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A structured pseudospectral method for \(\mathcal {H}_\infty \)-norm computation of large-scale descriptor systems

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Abstract

In this paper, we discuss the problem of computing the \({\mathcal {H}}_\infty \)-norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the \({\mathcal {H}}_\infty \)-norm and the structured complex stability radius of a corresponding matrix pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that, we use a new fast iterative scheme which is based on certain rank-1 perturbations of a matrix pencil. Finally, we analyze the performance of our algorithm by using real-world examples. In particular we compare our method with different other algorithms including a recently and independently derived method from Guglielmi, Gürbüzbalaban and Overton.

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Notes

  1. http://www.mpi-magdeburg.mpg.de/mpcsc/software/infnorm/.

  2. http://sites.google.com/site/rommes/software.

  3. http://www.slicot.org/index.php?site=benchmodred.

  4. http://www.slicot.org/.

  5. http://cims.nyu.edu/~mert/software/hinfinity.html.

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Acknowledgments

We thank Volker Mehrmann from TU Berlin for pointing at the difficulties arising if the perturbations make the transfer functions improper or not well-defined. We gratefully acknowledge the work done by our former intern Maximilian Bremer from the University of Texas at Austin by implementing a MATLAB code for the visualization of structured pseudospectra. Furthermore, we thank the two anonymous reviewers for their valuable and constructive comments that helped to significantly improve the quality of the paper.

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

    Peter Benner & Matthias Voigt

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  1. Peter Benner
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Correspondence to Matthias Voigt.

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Benner, P., Voigt, M. A structured pseudospectral method for \(\mathcal {H}_\infty \)-norm computation of large-scale descriptor systems. Math. Control Signals Syst. 26, 303–338 (2014). https://doi.org/10.1007/s00498-013-0121-7

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