How to take into account monotonicity (and other properties) in centroid approach to fuzzy and intuitionistic fuzzy control

In many application areas, there are skilled experts who excel in control and decision making. It is desirable to come up with an automated system that would use their skills to help others make similarly good decisions. Often, the experts can only formulate their skills in terms of rules that use imprecise ("fuzzy") words from natural language like "small". To transform these fuzzy rules into a precise control strategy, Zadeh designed special technique that he called fuzzy. This technique was later improved – e.g., by adding explicit information about what the experts consider not a good control; such addition is known as intuitionistic fuzzy technique. The problem that we consider in this paper in that in many cases, in addition to the fuzzy expert rules, we also have some extra knowledge about the function \overline{u}(x) – the function that describes what control to apply for a given input x. For example, it is often reasonable to require that this function is increasing: the larger x, the more control we should apply. In this paper, we use the general decision theory technique to show how this additional information can be incorporated into fuzzy and intuitionistic fuzzy control.