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

Neelam Choudhary, Nicolas Gillis, Punit Sharma

TL;DR

The paper develops a dissipative-Hamiltonian (DH) framework that characterizes the set of Ω-stable matrices for any LMI region Ω, expressing A as A=(J-R)Q with J^T=-J, R^T=R, and Q≻0, under LMIs MΩ(J,R,Q)≺0. This unifies and extends previous results to generic LMI regions and, in particular, provides explicit DH constraints for a wide range of region shapes, including conic sectors, disks, strips, ellipsoids, parabolas, and hyperbolas, with extensions to complex and extended LMI regions. The framework enables solving the nearest Ω-stable matrix problem via a convex-LMI-based reformulation and a nonconvex gradient-descent algorithm that factors the search over DH triplets, with SDP-based projections guiding feasibility. Numerical experiments illustrate the method on multi-region intersections, achieving Ω-stable nearest approximants with nontrivial accuracy in reasonable compute times. The results advance robust near-stability synthesis for LTI systems, model reduction, and control design by providing a systematic, scalable approach to project matrices onto complex stability regions expressed as 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 $Ω$-stable matrix problem: given an LMI region $Ω\subseteq \mathbb{C}$ and a matrix $A \in \mathbb{C}^{n,n}$, find the nearest matrix to $A$ whose eigenvalues belong to $Ω$. Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.

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

TL;DR

The paper develops a dissipative-Hamiltonian (DH) framework that characterizes the set of Ω-stable matrices for any LMI region Ω, expressing A as A=(J-R)Q with J^T=-J, R^T=R, and Q≻0, under LMIs MΩ(J,R,Q)≺0. This unifies and extends previous results to generic LMI regions and, in particular, provides explicit DH constraints for a wide range of region shapes, including conic sectors, disks, strips, ellipsoids, parabolas, and hyperbolas, with extensions to complex and extended LMI regions. The framework enables solving the nearest Ω-stable matrix problem via a convex-LMI-based reformulation and a nonconvex gradient-descent algorithm that factors the search over DH triplets, with SDP-based projections guiding feasibility. Numerical experiments illustrate the method on multi-region intersections, achieving Ω-stable nearest approximants with nontrivial accuracy in reasonable compute times. The results advance robust near-stability synthesis for LTI systems, model reduction, and control design by providing a systematic, scalable approach to project matrices onto complex stability regions expressed as 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 -stable matrix problem: given an LMI region and a matrix , find the nearest matrix to whose eigenvalues belong to . Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.
Paper Structure (13 sections, 6 theorems, 28 equations, 2 figures, 1 table)

This paper contains 13 sections, 6 theorems, 28 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contribution and outline of the paper
  3. Notation
  4. DH characterization of matrices with eigenvalues in generic LMI regions
  5. Special LMI regions and DH parametrization
  6. Application of the DH characterization of LMI regions: Nearest $\Omega$-stable matrix problem
  7. Reformulation
  8. Algorithm
  9. Numerical illustrations
  10. Example 1
  11. Example 2
  12. Extended LMI regions for complex matrices
  13. DH characterization of the horizontal strip with two $n \times n$ LMIs

Key Result

Theorem 1

ChilG96 Let $\Omega$ be an LMI region given by eq:eqlmidef1 and let $A \in {\mathbb R}^{n,n}$. Then $A$ is $\Omega$-stable if and only if there exists a symmetric matrix $X \in {\mathbb R}^{n,n}$ such that $X \succ 0$ and

Figures (2)

  • Figure 4.1: Eigenvalues of $A$, and of its $\Omega$-stable approximation $\tilde{A} = (J- R) Q$, where $\Omega$ is the intersection of a vertical strip, a horizontal strip, and a left and a right parabolic region.
  • Figure 4.2: Eigenvalues of $A$, and of its $\Omega$-stable approximation $\tilde{A} = (J- R) Q$, where $\Omega$ is the intersection of an ellipse, a right conic sector and a left hyperbolic region.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2: LMI Region, ChilG96
  • Definition 3: DH matrix
  • Theorem 1
  • Theorem 2
  • Remark 1: Non-uniqueness of the DH characterization
  • Remark 2: Code
  • Definition 4: Extended LMI Regions
  • Definition 5
  • Theorem 3
  • ...and 3 more