Algorithm to Compute a Kharitonov-Type Sector Containing All Roots of Hurwitz Interval Polynomials

TL;DR

This paper introduces a Kharitonov-type algorithm for complex interval Hurwitz polynomials that efficiently determines if all roots lie within a specific angular sector, using a finite set of polynomial evaluations and a bisection refinement.

Contribution

It develops a novel algorithm for sector containment of roots in interval Hurwitz polynomials, reducing computational effort with a finite polynomial set and a refinement procedure.

Findings

The algorithm requires up to sixteen polynomials for complex coefficients.

It can refine the sector to arbitrary accuracy through bisection.

Numerical results show the minimal sector often matches vertex polynomial bounds.

Abstract

This paper presents a Kharitonov-type algorithm for complex interval Hurwitz polynomials that determines whether all roots of a given interval polynomial lie within a prescribed angular sector of the complex plane. The method requires evaluating a finite set of additional Kharitonov polynomials. For complex coefficient uncertainty, up to sixteen such polynomials are sufficient, while in the real-coefficient case up to eight are needed. A bisection-based refinement procedure is introduced to compute a containing sector that encloses the angles of all roots. The algorithm progressively tightens the sector bounds and can achieve arbitrarily small accuracy. In the real-coefficient case, the symmetry of the construction allows the real Kharitonov result to be derived directly from the complex case. Numerical experiments suggest that the minimal containing sector coincides with the sector determined by the vertex polynomials, or possibly by a subset of them.