Abstract

In this paper, we extend the original criss-cross algorithms for computing the $\varepsilon$-pseudospectral abscissa and radius to general spectral value sets. By proposing new root-finding-based strategies for the horizontal/radial search subphases, we significantly reduce the number of expensive Hamiltonian eigenvalue decompositions incurred, which typically translates to meaningful speedups in overall computation times. Furthermore, and partly necessitated by our root-finding approach, we develop a new way of handling the singular pencils or problematic interior searches that can arise when computing the $\varepsilon$-spectral value set radius. Compared to would-be direct extensions of the original algorithms, that is, without our additional modifications, our improved criss-cross algorithms are not only noticeably faster but also more robust and numerically accurate, for both spectral value set and pseudospectral problems.

Keywords

  1. pseudospectra
  2. robust stability
  3. stability radius
  4. Hamiltonian
  5. symplectic
  6. H-infinity norm

MSC codes

  1. 15A18
  2. 15A22
  3. 93B40
  4. 93B60
  5. 93D09
  6. 93D20

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References

1.
P. Benner, V. Mehrmann, V. Sima, S. V. Huffel, and A. Varga, SLICOT - a subroutine library in systems and control theory, in Applied and Computational Control, Signals, and Circuits, B. N. Datta, ed., vol. 1, Birkhäuser, Boston, MA, 1999, ch. 10, pp. 499--539.
2.
P. Benner and T. Mitchell, Faster and more accurate computation of the $\mathcal{H}_\infty$ norm via optimization, SIAM J. Sci. Comput., 40 (2018), pp. A3609--A3635.
3.
P. Benner and M. Voigt, A structured pseudospectral method for $\mathcal{H}_\infty$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), pp. 303--338.
4.
J. V. Burke, A. S. Lewis, and M. L. Overton, Robust stability and a criss-cross algorithm for pseudospectra, IMA J. Numer. Anal., 23 (2003), pp. 359--375.
5.
R. Byers, A bisection method for measuring the distance of a stable matrix to matrices unstable matrices, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 875--881.
6.
L. Dai, Singular Control Systems, Lect. Notes Control Inf. Sci. 118, Springer-Verlag, Berlin, 1989.
7.
F. Freitas, J. Rommes, and N. Martins, Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans. Power Syst., 23 (2008), pp. 1258--1270.
8.
N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, Fast approximation of the $H_\infty$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737.
9.
N. Guglielmi and M. L. Overton, Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192.
10.
D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer-Verlag, Berlin, 2005.
11.
D. Hinrichsen and N. K. Son, Stability radii of linear discrete-time systems and symplectic pencils, Internat. J. Robust Nonlinear Control, 1 (1991), pp. 79--97.
12.
P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), pp. 377--387.
13.
A. Laub, Efficient multivariable frequency response computations, IEEE Trans. Automat. Control, 26 (1981), pp. 407--408.
14.
E. Mengi, Measures for Robust Stability and Controllability, Ph.D. thesis, New York University, New York, NY, 2006.
15.
E. Mengi and M. L. Overton, Software for Robust Stability and Controllability, http://home.ku.edu.tr/~emengi/software/robuststability.html.
16.
E. Mengi and M. L. Overton, Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix, IMA J. Numer. Anal., 25 (2005), pp. 648--669.
17.
T. Mitchell and M. L. Overton, Hybrid expansion-contraction: A robust scaleable method for approximating the $H_\infty$ norm, IMA J. Numer. Anal., 36 (2016), pp. 985--1014.
18.
M. L. Overton and R. S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 697--718.
19.
L. N. Trefethen, Computation of pseudospectra, Acta Numer., 8 (1999), pp. 247--295.
20.
L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.
21.
L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), pp. 578--584.
22.
P. Van Dooren and M. Verhaegen, On the use of unitary state-space transformations, in Linear Algebra and Its Role in Systems Theory (Brunswick, Maine, 1984), Contemp. Math. 47, American Mathematical Society, Providence, RI, 1985, pp. 447--463.
23.
C. F. Van Loan, How near is a stable matrix to an unstable matrix?, in Linear Algebra and Its Role in Systems Theory (Brunswick, ME, 1984), Contemp. Math. 47, American Mathematical Society, Providence, RI, 1985, pp. 465--478.
25.
T. G. Wright and L. N. Trefethen, Large-scale computation of pseudospectra using ARPACK and Eigs, SIAM J. Sci. Comput., 23 (2001), pp. 591--605.

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Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1325 - 1352
ISSN (online): 1095-7162

History

Submitted: 2 January 2018
Accepted: 29 August 2019
Published online: 14 November 2019

Keywords

  1. pseudospectra
  2. robust stability
  3. stability radius
  4. Hamiltonian
  5. symplectic
  6. H-infinity norm

MSC codes

  1. 15A18
  2. 15A22
  3. 93B40
  4. 93B60
  5. 93D09
  6. 93D20

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