Approximating the real structured stability radius with frobenius-norm bounded perturbations

N. Guglielmi, M. Gürbüzbalaban, T. Mitchell, Michael Overton

Research output: Contribution to journalArticle

Abstract

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

Original languageEnglish (US)
Pages (from-to)1323-1353
Number of pages31
JournalSIAM Journal on Matrix Analysis and Applications
Volume38
Issue number4
DOIs
StatePublished - Jan 1 2017

Fingerprint

Stability Radius
Frobenius norm
Perturbation
Pseudospectra
Abscissa
Linear Dynamical Systems
Continuous-time Systems
Output Feedback
Sparse matrix
Large-scale Systems
Discrete-time Systems
Modulus
Radius
Upper bound
Eigenvalue
Norm
Costs

Keywords

  • H (H-infinity) norm
  • Linear dynamical systems
  • Robust stability
  • Spectral value sets
  • Structured pseudospectra

ASJC Scopus subject areas

  • Analysis

Cite this

Approximating the real structured stability radius with frobenius-norm bounded perturbations. / Guglielmi, N.; Gürbüzbalaban, M.; Mitchell, T.; Overton, Michael.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 38, No. 4, 01.01.2017, p. 1323-1353.

Research output: Contribution to journalArticle

@article{bf73fa6992e04c419e12810dc084f20f,
title = "Approximating the real structured stability radius with frobenius-norm bounded perturbations",
abstract = "We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.",
keywords = "H (H-infinity) norm, Linear dynamical systems, Robust stability, Spectral value sets, Structured pseudospectra",
author = "N. Guglielmi and M. G{\"u}rb{\"u}zbalaban and T. Mitchell and Michael Overton",
year = "2017",
month = "1",
day = "1",
doi = "10.1137/16M1110169",
language = "English (US)",
volume = "38",
pages = "1323--1353",
journal = "SIAM Journal on Matrix Analysis and Applications",
issn = "0895-4798",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

TY - JOUR

T1 - Approximating the real structured stability radius with frobenius-norm bounded perturbations

AU - Guglielmi, N.

AU - Gürbüzbalaban, M.

AU - Mitchell, T.

AU - Overton, Michael

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

AB - We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost is dominated by the computation of the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

KW - H (H-infinity) norm

KW - Linear dynamical systems

KW - Robust stability

KW - Spectral value sets

KW - Structured pseudospectra

UR - http://www.scopus.com/inward/record.url?scp=85040307474&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040307474&partnerID=8YFLogxK

U2 - 10.1137/16M1110169

DO - 10.1137/16M1110169

M3 - Article

AN - SCOPUS:85040307474

VL - 38

SP - 1323

EP - 1353

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 4

ER -