Calculations of Zadeh’s Extension of Piecewise Linear Functions

Abstract

Zadeh’s extension principle is one of the most classical techniques in fuzzy set theory. It is a tool which, for example, can naturally extend a real-valued continuous map to a map having fuzzy sets as its arguments. Theoretically, it is a nice mathematical tool used in many theories, e.g. in studies on fuzzy dynamical systems. However, concrete calculations or even approximations can be very difficult in general and, consequently, many approaches trying to solve this problem appeared. In this work we present a novel algorithm which can compute Zadeh’s extension of given continuous piecewise linear functions. Among other things, an advantage of this approach is that, unlike almost all former approaches, it can deal with discontinuities which naturally appear in simulations of fuzzy dynamical systems.