### 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 language | English (US) |
---|---|

Pages (from-to) | 1323-1353 |

Number of pages | 31 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*38*(4), 1323-1353. https://doi.org/10.1137/16M1110169

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

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 38, no. 4, pp. 1323-1353. https://doi.org/10.1137/16M1110169

}

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 -