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Robust Schur Stability of a Polynomial Matrix Family

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

The problem of robust Schur stability of a polynomial matrix family is considered as that of discovering the structure of the stability domain in parameter space. The algorithms are proposed for establishing whether or not any given box in the parameter space belongs to this domain, and for finding the distance to instability from any internal point of the domain to its boundary. The treatment is performed in the ideology of analytical algorithm for elimination of variables and localization of zeros of algebraic systems. Some examples are given.

Supported by RFBR, project No. 17-29-04288.

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Notes

  1. 1.

    \({\mathcal P}(A_1,\dots ,A_K):=\sum _{j=1}^{K-1} {\mathcal P}(A_j,A_{j+1})\) where \( {\mathcal P}(A_j,A_{j+1}) :=1 \) if \( A_jA_{j+1}>0 \) and \( {\mathcal P}(A_j,A_{j+1}) :=0 \) if \( A_jA_{j+1}<0 \). \({\mathcal V}(A_1,\dots ,A_K) \) is defined similarly with \( {\mathcal V}(A_j,A_{j+1}):=1-{\mathcal P}(A_j,A_{j+1}) \).

  2. 2.

    We treat the matrix entries as rational fractions.

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Acknowledgments

The authors are grateful to Prof Evgenii Vorozhtzov and to the anonimous referees for valuable suggestions that helped to improve the quality of the paper.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, St. Petersburg State University, 7–9 Universitetskaya nab., St. Petersburg, 199034, Russia

    Elizaveta Kalinina, Yuri Smol’kin & Alexei Uteshev

Authors
  1. Elizaveta Kalinina
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  2. Yuri Smol’kin
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  3. Alexei Uteshev
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Corresponding author

Correspondence to Elizaveta Kalinina .

Editor information

Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Kalinina, E., Smol’kin, Y., Uteshev, A. (2019). Robust Schur Stability of a Polynomial Matrix Family. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_18

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_18

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