Construction of Granular Derivatives and Solution of Granular Initial Value Problem

Abstract

Rule based derivatives are considered. The methods of construction of granular differentials correspondent to rules are discussed. These differentials are determined by granular slope values given by rules and represented by fuzzy linear functions called directions. The method of solution of initial value problem based on the rule based derivative and on initial condition “If X is X ₀ then Y is Y ₀ ”, where X ₀ and Y ₀ are fuzzy numbers, is proposed. This method uses a granulation of slope values, an extension of fuzzy set Y ₀ in directions defined by granular slopes, a cylindrical extension of fuzzy constraints given in the left sides of rules, a concatenation and an aggregation of function values extracted from rules. The solution is represented by fuzzy relation R defined on X×Y. The value Y(X*) for given fuzzy value X* is obtained as a result of cylindrical extension of X, its intersection with R and projection of result on axes Y. The fuzzy set Y(X*) may be re-translated in linguistic form or converted into real number as a result of defuzzification procedure. The proposed approach t o solution o f granular initial value problem is illustrated on example. The possibilities of construction of granular derivatives from expert and from data are discussed.

Research supported in part by RFBR Grant 02-01-00092 and the BISC Program of UC Berkeley.