How Accurate Are Fuzzy Control Recommendations: Interval-Valued Case

Abstract

As a result of applying fuzzy rules, we get a fuzzy set describing possible control values. In automatic control systems, we need to defuzzify this fuzzy set, i.e., to transform it to a single control value. One of the most frequently used defuzzification techniques is centroid defuzzification. From the practical viewpoint, an important question is: how accurate is the resulting control recommendation? The more accurately we need to implement the control, the more expensive the resulting controller. The possibility to gauge the accuracy of the fuzzy control recommendation follows from the fact that, from the mathematical viewpoint, centroid defuzzification is equivalent to transforming the fuzzy set into a probability distribution and computing the mean value of control. In view of this interpretation, a natural measure of accuracy of a fuzzy control recommendation is the standard deviation of the corresponding random variable. Computing this standard deviation is straightforward for the traditional [0, 1]-based fuzzy logic, in which all experts’ degree of confidence are represented by numbers from the interval [0, 1]. In practice, however, an expert usually cannot describe his/her degree of confidence by a single number, a more appropriate way to describe his/her confidence is by allowing to mark an interval of possible degrees. In this paper, we provide an efficient algorithm for estimating the accuracy of fuzzy control recommendations under such interval-valued fuzzy uncertainty.