Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

Authors

Christian Mehl, Volker Mehrmann, Punit Sharma

Abstract

Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical systems. One of the major advantages of the DH formulation is that system properties are encoded in an algebraic way. For instance, the algebraic structure of DH systems guarantees that the system is automatically stable. In this paper the question is discussed when a linear constant coefficient DH system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well understood. In this paper, explicit formulas for this distance under structure-preserving perturbations are determined. It is also shown (via numerical examples) that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbations, since the distance to instability can be much larger when structure-preserving perturbations are considered.

Citation

• Journal: SIAM Journal on Matrix Analysis and Applications
• Year: 2016
• Volume: 37
• Issue: 4
• Pages: 1625–1654
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/16m1067330

BibTeX

@article{Mehl_2016,
  title={{Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations}},
  volume={37},
  ISSN={1095-7162},
  DOI={10.1137/16m1067330},
  number={4},
  journal={SIAM Journal on Matrix Analysis and Applications},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Mehl, Christian and Mehrmann, Volker and Sharma, Punit},
  year={2016},
  pages={1625--1654}
}

Download the bib file

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