Model Inversion Using Extended Gradual Interval Arithmetic

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 like their fuzzy extension, gives results that are more imprecise than necessary and, in some cases, even counterintuitive. 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 the inversion of a linear regressive model and to a control problem that is based on the inversion of a linear model.