Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

M A Freitag, A Spence, Paul Van Dooren

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Abstract


We propose a fast algorithm to calculate the $H_{\infty}$-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189--3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737].
Original languageEnglish
Pages (from-to)619-635
Number of pages17
JournalSIAM Journal On Matrix Analysis and Applications (SIMAX)
Volume35
Issue number2
DOIs
Publication statusPublished - 15 May 2014

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Determinant
Norm
Linear Systems
Jordan Block
Hamiltonian Matrix
Singular Values
Transfer Matrix
Linear algebra
Transfer Function
Fast Algorithm
Eigenvalue Problem
Critical value
Two Parameters
Eigenvalue
Calculate
Numerical Examples

Keywords

  • Newton method
  • transfer matrix
  • Hamiltonian matrix
  • H∞-norm

Cite this

Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method. / Freitag, M A; Spence, A; Van Dooren, Paul.

In: SIAM Journal On Matrix Analysis and Applications (SIMAX), Vol. 35, No. 2, 15.05.2014, p. 619-635.

Research output: Contribution to journalArticle

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N2 - We propose a fast algorithm to calculate the $H_{\infty}$-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189--3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737].

AB - We propose a fast algorithm to calculate the $H_{\infty}$-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189--3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737].

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