Stability radii for real linear Hamiltonian systems with perturbed dissipation

Authors

Christian Mehl, Volker Mehrmann, Punit Sharma

Abstract

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability ( real stability radius ) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

Keywords

Dissipative Hamiltonian system; Port-Hamiltonian system; Real distance to instability; Real structured distance to instability; Restricted real distance to instability; 93D20; 93D09; 65F15; 15A21; 65L80; 65L05; 34A30

Citation

• Journal: BIT Numerical Mathematics
• Year: 2017
• Volume: 57
• Issue: 3
• Pages: 811–843
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s10543-017-0654-0

BibTeX

@article{Mehl_2017,
  title={{Stability radii for real linear Hamiltonian systems with perturbed dissipation}},
  volume={57},
  ISSN={1572-9125},
  DOI={10.1007/s10543-017-0654-0},
  number={3},
  journal={BIT Numerical Mathematics},
  publisher={Springer Science and Business Media LLC},
  author={Mehl, Christian and Mehrmann, Volker and Sharma, Punit},
  year={2017},
  pages={811--843}
}

Download the bib file

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