Computation of structured stability radii for Dissipative-Hamiltonian systems
Peter Benner, Volker Mehrmann, Anshul Prajapati, Punit Sharma
TL;DR
This work develops explicit, computable formulas for structured stability radii of Dissipative Hamiltonian DH systems under structure-preserving perturbations to the matrices J and R, framing the radii as minimizations of generalized Rayleigh quotients and, in some cases, as nonlinear eigenvalue problems (NEPv). The authors derive distinct expressions for three perturbation classes (S_d, S_i, S) and connect the inner optimizations to Joint Numerical Range (JNR) formulations, enabling efficient computation via NEPv-based methods. Numerical experiments reveal that structure-preserving perturbations yield significantly larger distances to instability than unstructured perturbations, underscoring the robustness of DH representations. The results also illuminate procedures for optimally robust representations of stable systems and outline open questions for joint perturbations involving J, R, and Q.
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.