Finding the nearest passive or non-passive system via Hamiltonian eigenvalue optimization

TL;DR

This paper introduces algorithms based on Hamiltonian eigenvalue optimization to find the nearest passive or non-passive system to a given linear system, addressing structured perturbations and large-scale problems.

Contribution

It develops two-level eigenvalue optimization algorithms for passive system proximity, including a low-rank variant for large systems, with convergence analysis and numerical validation.

Findings

Algorithms effectively compute nearest passive systems.

Quadratic convergence observed in eigenvalue coalescence cases.

Low-rank approach improves scalability for large systems.

Abstract

We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured perturbations), and also a closely related algorithm for computing the structured distance of a given passive system to non-passivity. Both problems are addressed by solving eigenvalue optimization problems for Hamiltonian matrices that are constructed from perturbed system matrices. The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow. They optimize over the perturbation size in the outer iteration, which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis. For large systems, we propose a variant of the algorithm that takes advantage of the inherent low-rank structure of the problem. Numerical experiments illustrate the behavior of the proposed algorithms.