zbMATH — the first resource for mathematics

\(\mathcal{L}_{\infty}\)-norm computation for large-scale descriptor systems using structured iterative eigensolvers. (English) Zbl 1405.93028
Summary: In this article, we discuss a method for computing the \(\mathcal{L}_\infty\)-norm for transfer functions of descriptor systems using structured iterative eigensolvers. In particular, the algorithm computes some desired imaginary eigenvalues of an even matrix pencil and uses them to determine an upper and lower bound to the \(\mathcal{L}_\infty\)-norm. Finally, we compare our method to a previously developed algorithm using pseudopole sets. Numerical examples demonstrate the reliability and accuracy of the new method along with a significant drop in the runtime.

93A15 Large-scale systems
93B40 Computational methods in systems theory (MSC2010)
15A22 Matrix pencils
93-04 Software, source code, etc. for problems pertaining to systems and control theory
93C05 Linear systems in control theory
93B55 Pole and zero placement problems
93B60 Eigenvalue problems
93C15 Control/observation systems governed by ordinary differential equations
PDF BibTeX Cite
Full Text: DOI
[1] N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner and M. Voigt, Large-scale computation of \begin{document} \(\mathcal{L}_∞\) \end{document}-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., Accepted for publication. · Zbl 1379.65020
[2] P. Benner, R. Byers, P. Losse, V. Mehrmann and H. Xu, Numerical solution of real skew-Hamiltonian/Hamiltonian eigenproblems, 2007, Unpublished report.
[3] P. Benner; R. Byers; V. Mehrmann; H. Xu, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24, 165, (2002) · Zbl 1035.49022
[4] P. Benner; C. Effenberger, A rational SHIRA method for the Hamiltonian eigenvalue problem, Taiwanese J. Math., 14, 805, (2010) · Zbl 1207.65037
[5] P. Benner; H. Faßbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263, 75, (1997) · Zbl 0884.65028
[6] P. Benner; H. Faßbender; M. Stoll, A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process, Linear Algebra Appl., 435, 578, (2011) · Zbl 1219.65040
[7] P. Benner; V. Sima; M. Voigt, \begin{document} \(\mathcal{L}_∞\) \end{document}-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57, 233, (2012) · Zbl 1369.93174
[8] P. Benner, V. Sima and M. Voigt, Robust and efficient algorithms for \begin{document} \(\mathcal{L}_∞\) \end{document}-norm computation for descriptor systems, in Proc. 7th IFAC Symposium on Robust Control Design, IFAC, Aalborg, Denmark, 2012,195-200.
[9] P. Benner, V. Sima and M. Voigt, Algorithm 961 -Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems, ACM Trans. Math. Software, 42, Paper 24.
[10] P. Benner; M. Voigt, A structured pseudospectral method for \begin{document} \(\mathcal{H}_∞\) \end{document}-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26, 303, (2014) · Zbl 1290.93083
[11] S. Boyd; V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \begin{document} \(L_∞\) \end{document}-norm, Systems Control Lett., 15, 1, (1990) · Zbl 0704.93014
[12] S. Boyd; V. Balakrishnan; P. Kabamba, A bisection method for computing the \begin{document} \(H_{∞}\) \end{document} norm of a transfer matrix and related problems, Math. Control Signals Systems, 2, 207, (1989) · Zbl 0674.93020
[13] N. A. Bruinsma; M. Steinbuch, A fast algorithm to compute the \begin{document} \(\mathcal{H}_∞\) \end{document}-norm of a transfer function matrix, Systems Control Lett., 14, 287, (1990) · Zbl 0699.93021
[14] J. V. Burke, D. Henrion, A. S. Lewis and M. L. Overton, HIFOO-A MATLAB package for fixed-order controller design and \begin{document} \(H_∞\) \end{document} optimization, in Proc. 5th IFAC Syposium on Robust Control Design, Toulouse, France, 2006.
[15] Y. Chahlaoui and P. Van Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, Technical Report 2002-2,2002, Available from http://www.slicot.org/index.php?site=benchmodred. · Zbl 1100.93006
[16] M. A. Freitag; A. Spence; P. V. Dooren, Calculating the \begin{document} \(H_∞\) \end{document}-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35, 619, (2014) · Zbl 1305.65161
[17] F. Freitas; J. Rommes; N. Martins, Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans. Power Syst., 23, 1258, (2008)
[18] N. Guglielmi; M. Gürbüzbalaban; M. L. Overton, Fast approximation of the \begin{document} \(H_∞\) \end{document} norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34, 709, (2013) · Zbl 1271.93057
[19] P. Losse; V. Mehrmann; L. Poppe; T. Reis, The modified optimal \begin{document} \(\mathcal H_∞\) \end{document} control problem for descriptor systems, SIAM J. Control Optim., 47, 2795, (2008) · Zbl 1176.93023
[20] N. Martins; P. C. Pellanda; J. Rommes, Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans. Power Syst., 22, 1657, (2007)
[21] V. Mehrmann; C. Schröder; V. Simoncini, An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl., 436, 4070, (2012) · Zbl 1247.65050
[22] V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, in Dimension Reduction of Large-Scale Systems (eds. P. Benner, V. Mehrmann and D. Sorensen), vol. 45 of Lecture Notes Comput. Sci. Eng., Springer-Verlag, Berlin, Heidelberg, New York, 2005, chapter 3, 89-116. · Zbl 1107.93013
[23] T. Mitchell and M. L. Overton, Fixed low-order controller design and \begin{document} \( H_∞\) \end{document} optimization for large-scale dynamical systems, in Proc. 8th IFAC Symposium on Robust Control Design, Bratislava, Slovakia, 2015, 25-30.
[24] T. Mitchell; M. L. Overton, Hybrid expansion-contraction: a robust scaleable method for approxiating the \begin{document} \(H_∞\) \end{document} norm, IMA J. Numer. Anal., 36, 985, (2016) · Zbl 1433.93100
[25] J. Rommes, Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems \begin{document} \(Ax = λ Bx\) \end{document} with singular \begin{document} \(B\) \end{document}, Math. Comp., 77, 995, (2008) · Zbl 1133.65020
[26] J. Rommes; N. Martins, Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21, 1471, (2006)
[27] J. Rommes; G. L. G. Sleijpen, Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 30, 346, (2008) · Zbl 1165.65016
[28] A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems, in Recent Advances in Iterative Methods (eds. G. Golub, A. Greenbaum and M. Luskin), vol. 60 of IMA Vol. Math. Appl., Springer-Verlag, New York, 1994,149-164. · Zbl 0803.65045
[29] C. Schröder, Private communication, 2013.
[30] M. Voigt, On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems, Logos-Verlag, Berlin, 2015, Also as Dissertation, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, 2015. · Zbl 1329.49001
[31] K. Zhou and J. C. Doyle, Essentials of Robust Control, Hemel Hempstead: Prentice Hall, 1997. · Zbl 0890.93003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.