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

  • * Corresponding author: Matthias Voigt

    * 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

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  • 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.

    Mathematics Subject Classification: Primary: 93A15; Secondary: 15A22, 65F15, 93C05.

    Citation:

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  • Figure 1.  Eigenvalues of a skew-Hamiltonian/Hamiltonian matrix pencil $\mathcal{H}_\gamma(s)$

    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.

    Figure 3.  linorm output for the bips07 3078 example

    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.

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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