings of th 2nd IEEE conference on fuzzy systems, March 1993, San Francisco, CA, USA. New York, NY: Wiley. 91-116 (1995).
The present work gives a survey on some aspects of fuzzy controllers from a mathematical point of view. A fuzzy controller is usually a rule-based system that consists of the fuzzification, the fuzzy inference, and the defuzzification. A fuzzy controller defines a control function such that it assigns to each input value a corresponding unique output value.
First, the concept of equality relations and fuzzy points is introduced. The equivalence of equality relations and pseudometrics, characterizing fuzzy points by equality relations, is emphasized. A necessary and sufficient condition for a family of fuzzy sets to form a partition consisting of fuzzy points are given.
Exact mathematical definitions of the Mamdani and Sugeno controllers are considered. The close connection between these definitions is shown. Also in view of the compositional rule of inference, there is a close relation to a certain type of implication.
With the help of the classical Stone-Weierstrass Theorem, it can be shown that continuous functions on compact subsets of the \(m\)-dimensional Euclidean space can be approximated uniformly by a suitable Sugeno controller. This existence result is completed by some theorems which give more constructive methods of approximating control functions by a Sugeno controller.
For the entire collection see [Zbl 0834.00063].

MSC:

93C42 Fuzzy control/observation systems
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