A New Derivation of Centroid Defuzzi cation

We describe a new symmetry-based derivation of centroid defuzziication. The Need for Defuzziication. Fuzzy logic and fuzzy control start with the knowledge expressed by experts in terms of words from a natural language, and end up with control or decision recommendations; see, e.g., 1, 2, 4]. As a result of the standard fuzzy control methodology, we get a fuzzy set (membership function) (u) which describes, for each possible control value u, how reasonable it is to use this particular value. In automatic control applications, we want to transform this fuzzy recommendation into a single value u of the control that will actually be applied. This transformation from a fuzzy set to a (non-fuzzy) number is called a defuzziication. What defuzziication should we apply? The Main Idea Behind the Standard Choice of a Defuzziication. In order to nd out what defuzziication is the best, let us recall the meaning of the values (u). In fuzzy control, the values (u) are obtained indirectly, by processing the expert rules formulated for the general inputs. However, we could, in principle, have obtained them directly, by asking experts about the possible controls for this very situation, and then by applying known elicitation techniques to get the degrees of conndence (u). There are two major elicitation methods: estimating on a scale and polling (there also exist other, less frequently used methods such as methods based on betting). Estimation on a scale method does not explain why this or that number is chosen by an expert, so if we assume that (u) is obtained by this method, this does not help much in guring out how we can defuzzify this information. The second, polling method, as we will see, turns out to be much more helpful. According to this second elicitation method, to get a value (u), we poll several (N) experts and then deene (u) as the ratio M(u)=N, where M(u) is the total number of experts who believe that for this particular situation, u is a reasonable control value. The function (u) is usually diierent from 0 for innnitely many diierent values u. However, in reality, we can only ask experts about nitely many diierent values. So, to use this interpretation, let us assume that there are only nitely many possible control values.