Model Inversion using Extended Gradual Intervals Arithmetic

Abstract

Recently, gradual numbers have been introduced as a means of extending standard interval computation methods to fuzzy and gradual intervals. However, it is well known that the practical use of standard interval arithmetic operators, just as their fuzzy extension, gives results more imprecise than necessary. In this paper, we combine the concepts of gradual numbers and Kaucher arithmetic on extended intervals to define extended gradual interval arithmetic where subtraction and division operators are respectively the inverse operators of the addition and the multiplication. They are applied to a control problem based on the inversion of a linear model.

A fuzzy metric on the space of fuzzy sets

“…The authors claimed that the concept of gradual numbers is a missing primitive concept in fuzzy set theory. In the brief time since their introduction, gradual numbers have been researched by many authors (see, e.g., [3,9,10,14,16,17]). In particular, a fuzzy number can be denoted as a crisp interval of gradual numbers which can be bounded by two special gradual numbers.…”

“…LetÃ,B andC be in F c (R). Then(1)d H (Ã,B) 0 ;(2)d H (Ã,B) =0 if and only ifà =B;(3)d H (Ã,B) =d H (B,Ã); (4)d H (Ã,B) d H (Ã,C) +d H (C,B).Proof (1). and(3)are obvious.…”