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Logic Journal of IGPL 2005 13(5):525-541; doi:10.1093/jigpal/jzi040
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© The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

Mathematical Fuzzy Control. A Survey of Some Recent Results

Siegfried Gottwald

Leipzig University, Institute for Logic and Philosophy of Science, Beethovenstr. 15, 04107 Leipzig, Germany. E-mail: gottwald{at}uni-leipzig.de

The core point of fuzzy control approaches are finite lists of linguistic control rules. For computer-based automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm should yield a fuzzy subset of the output space of the control problem if confronted with a fuzzy subset of the input space.

This paper surveys mathematical problems which are connected with, and arose out of these basic ideas. The main formal tools used in these mathematical considerations are fuzzy sets and fuzzy relations together with some generalized, viz. many-valued logic which underlies these considerations.

And the essential way of understanding the mathematical context of fuzzy control is to look at it as an interpolation problem: one has to determine a fuzzy control function out of a finite list of interpolation nodes.

Key Words: fuzzy relation equations, solvability criteria, solvability index, pseudo-solutions, optimality of pseudo-solutions, approximation and interpolation, compositional rule of inference, fuzzy control

Received 13 October 2004.


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