Schur Stability of Matrix Segment via Bialternate Product

TL;DR

This paper investigates the robust Schur stability of matrix segments using the bialternate product, providing conditions based on eigenvalues of constructed matrices, with practical examples demonstrating the results.

Contribution

It introduces a novel approach to assess robust Schur stability of matrix segments via eigenvalue analysis of specific matrices, extending stability criteria to rank-one matrix differences.

Findings

Reduced stability problem to eigenvalue conditions of three matrices

Provided necessary and sufficient conditions for convex combinations of matrices

Demonstrated equivalence of stability in convex hulls and segments

Abstract

In this study, the problem of robust Schur stability of $n\times n$ dimensional matrix segments by using the bialternate product of matrices is considered. It is shown that the problem can be reduced to the existence of negative eigenvalues of two of three specially constructed matrices and the existence of eigenvalues belonging to the interval $[1,\infty)$ of the third matrix. A necessary and sufficient condition is given for the convex combinations of two stable matrices with rank one difference to be robust Schur stable. It is shown that the robust stability of the convex hull of a finite number of matrices whose two-by-two differences are of rank $1$ is equivalent to the robust stability of the segments formed by these matrices. Examples of applying the obtained results are given.