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On approximating the nearest Ω-stable matrix

Neelam Choudhary, Nicolas Gillis, Punit Sharma

TL;DR

The paper tackles the problem of finding the nearest matrix $X$ to a given $A$ whose eigenvalues reside in a prescribed region $\Omega$ formed by intersections of conic sectors, vertical strips, and disks, thereby generalizing continuous- and discrete-time stability. It adopts a dissipative Hamiltonian parametrization $A=(J-R)Q$ and derives region-specific LMIs to enforce $\Omega$-stability, reformulating the problem into a convex-feasible-set optimization via $P=Q^{-1}$ and closures of the feasible sets. A block coordinate descent algorithm is developed to minimize $\|A-(J-R)P^{-1}\|_F^2$ over the nonconvex objective with a convex LMIs-feasible set, and two initialization strategies (identity-based and LMI-relaxation-based) are proposed to improve robustness. Numerical experiments on synthetic data and a discrete-time stability case demonstrate the method’s ability to produce nearby $\Omega$-stable matrices and illustrate the influence of initialization on solution quality. The approach provides a flexible framework for stable system identification and design when specific eigenvalue-region constraints are required.

Abstract

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω, within the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips and disks. We refer to this problem as the nearest Ω-stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time-invariant systems. In order to achieve this goal, we parametrize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.

On approximating the nearest Ω-stable matrix

TL;DR

The paper tackles the problem of finding the nearest matrix to a given whose eigenvalues reside in a prescribed region formed by intersections of conic sectors, vertical strips, and disks, thereby generalizing continuous- and discrete-time stability. It adopts a dissipative Hamiltonian parametrization and derives region-specific LMIs to enforce -stability, reformulating the problem into a convex-feasible-set optimization via and closures of the feasible sets. A block coordinate descent algorithm is developed to minimize over the nonconvex objective with a convex LMIs-feasible set, and two initialization strategies (identity-based and LMI-relaxation-based) are proposed to improve robustness. Numerical experiments on synthetic data and a discrete-time stability case demonstrate the method’s ability to produce nearby -stable matrices and illustrate the influence of initialization on solution quality. The approach provides a flexible framework for stable system identification and design when specific eigenvalue-region constraints are required.

Abstract

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω, within the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips and disks. We refer to this problem as the nearest Ω-stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time-invariant systems. In order to achieve this goal, we parametrize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.

Paper Structure

This paper contains 14 sections, 7 theorems, 43 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline and contribution
  4. DH matrices and $\Omega$-stability
  5. Parametrization for conic sectors $\Omega_C$
  6. Parametrization for vertical strips $\Omega_V$
  7. Parametrization for disks $\Omega_D$
  8. Reformulation of the nearest $\Omega$-stable matrix problem
  9. Algorithm and numerical experiments
  10. Block coordinate descent method for \ref{['finalform']}
  11. Initialization
  12. Synthetic data sets
  13. Discrete-time stability: $\Omega = \Omega_D(0,1)$
  14. Conclusion and further work

Key Result

Lemma 1

Let $A=(J-R)Q$, where $J,R,Q \in {\mathbb R}^{n,n}$ such that $J^T=-J$, $R^T=R$, and $Q^T=Q$ is invertible. Let $\lambda \in {\mathbb C}$, and $v \in {\mathbb C}^{n}\setminus \{0\}$ be such that $v^*A=\lambda v^*$. Then

Figures (3)

  • Figure 1.1: Illustration of $\Omega = \{ x+iy \ | \sin(\theta) x < \cos(\theta) y < -\sin(\theta) x,\, x<-h<0,\, |x+iy|<r \} = \Omega_C(0,\theta) \cap \Omega_V(h,+\infty) \cap \Omega_D(0,r)$.
  • Figure 4.1: On the left: evolution of the relative error for the different initializations. On the right: eigenvalues of $A$, $(J_i^*-R_i^*)Q_i^*$, $(J_x^*-R_x^*)Q_x^*$ and $(J_t^*-R_t^*)Q_t^*$. The shaded area is the set $\Omega = \Omega_C(3\pi/8) \cap \Omega_V(0.5,1.75) \cap \Omega_D(1,3)$.
  • Figure 4.2: Eigenvalues of $A$, $A_+$, $A_b$, $(J_i^*-R_i^*)Q_i^*$ and $(J_x^*-R_x^*)Q_x^*$.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 3
  • Theorem 4