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March  2018, 8(1): 119-133. doi: 10.3934/naco.2018007

$\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers

Peter Benner 1, , Ryan Lowe 2, and  Matthias Voigt 3,,

1. 

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany
2. 

McGill University, Reasoning and Learning Laboratory, 3480 University Street, H3A 0E9 Montreal, Quebec, Canada
3. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author: Matthias Voigt

The main parts of this work were developed while the second author visited the third author as an intern with the RISE program of the Deutscher Akademischer Austauschdienst (DAAD) at the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg in summer 2013. We gratefully thank the MPI Magdeburg and the DAAD for their financial support

Received  April 2017 Revised  January 2018 Published  March 2018

In this article, we discuss a method for computing the $\mathcal{L}_∞$-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 $\mathcal{L}_∞$-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.

Keywords: Arnoldi methods, descriptor systems, dominant poles, even matrix pencils, $\mathcal{L}_∞$-norm.
Mathematics Subject Classification: Primary: 93A15; Secondary: 15A22, 65F15, 93C05.
Citation: Peter Benner, Ryan Lowe, Matthias Voigt. $\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 119-133. doi: 10.3934/naco.2018007
References:
[1]

N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner and M. Voigt, Large-scale computation of $\mathcal{L}_∞$-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., Accepted for publication. Google Scholar

[2]

P. Benner, R. Byers, P. Losse, V. Mehrmann and H. Xu, Numerical solution of real skew-Hamiltonian/Hamiltonian eigenproblems, 2007, Unpublished report. Google Scholar

[3]

P. Benner, R. Byers, V. Mehrmann and H. Xu, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), 165-190.  doi: 10.1137/S0895479800367439.  Google Scholar

[4]

P. Benner and C. Effenberger, A rational SHIRA method for the Hamiltonian eigenvalue problem, Taiwanese J. Math., 14 (2010), 805-823.  doi: 10.11650/twjm/1500405868.  Google Scholar

[5]

P. Benner and H. Faßbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263 (1997), 75-111.  doi: 10.1016/S0024-3795(96)00524-1.  Google Scholar

[6]

P. Benner, H. Faßbender and M. Stoll, A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process, Linear Algebra Appl., 435 (2011), 578-600.  doi: 10.1016/j.laa.2010.04.048.  Google Scholar

[7]

P. Benner, V. Sima and M. Voigt, $\mathcal{L}_∞$-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57 (2012), 233-238.  doi: 10.1109/TAC.2011.2161833.  Google Scholar

[8]

P. Benner, V. Sima and M. Voigt, Robust and efficient algorithms for $\mathcal{L}_∞$-norm computation for descriptor systems, in Proc. 7th IFAC Symposium on Robust Control Design, IFAC, Aalborg, Denmark, 2012,195-200. doi: 10.3182/20120620-3-DK-2025.00114.  Google Scholar

[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. doi: 10.1145/2818313.  Google Scholar

[10]

P. Benner and M. Voigt, A structured pseudospectral method for $\mathcal{H}_∞$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), 303-338.  doi: 10.1007/s00498-013-0121-7.  Google Scholar

[11]

S. Boyd and V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $L_∞$-norm, Systems Control Lett., 15 (1990), 1-7.  doi: 10.1016/0167-6911(90)90037-U.  Google Scholar

[12]

S. Boyd, V. Balakrishnan and P. Kabamba, A bisection method for computing the $H_{∞}$ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), 207-219.  doi: 10.1007/BF02551385.  Google Scholar

[13]

N. A. Bruinsma and M. Steinbuch, A fast algorithm to compute the $\mathcal{H}_∞$-norm of a transfer function matrix, Systems Control Lett., 14 (1990), 287-293.  doi: 10.1016/0167-6911(90)90049-Z.  Google Scholar

[14]

J. V. Burke, D. Henrion, A. S. Lewis and M. L. Overton, HIFOO-A MATLAB package for fixed-order controller design and $H_∞$ optimization, in Proc. 5th IFAC Syposium on Robust Control Design, Toulouse, France, 2006. doi: 10.3182/20060705-3-FR-2907.00059.  Google Scholar

[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. Google Scholar

[16]

M. A. Freitag, A. Spence and P. V. Dooren, Calculating the $H_∞$-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35 (2014), 619-634.  doi: 10.1137/130933228.  Google Scholar

[17]

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), 1258-1270.  doi: 10.1109/TPWRS.2008.926693.  Google Scholar

[18]

N. Guglielmi, M. Gürbüzbalaban and M. L. Overton, Fast approximation of the $H_∞$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), 709-737.  doi: 10.1137/120875752.  Google Scholar

[19]

P. Losse, V. Mehrmann, L. Poppe and T. Reis, The modified optimal $\mathcal H_∞$ control problem for descriptor systems, SIAM J. Control Optim., 47 (2008), 2795-2811.  doi: 10.1137/070710093.  Google Scholar

[20]

N. Martins, P. C. Pellanda and J. Rommes, Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans. Power Syst., 22 (2007), 1657-1664.  doi: 10.1109/TPWRS.2007.907526.  Google Scholar

[21]

V. Mehrmann, C. Schröder and V. Simoncini, An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl., 436 (2012), 4070-4087.  doi: 10.1016/j.laa.2009.11.009.  Google Scholar

[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. doi: 10.1007/3-540-27909-1_3.  Google Scholar

[23]

T. Mitchell and M. L. Overton, Fixed low-order controller design and $ H_∞$ optimization for large-scale dynamical systems, in Proc. 8th IFAC Symposium on Robust Control Design, Bratislava, Slovakia, 2015, 25-30. doi: 10.1016/j.ifacol.2015.09.428.  Google Scholar

[24]

T. Mitchell and M. L. Overton, Hybrid expansion-contraction: a robust scaleable method for approxiating the $H_∞$ norm, IMA J. Numer. Anal., 36 (2016), 985-1014.  doi: 10.1093/imanum/drv046.  Google Scholar

[25]

J. Rommes, Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax = λ Bx$ with singular $B$, Math. Comp., 77 (2008), 995-1015.  doi: 10.1090/S0025-5718-07-02040-6.  Google Scholar

[26]

J. Rommes and N. Martins, Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), 1471-1487.  doi: 10.1109/TPWRS.2006.881154.  Google Scholar

[27]

J. Rommes and G. L. G. Sleijpen, Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 30 (2008), 346-363.  doi: 10.1137/060671401.  Google Scholar

[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. doi: 10.1007/978-1-4613-9353-5_10.  Google Scholar

[29]

C. Schröder, Private communication, 2013. Google Scholar

[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. Google Scholar

[31]

K. Zhou and J. C. Doyle, Essentials of Robust Control, Hemel Hempstead: Prentice Hall, 1997. Google Scholar

show all references

References:
[1]

N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner and M. Voigt, Large-scale computation of $\mathcal{L}_∞$-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., Accepted for publication. Google Scholar

[2]

P. Benner, R. Byers, P. Losse, V. Mehrmann and H. Xu, Numerical solution of real skew-Hamiltonian/Hamiltonian eigenproblems, 2007, Unpublished report. Google Scholar

[3]

P. Benner, R. Byers, V. Mehrmann and H. Xu, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), 165-190.  doi: 10.1137/S0895479800367439.  Google Scholar

[4]

P. Benner and C. Effenberger, A rational SHIRA method for the Hamiltonian eigenvalue problem, Taiwanese J. Math., 14 (2010), 805-823.  doi: 10.11650/twjm/1500405868.  Google Scholar

[5]

P. Benner and H. Faßbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263 (1997), 75-111.  doi: 10.1016/S0024-3795(96)00524-1.  Google Scholar

[6]

P. Benner, H. Faßbender and M. Stoll, A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process, Linear Algebra Appl., 435 (2011), 578-600.  doi: 10.1016/j.laa.2010.04.048.  Google Scholar

[7]

P. Benner, V. Sima and M. Voigt, $\mathcal{L}_∞$-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57 (2012), 233-238.  doi: 10.1109/TAC.2011.2161833.  Google Scholar

[8]

P. Benner, V. Sima and M. Voigt, Robust and efficient algorithms for $\mathcal{L}_∞$-norm computation for descriptor systems, in Proc. 7th IFAC Symposium on Robust Control Design, IFAC, Aalborg, Denmark, 2012,195-200. doi: 10.3182/20120620-3-DK-2025.00114.  Google Scholar

[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. doi: 10.1145/2818313.  Google Scholar

[10]

P. Benner and M. Voigt, A structured pseudospectral method for $\mathcal{H}_∞$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), 303-338.  doi: 10.1007/s00498-013-0121-7.  Google Scholar

[11]

S. Boyd and V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $L_∞$-norm, Systems Control Lett., 15 (1990), 1-7.  doi: 10.1016/0167-6911(90)90037-U.  Google Scholar

[12]

S. Boyd, V. Balakrishnan and P. Kabamba, A bisection method for computing the $H_{∞}$ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), 207-219.  doi: 10.1007/BF02551385.  Google Scholar

[13]

N. A. Bruinsma and M. Steinbuch, A fast algorithm to compute the $\mathcal{H}_∞$-norm of a transfer function matrix, Systems Control Lett., 14 (1990), 287-293.  doi: 10.1016/0167-6911(90)90049-Z.  Google Scholar

[14]

J. V. Burke, D. Henrion, A. S. Lewis and M. L. Overton, HIFOO-A MATLAB package for fixed-order controller design and $H_∞$ optimization, in Proc. 5th IFAC Syposium on Robust Control Design, Toulouse, France, 2006. doi: 10.3182/20060705-3-FR-2907.00059.  Google Scholar

[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. Google Scholar

[16]

M. A. Freitag, A. Spence and P. V. Dooren, Calculating the $H_∞$-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35 (2014), 619-634.  doi: 10.1137/130933228.  Google Scholar

[17]

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), 1258-1270.  doi: 10.1109/TPWRS.2008.926693.  Google Scholar

[18]

N. Guglielmi, M. Gürbüzbalaban and M. L. Overton, Fast approximation of the $H_∞$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), 709-737.  doi: 10.1137/120875752.  Google Scholar

[19]

P. Losse, V. Mehrmann, L. Poppe and T. Reis, The modified optimal $\mathcal H_∞$ control problem for descriptor systems, SIAM J. Control Optim., 47 (2008), 2795-2811.  doi: 10.1137/070710093.  Google Scholar

[20]

N. Martins, P. C. Pellanda and J. Rommes, Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans. Power Syst., 22 (2007), 1657-1664.  doi: 10.1109/TPWRS.2007.907526.  Google Scholar

[21]

V. Mehrmann, C. Schröder and V. Simoncini, An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl., 436 (2012), 4070-4087.  doi: 10.1016/j.laa.2009.11.009.  Google Scholar

[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. doi: 10.1007/3-540-27909-1_3.  Google Scholar

[23]

T. Mitchell and M. L. Overton, Fixed low-order controller design and $ H_∞$ optimization for large-scale dynamical systems, in Proc. 8th IFAC Symposium on Robust Control Design, Bratislava, Slovakia, 2015, 25-30. doi: 10.1016/j.ifacol.2015.09.428.  Google Scholar

[24]

T. Mitchell and M. L. Overton, Hybrid expansion-contraction: a robust scaleable method for approxiating the $H_∞$ norm, IMA J. Numer. Anal., 36 (2016), 985-1014.  doi: 10.1093/imanum/drv046.  Google Scholar

[25]

J. Rommes, Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax = λ Bx$ with singular $B$, Math. Comp., 77 (2008), 995-1015.  doi: 10.1090/S0025-5718-07-02040-6.  Google Scholar

[26]

J. Rommes and N. Martins, Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), 1471-1487.  doi: 10.1109/TPWRS.2006.881154.  Google Scholar

[27]

J. Rommes and G. L. G. Sleijpen, Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 30 (2008), 346-363.  doi: 10.1137/060671401.  Google Scholar

[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. doi: 10.1007/978-1-4613-9353-5_10.  Google Scholar

[29]

C. Schröder, Private communication, 2013. Google Scholar

[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. Google Scholar

[31]

K. Zhou and J. C. Doyle, Essentials of Robust Control, Hemel Hempstead: Prentice Hall, 1997. Google Scholar

Figure 1.  Eigenvalues of a skew-Hamiltonian/Hamiltonian matrix pencil $\mathcal{H}_\gamma(s)$
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Figure 2.  Maximum singular value plot for $ \mathtt{peec}$ example. Note that not all peaks are captured due to plotting resolution. The red cirlce corresponds to the computed value of the $\mathcal{L}_\infty$-norm.
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Figure 3.  linorm output for the bips07 3078 example
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Figure 4.  Illustration of the convergence of the $ \mathtt{bips07\_3078} $ example. The red dashed lines correspond to the different values of $\gamma$ used during the iteration and the dashed black line depicts the midpoint used after the first iteration. Moreover, the black crosses correspond to the computed eigenvalues obtained from even IRA, whereas the red circle shows the computed value of the $\mathcal{L}_\infty$ -norm.
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Table 1.  Most important $\mathtt{linorm}$ design parameters contained in $ \mathtt{options} $ struct
parameter description default value
$\mathtt{tol}$ desired relative accuracy of $\mathcal{L}_\infty$-norm value 1e-06
$\mathtt{npolinit}$ number of dominant poles computed in initial stage 20
$\mathtt{domtol}$ relative cutoff value for the dominance of the poles to 0.5
determine the initial eigenvalues
$\mathtt{shifttol}$ minimum relative distance between two subsequent shifts 0.01
$\mathtt{mmax}$ maximum search space dimension even IRA 8
$\mathtt{maxcycles}$ maximum number of restarts for even IRA 30
$\mathtt{nwanted}$ number of eigenvalues calculated per shift in even IRA 4
$\mathtt{eigtol}$ tolerance on the eigenvalue residual for even IRA 1e-06
$\mathtt{sweeptol}$ relative eigenvalue distance for frequency sweep 1e-05
$\mathtt{shiftmv}$ relative shift displacement 5e-04
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parameter description default value
$\mathtt{tol}$ desired relative accuracy of $\mathcal{L}_\infty$-norm value 1e-06
$\mathtt{npolinit}$ number of dominant poles computed in initial stage 20
$\mathtt{domtol}$ relative cutoff value for the dominance of the poles to 0.5
determine the initial eigenvalues
$\mathtt{shifttol}$ minimum relative distance between two subsequent shifts 0.01
$\mathtt{mmax}$ maximum search space dimension even IRA 8
$\mathtt{maxcycles}$ maximum number of restarts for even IRA 30
$\mathtt{nwanted}$ number of eigenvalues calculated per shift in even IRA 4
$\mathtt{eigtol}$ tolerance on the eigenvalue residual for even IRA 1e-06
$\mathtt{sweeptol}$ relative eigenvalue distance for frequency sweep 1e-05
$\mathtt{shiftmv}$ relative shift displacement 5e-04
Table 2.  Numerical results for 33 test examples
# example n m p computed $\mathcal{H}_\infty$-norm optimal frequency $\omega_\text{opt}$ time in s
pseudopoles matrix pencil pseudopoles matrix pencil pseudopoles matrix pencil
1 $ \mathtt{build}$ 48 1 1 5.27633e-03 5.27634e-03 5.20608e+00 5.20584e+00 1.54 0.54
2 $ \mathtt{pde}$ 84 1 1 1.08358e+01 1.08358e+01 0.00000e+00 0.00000e+00 2.08 0.61
3 $ \mathtt{CDplayer}$ 120 2 2 2.31982e+06 2.31982e+06 2.25682e+01 2.25682e+01 2.70 0.54
4 $ \mathtt{iss}$ 270 3 3 1.15887e-01 1.15887e-01 7.75093e-01 7.75089e-01 2.69 0.61
5 $ \mathtt{beam}$ 348 1 1 4.55487e+03 4.55488e+03 1.04575e-01 1.04573e-01 50.22 38.15
6 $ \mathtt{S10PI\_n1}$ 528 1 1 3.97454e+00 3.97454e+00 7.53151e+03 7.53152e+03 1.78 0.79
7 $ \mathtt{S20PI\_n1}$ 1028 1 1 3.44317e+00 3.44317e+00 7.61831e+03 7.61835e+03 4.22 1.33
8 $ \mathtt{S40PI\_n1}$ 2028 1 1 3.34732e+00 3.34732e+00 6.95875e+03 6.95873e+03 4.77 2.12
9 $ \mathtt{S80PI\_n1}$ 4028 1 1 3.37016e+00 3.37016e+00 6.96149e+03 6.96147e+03 10.50 4.80
10 $ \mathtt{M10PI\_n1}$ 528 3 3 4.05662e+00 4.05662e+00 7.53181e+03 7.53182e+03 3.08 1.29
11 $ \mathtt{M20PI\_n1}$ 1028 3 3 3.87260e+00 3.87260e+00 5.06412e+03 5.06412e+03 3.63 1.62
12 $ \mathtt{M40PI\_n1}$ 2028 3 3 3.81767e+00 3.81767e+00 5.07107e+03 5.07107e+03 5.83 2.83
13 $ \mathtt{M80PI\_n1}$ 4028 3 3 3.80375e+00 3.80376e+00 5.07279e+03 5.07279e+03 10.88 5.23
14 $ \mathtt{peec}$ 480 1 1 3.52651e-01 3.52541e-01 5.46349e+00 5.46349e+00 20.80 9.31
15 $ \mathtt{S10PI\_n}$ 682 1 1 3.97454e+00 3.97454e+00 7.53151e+03 7.53152e+03 2.29 1.03
16 $ \mathtt{S20PI\_n}$ 1182 1 1 3.44317e+00 3.44317e+00 7.61831e+03 7.61835e+03 4.67 1.53
17 $ \mathtt{S40PI\_n}$ 2182 1 1 3.34732e+00 3.34732e+00 6.95875e+03 6.95873e+03 5.35 2.47
18 $ \mathtt{S80PI\_n}$ 4182 1 1 3.37016e+00 3.37016e+00 6.96149e+03 6.96147e+03 10.41 4.51
19 $ \mathtt{M10PI\_n}$ 682 3 3 4.05662e+00 4.05662e+00 7.53181e+03 7.53182e+03 3.56 1.36
20 $ \mathtt{M20PI\_n}$ 1182 3 3 3.87260e+00 3.87260e+00 5.06412e+03 5.06412e+03 4.03 1.73
21 $ \mathtt{M40PI\_n}$ 2182 3 3 3.81767e+00 3.81767e+00 5.07107e+03 5.07107e+03 6.11 2.85
22 $ \mathtt{M80PI\_n}$ 4182 3 3 3.80375e+00 3.80376e+00 5.07279e+03 5.07279e+03 11.28 5.19
23 $ \mathtt{bips98\_606}$ 7135 4 4 2.01956e+02 2.01956e+02 3.81763e+00 3.81763e+00 35.43 14.43
24 $ \mathtt{bips98\_1142}$ 9735 4 4 1.60427e+02 1.60427e+02 4.93005e+00 4.93005e+00 69.65 16.37
25 $ \mathtt{bips98\_1450}$ 11305 4 4 1.97389e+02 1.97389e+02 5.64575e+00 5.64650e+00 61.65 17.91
26 $ \mathtt{bips07\_1693}$ 13275 4 4 2.04168e+02 2.04168e+02 5.53766e+00 5.53767e+00 167.10 31.98
27 $ \mathtt{bips07\_1998}$ 15066 4 4 1.97064e+02 1.97064e+02 6.39968e+00 6.39879e+00 102.11 29.99
28 $ \mathtt{bips07\_2476}$ 16861 4 4 1.89579e+02 1.89579e+02 5.88971e+00 5.89023e+00 146.18 31.62
29 $ \mathtt{bips07\_3078}$ 21128 4 4 2.09445e+02 2.09445e+02 5.55792e+00 5.55839e+00 91.05 34.73
30 $ \mathtt{xingo\_afonso\_itaipu}$ 13250 1 1 4.05605e+00 4.05605e+00 1.09165e+00 1.09165e+00 39.24 16.80
31 $ \mathtt{mimo8x8\_system}$ 13309 8 8 5.34292e-02 5.34293e-02 1.03313e+00 1.03308e+00 78.47 23.25
32 $ \mathtt{mimo28x28\_system}$ 13251 28 28 1.18618e-01 1.18618e-01 1.07935e+00 1.07935e+00 85.36 35.45
33 $ \mathtt{mimo46x46\_system}$ 13250 46 46 2.05631e+02 2.05631e+02 1.07908e+00 1.07908e+00 115.91 49.13
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# example n m p computed $\mathcal{H}_\infty$-norm optimal frequency $\omega_\text{opt}$ time in s
pseudopoles matrix pencil pseudopoles matrix pencil pseudopoles matrix pencil
1 $ \mathtt{build}$ 48 1 1 5.27633e-03 5.27634e-03 5.20608e+00 5.20584e+00 1.54 0.54
2 $ \mathtt{pde}$ 84 1 1 1.08358e+01 1.08358e+01 0.00000e+00 0.00000e+00 2.08 0.61
3 $ \mathtt{CDplayer}$ 120 2 2 2.31982e+06 2.31982e+06 2.25682e+01 2.25682e+01 2.70 0.54
4 $ \mathtt{iss}$ 270 3 3 1.15887e-01 1.15887e-01 7.75093e-01 7.75089e-01 2.69 0.61
5 $ \mathtt{beam}$ 348 1 1 4.55487e+03 4.55488e+03 1.04575e-01 1.04573e-01 50.22 38.15
6 $ \mathtt{S10PI\_n1}$ 528 1 1 3.97454e+00 3.97454e+00 7.53151e+03 7.53152e+03 1.78 0.79
7 $ \mathtt{S20PI\_n1}$ 1028 1 1 3.44317e+00 3.44317e+00 7.61831e+03 7.61835e+03 4.22 1.33
8 $ \mathtt{S40PI\_n1}$ 2028 1 1 3.34732e+00 3.34732e+00 6.95875e+03 6.95873e+03 4.77 2.12
9 $ \mathtt{S80PI\_n1}$ 4028 1 1 3.37016e+00 3.37016e+00 6.96149e+03 6.96147e+03 10.50 4.80
10 $ \mathtt{M10PI\_n1}$ 528 3 3 4.05662e+00 4.05662e+00 7.53181e+03 7.53182e+03 3.08 1.29
11 $ \mathtt{M20PI\_n1}$ 1028 3 3 3.87260e+00 3.87260e+00 5.06412e+03 5.06412e+03 3.63 1.62
12 $ \mathtt{M40PI\_n1}$ 2028 3 3 3.81767e+00 3.81767e+00 5.07107e+03 5.07107e+03 5.83 2.83
13 $ \mathtt{M80PI\_n1}$ 4028 3 3 3.80375e+00 3.80376e+00 5.07279e+03 5.07279e+03 10.88 5.23
14 $ \mathtt{peec}$ 480 1 1 3.52651e-01 3.52541e-01 5.46349e+00 5.46349e+00 20.80 9.31
15 $ \mathtt{S10PI\_n}$ 682 1 1 3.97454e+00 3.97454e+00 7.53151e+03 7.53152e+03 2.29 1.03
16 $ \mathtt{S20PI\_n}$ 1182 1 1 3.44317e+00 3.44317e+00 7.61831e+03 7.61835e+03 4.67 1.53
17 $ \mathtt{S40PI\_n}$ 2182 1 1 3.34732e+00 3.34732e+00 6.95875e+03 6.95873e+03 5.35 2.47
18 $ \mathtt{S80PI\_n}$ 4182 1 1 3.37016e+00 3.37016e+00 6.96149e+03 6.96147e+03 10.41 4.51
19 $ \mathtt{M10PI\_n}$ 682 3 3 4.05662e+00 4.05662e+00 7.53181e+03 7.53182e+03 3.56 1.36
20 $ \mathtt{M20PI\_n}$ 1182 3 3 3.87260e+00 3.87260e+00 5.06412e+03 5.06412e+03 4.03 1.73
21 $ \mathtt{M40PI\_n}$ 2182 3 3 3.81767e+00 3.81767e+00 5.07107e+03 5.07107e+03 6.11 2.85
22 $ \mathtt{M80PI\_n}$ 4182 3 3 3.80375e+00 3.80376e+00 5.07279e+03 5.07279e+03 11.28 5.19
23 $ \mathtt{bips98\_606}$ 7135 4 4 2.01956e+02 2.01956e+02 3.81763e+00 3.81763e+00 35.43 14.43
24 $ \mathtt{bips98\_1142}$ 9735 4 4 1.60427e+02 1.60427e+02 4.93005e+00 4.93005e+00 69.65 16.37
25 $ \mathtt{bips98\_1450}$ 11305 4 4 1.97389e+02 1.97389e+02 5.64575e+00 5.64650e+00 61.65 17.91
26 $ \mathtt{bips07\_1693}$ 13275 4 4 2.04168e+02 2.04168e+02 5.53766e+00 5.53767e+00 167.10 31.98
27 $ \mathtt{bips07\_1998}$ 15066 4 4 1.97064e+02 1.97064e+02 6.39968e+00 6.39879e+00 102.11 29.99
28 $ \mathtt{bips07\_2476}$ 16861 4 4 1.89579e+02 1.89579e+02 5.88971e+00 5.89023e+00 146.18 31.62
29 $ \mathtt{bips07\_3078}$ 21128 4 4 2.09445e+02 2.09445e+02 5.55792e+00 5.55839e+00 91.05 34.73
30 $ \mathtt{xingo\_afonso\_itaipu}$ 13250 1 1 4.05605e+00 4.05605e+00 1.09165e+00 1.09165e+00 39.24 16.80
31 $ \mathtt{mimo8x8\_system}$ 13309 8 8 5.34292e-02 5.34293e-02 1.03313e+00 1.03308e+00 78.47 23.25
32 $ \mathtt{mimo28x28\_system}$ 13251 28 28 1.18618e-01 1.18618e-01 1.07935e+00 1.07935e+00 85.36 35.45
33 $ \mathtt{mimo46x46\_system}$ 13250 46 46 2.05631e+02 2.05631e+02 1.07908e+00 1.07908e+00 115.91 49.13
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