Characterizing matrices with eigenvalues in an LMI region: A dissipative-Hamiltonian approach

TL;DR

This paper introduces a dissipative Hamiltonian framework to characterize matrices with eigenvalues in a specified LMI region, extending previous work and enabling applications like finding the nearest stable matrix.

Contribution

It generalizes the DH characterization of matrices with eigenvalues in LMI regions to broader classes, including complex regions defined by LMIs, enhancing analysis and control design tools.

Findings

Provides a DH-based characterization for matrices in any LMI region.

Enables solving the nearest $ ext{LMI}$-region-stable matrix problem.

Extends the framework to regions defined by complex LMIs.

Abstract

In this paper, we provide a dissipative Hamiltonian (DH) characterization for the set of matrices whose eigenvalues belong to a given LMI region. This characterization is a generalization of that of Choudhary et al. (Numer. Linear Algebra Appl., 2020) to any LMI region. It can be used in various contexts, which we illustrate on the nearest $\Omega$-stable matrix problem: given an LMI region $\Omega \subseteq \mathbb{C}$ and a matrix $A \in \mathbb{C}^{n,n}$, find the nearest matrix to $A$ whose eigenvalues belong to $\Omega$. Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.