Algorithm to Compute a Kharitonov-Type Sector Containing All Roots of Hurwitz Interval Polynomials
David Hertz
TL;DR
The paper addresses robust Hurwitz stability of interval polynomials by computing a Kharitonov-type sector that contains all root directions of $P(s;[{f a}+i{f b}])$ in the complex plane. For the complex case, it presents a practical algorithm based on Kharitonov's $8$ vertex polynomials (K8PT), requiring up to $16$ tests to certify that a sector $[0.5\pi+\alpha,1.5\pi-\beta]$ contains all roots, with a bisection scheme refining $\alpha$ and $\beta$. A real-case variant leverages symmetry to reduce the problem to a symmetric sector $[0.5\pi+\alpha,1.5\pi-\alpha]$ and tests rotated real coefficients via up to $4$ vertex polynomials (equivalently eight rotated complex tests) to obtain a suboptimal containing sector. Numerical examples indicate the minimal containing sector aligns with the vertex set, leading to conjectures that the exact minimal sector is determined by a predetermined vertex subset of size at most $2^{2(N+1)}$ in the complex case and $2^{N+1}$ in the real case; the study also demonstrates further tightening via recursive HertzND methods.
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.