Robust D-stability of sets of polynomials with affine coefficients

Abstract

The determination of whether all roots of a set of polynomials P with affine coefficients are inside a D region is a fundamental problem for control engineers. In this paper we give easy testable sufficient conditions to solve the problem under the assumption that all coefficients of each element of P are affine functions over a common compact and convex multidimensional domain K. The kind of admissible D regions where the method is applicable include the open left complex plane and the unit circle. The searching for roots on the boundary of D is done by partitioning the domain in a finite number of subintervals. The value set of P on each point of each subinterval is encapsulated on a rectangle whose vertices are a kind of Kharitonov polynomials. If one of these four polynomials is not D stable, the analysis of part of the exposed edges of the value set (on that subinterval) is required.