Computation of structured stability radii for Dissipative-Hamiltonian systems

TL;DR

This paper develops explicit formulas for computing the structured stability radii of Dissipative Hamiltonian systems, demonstrating their robustness under structure-preserving perturbations through theoretical reformulations and numerical experiments.

Contribution

It introduces explicit formulas for stability radii of DH systems and reformulates the problem using Rayleigh quotients, enhancing understanding of their robustness.

Findings

Structured stability radii can be explicitly computed for DH systems.

DH systems exhibit greater robustness under structure-preserving perturbations.

Numerical experiments confirm the theoretical robustness advantages.

Abstract

We study linear time-invariant Dissipative Hamiltonian (DH) systems arising in energy-based modeling of dynamical systems. An advantage of DH systems is that they are always stable due to the structure of their coefficient matrices, and, under further weak conditions, even asymptotically stable. In this paper, we discuss the computation of the stability radii for a given asymptotically stable DH system; i.e., the smallest structured perturbation that puts a DH system on the boundary of the region of asymptotic stability, so that it has purely imaginary eigenvalues. We obtain explicit computable formulas for various structured stability radii. For this, the problem of computing stability radii is reformulated in terms of minimizing the Rayleigh quotient of a Hermitian matrix or the sum of two generalized Rayleigh quotients of Hermitian semidefinite matrices. This reformulation results in the problem of minimizing the largest eigenvalue of an eigenvector-dependent Hermitian matrix or minimizing the smallest eigenvalue of a Hermitian matrix which depends on the eigenvector. It is also demonstrated (via numerical experiments) that, under structure-preserving perturbations, the asymptotic stability of a DH system is much more robust than under general perturbations, since the distance to instability is typically much larger when structure-preserving perturbations are considered. Finally, similar results are obtained for optimally robust representations of stable systems.