Mathematical Fuzzy Control. A Survey of Some Recent Results

The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased 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.