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Preprint 9-2018

Computation of Stability Radii for Large-Scale Dissipative Hamiltonian Systems

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Author(s) : Nicat Aliyev , Volker Mehrmann , Emre Mengi

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 9-2018

MSC 2000

65F15 Eigenvalues, eigenvectors
93D09 Robust stability

Abstract :
A linear time-invariant dissipative Hamiltonian (DH) system $\dot x = (J-R)Q x$, with a skew-Hermitian $J$, an Hermitian positive semi-definite $R$, and an Hermitian positive definite $Q$, is always Lyapunov stable and under weak further conditions even asymptotically stable. In various applications there is uncertainty on the system matrices $J, R, Q$, and it is desirable to know whether the system remains asymptotically stable uniformly against all possible uncertainties within a given perturbation set. Such robust stability considerations motivate the concept of stability radius for DH systems, i.e., what is the maximal perturbation permissible to the coefficients $J, R, Q$, while preserving the asymptotic stability. We consider two stability radii, the unstructured one where $J, R, Q$ are subject to unstructured perturbation, and the structured one where the perturbations preserve the DH structure. We employ characterizations for these radii that have been derived recently in SIAM J. Matrix Anal. Appl., 37, pp. 1625-1654, 2016 and propose new algorithms to compute these stability radii for large scale problems by tailoring subspace frameworks thatear rate in theory. At every iteration, they first solve a reduced problem and then expand the subspaces in order to attain certain Hermite interpolation properties between the full andreduced problems. The reduced problems are solved by means of the adaptations of existing level-set algorithms for ${\mathcal H}_\infty$-norm computation in the unstructured case, while, for the structured radii, we benefit from algorithms that approximate the objective eigenvalue function with a piece-wise quadratic global underestimator. The performance of the new approaches is illustrated with several examples including a system that arises from a finite-element modeling of an industrial disk brake.

Keywords : Linear Time-Invariant Dissipative Hamiltonian System, Port-Hamiltonian system, Robust Stability, Stability Radius, Eigenvalue Optimization, Subspace Projection, Structure Preserving Subspace Framework, Hermite Interpolation

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