Approximation of stability radii for large-scale dissipative Hamiltonian systems
type: Item
instance of: scholarly article
title: Approximation of stability radii for large-scale dissipative Hamiltonian systems
zbMATH Open document ID: 1436.65206
author: Nicat Aliyev
author: Emre Mengi
author: Volker Mehrmann
published in: Advances in Computational Mathematics
publication date: 2020-04-15T00:00:00Z
review text: Linear dissipative Hamiltonian (DH) systems of the form \[ \dot{x} = (J-R)Qx \] with a skew-Hermitian matrix \(J\), a Hermitian positive semidefinite matrix \(R\), and a Hermitian positive definite matrix \(Q\), arise as homogeneous parts of port-Hamiltonian systems without inputs and outputs. These systems are Lyapunov stable and under additional weak conditions even asymptotically stable. However, the matrix \((J - R)Q\) may have purely imaginary eigenvalues, implying that DH systems are not necessarily asymptotically stable. Small perturbations may cause the system to become unstable. Asymptotic stability can only be guaranteed when the stability radius of the system is reasonably large. The authors propose subspace frameworks for the estimation of the stability radii for large scale dissipative Hamiltonian systems. At every iteration, the DH system is projected to small subspaces by preserving its structure. Then the corresponding stability radii are computed for the reduced system by exploiting its eigenvalue optimization characterization. The subspaces used in the projections are expanded to attain the Hermite interpolation properties between the objective eigenvalue function of the full and the reduced problems. The authors show that the stability radii are accurately and efficiently estimated for the DH systems of small and moderate size and for large scale finite-element model of an industrial disk brake. MATLAB implementations of the algorithms and subspace frameworks, as well as some of the data, are here presented.
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 37M21
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 90C26
zbMATH DE Number: 7189692
zbMATH Keywords: stability radius
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: robust stability
zbMATH Keywords: eigenvalue optimization
zbMATH Keywords: subspace projection
zbMATH Keywords: dissipative Hamiltonian system
reviewed by: Bülent Karasözen
describes a project that uses: CVX
MaRDI profile type: Publication
full work available at URL: s10444-020-09763-5
OpenAlex ID: W3004762274
cites work: Large-Scale Computation of $\mathcal{L}_\infty$-Norms by a Greedy Subspace Method
cites work: A Subspace Framework for ${\mathcal H}_\infty$-Norm Minimization
cites work: \(\mathcal{L}_{\infty}\)-norm computation for large-scale descriptor systems using structured iterative eigensolvers
cites work: Faster and More Accurate Computation of the $\mathcal{H}_\infty$ Norm via Optimization
cites work: A structured pseudospectral method for \(\mathcal {H}_{\infty}\)-norm computation of large-scale descriptor systems
cites work: A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \(L_{\infty}\)-norm
cites work: A fast algorithm to compute the \(H_{\infty}\)-norm of a transfer function matrix
cites work: Calculating the \(H_{\infty}\)-norm using the implicit determinant method
cites work: Model Reduction of MIMO Systems via Tangential Interpolation
cites work: Numerical methods for parametric model reduction in the simulation of disk brake squeal
cites work: Graph Implementations for Nonsmooth Convex Programs
cites work: Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems
cites work: Fast approximation of the \(H_\infty\) norm via optimization over spectral value sets
cites work: Linear port-Hamiltonian systems on infinite-dimensional spaces.
cites work: Perturbation theory for linear operators.
cites work: On eigenvalues of matrices dependent on a parameter
cites work: Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations
cites work: Numerical optimization of eigenvalues of Hermitian matrix functions
cites work: Hybrid expansion–contraction: a robust scaleable method for approximating theH∞norm
cites work: Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos
cites work: Q5571936
cites work: Port-Hamiltonian Systems Theory: An Introductory Overview
cites work: Q3697166
DOI: 10.1007/S10444-020-09763-5
Recommended article: Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations
Recommended article: Stability radii for real linear Hamiltonian systems with perturbed dissipation
Recommended article: Stability theory for dissipatively perturbed hamiltonian systems
Recommended article: Stabilization of solutions of dissipative Hamiltonian systems
Recommended article: Stability theory for linear dissipative Hamiltonian systems
Recommended article: Asymptotic stability for perturbed Hamiltonian systems
Recommended article: The stability of multidimensional Hamiltonian systems
Recommended article: Q4287413
Recommended article: Asymptoic Stability of Dissipated Hamilton–Poisson Systems
Recommended article: On the problem of stability for near to integrable Hamiltonian systems
description: scientific article; zbMATH DE number 7189692