An iterative method for computing robustness of polynomial stability

We propose a method for computing the distance of a stable polynomial to the set of unstable ones (both in the Hurwitz and in the Schur case). The method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial. We first introduce the structured ε-pseudospectrum of a companion matrix and write a system of ordinary differential equations which maximize the real part (or the absolute value) of elements of the structured ε-pseudospectrum and then exploit the knowledge of the derivative of the maximizers with respect to ε to devise a quadratically convergent iteration. Furthermore we use a variant of the same ODEs to compute the boundary of structured pseudospectra and compare them to unstructured ones. An extension to constrained perturbations is also considered.