Fast approximation of the H∞ norm via optimization over spectral value sets

Nicola Guglielmi, Mert Gürbüzbalaban, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

The H norm of a transfer matrix function for a control system is the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the stability region for the dynamical system (the left half-plane in the continuous-time case and the unit disk in the discrete-time case). After deriving some fundamental properties of spectral value sets, particularly the intricate relationship between the singular vectors of the transfer matrix and the eigenvectors of the corresponding perturbed system matrix, we extend an algorithm recently introduced by Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for approximating the maximal real part or modulus of points in a matrix pseudospectrum to spectral value sets, characterizing its fixed points. We then introduce a Newton-bisection method to approximate the H norm, for which each step requires optimization of the real part or the modulus over an ε-spectral value set. Although the algorithm is guaranteed only to find lower bounds on the H norm, it typically finds good approximations in cases where we can test this. It is much faster than the standard Boyd- Balakrishnan-Bruinsma-Steinbuch algorithm to compute the H norm when the system matrices are large and sparse and the number of inputs and outputs is small. The main work required by the algorithm is the computation of the spectral abscissa or radius of a sequence of matrices that are rank-one perturbations of a sparse matrix.

Original languageEnglish (US)
Pages (from-to)709-737
Number of pages29
JournalSIAM Journal on Matrix Analysis and Applications
Volume34
Issue number2
DOIs
StatePublished - 2013

Fingerprint

Norm
Optimization
Approximation
Transfer Matrix
Modulus
Rank One Perturbation
Bisection Method
Pseudospectra
Abscissa
Singular Vectors
Stability Region
Perturbed System
Matrix Function
Sparse matrix
Half-plane
Newton Methods
Transfer Function
Unit Disk
Eigenvector
Continuous Time

Keywords

  • H norm
  • Large and sparse systems
  • Linear dynamical systems
  • Newton-bisection method
  • Pseudospectra
  • Spectral value set abscissa
  • Spectral value set radius
  • Spectral value sets
  • Stability radius
  • Transfer matrix

ASJC Scopus subject areas

  • Analysis

Cite this

Fast approximation of the H∞ norm via optimization over spectral value sets. / Guglielmi, Nicola; Gürbüzbalaban, Mert; Overton, Michael L.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 2, 2013, p. 709-737.

Research output: Contribution to journalArticle

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