Distance to the Nearest Stable Metzler Matrix

TL;DR

This paper develops a novel algorithm to find the closest Metzler matrix to a given matrix, leveraging dissipative Hamiltonian theory to address a non-convex optimization problem with applications in scalable control systems.

Contribution

It introduces a block coordinate descent method combining quadratic and semidefinite programs, improving tractability for nearest Metzler matrix computation.

Findings

Algorithm effectively computes nearest Metzler matrices.

Utilizes dissipative Hamiltonian theory for problem characterization.

Enhances computational efficiency with recent diagonal dominance results.

Abstract

This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and control with regard to scalability. This motivates the question, how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix are we? Dropping the Metzler constraint, this problem has recently been studied using the theory of dissipative Hamiltonian (DH) systems, which provide a helpful characterization of the feasible set of stable matrices. This work uses the DH theory to provide a block coordinate descent algorithm consisting of a quadratic program with favourable structural properties and a semidefinite program for which recent diagonal dominance results can be used to improve tractability.