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Robust Schur Stability of a Polynomial Matrix Family

  • Elizaveta KalininaEmail author
  • Yuri Smol’kin
  • Alexei Uteshev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

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.

Keywords

Matrix polynomials Robust schur stability Parameters Discriminant 

Notes

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.

References

  1. 1.
    Ackermann, J.: Robust Control: The Parameter Space Approach. Springer-Verlag, London (2002)CrossRefGoogle Scholar
  2. 2.
    Akritas, A.G.: Elements of Computer Algebra with Applications. Wiley, New York (1989)zbMATHGoogle Scholar
  3. 3.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer-Verlag, Heidelberg (2010)zbMATHGoogle Scholar
  4. 4.
    Bhattacharya, S.P., Chapellat, H., Keel, L.: Robust Control: The Parametric Approach. Prentice-Hall, New Jersey (1995)zbMATHGoogle Scholar
  5. 5.
    Bikker, P., Uteshev, A.Y.: On the Bézout construction of the resultant. J. Symb. Comput. 28, 45–88 (1999)CrossRefGoogle Scholar
  6. 6.
    Büyükköroǧlu, T., Çelebi, G., Dzhafarov, V.: On the robust stability of polynomial matrix families. Electron. J. Linear Algebra 30, 905–915 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chesi, G.: Exact robust stability analysis of uncertain systems with a scalar parameter via LMIs. Automatica 49, 1083–1086 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chilali, M., Gahinet, P., Apkarian, P.: Robust pole placement in LMI regions. IEEE Trans. Automat. Control. 44(12), 2257–2270 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dzhafarov, V., Büyükköroǧlu, T.: On nonsingularity of a polytope of matrices. Linear Algebra Appl. 429, 1174–1183 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2005)zbMATHGoogle Scholar
  12. 12.
    Faddeev, D.K., Faddeeva, V.N.: Computational Methods of Linear Algebra. Freeman, San Francisco (1963)zbMATHGoogle Scholar
  13. 13.
    Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29(6), 379–419 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gantmakher, F.R.: The Theory of Matrices, vol. I, II. Chelsea, New York (1959)Google Scholar
  15. 15.
    Henrion, O., Bachelier, O., Sebek, M.: \(D\)-stability of polynomial matrices. Int. J. Control 74(8), 845–856 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Joachimsthal, F.: Bemerkungen über den Sturmschen Satz. J. Reine Angew. Math. 48, 386–416 (1854)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jury, E.I.: Innors and Stability of Dynamic Systems. Wiley, New York (1974)Google Scholar
  18. 18.
    Kalinina, E.A.: Stability and D-stability of the family of real polynomials. Linear Algebra Appl. 438, 2635–2650 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Keel, L.H., Bhattacharya, S.P.: Robust stability via sign-definite decomposition. IEEE Trans. Automat. Control 56(1), 140–145 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krein, M.G., Naimark, M.A.: The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear Multilinear A. 10, 265–308 (1981)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kunz, E.: Introduction to Plane Algebraic Curves. Birkhäuser, Boston (2005)zbMATHGoogle Scholar
  22. 22.
    Lee, M. M.-D.: Factorization of Multivariate Polynomials. Technische Universität Kaiserslautern (2013)Google Scholar
  23. 23.
    Lenstra, A.K.: Factoring multivariate polynomials over finite fields. J. Comput. Syst. Sci. 30(2), 235–248 (1985)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mishra, B.: Algorithmic Algebra. Springer-Verlag, New York (1993)CrossRefGoogle Scholar
  25. 25.
    Schur, J.: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math., 147, 205–232 (1917), 148, 122–145 (1918)Google Scholar
  26. 26.
    Skalna, I.: Parametric Interval Algebraic Systems. Springer, Cham (2018)CrossRefGoogle Scholar
  27. 27.
    Uteshev, A.Y., Shulyak, S.G.: Hermite’s method of separation of solutions of systems of algebraic equations and its applications. Linear Algebra Appl. 177, 49–88 (1992)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Smith III, J.O.: Introduction to Digital Filters: with Audio Applications. W3K Publishing, USA (2007)Google Scholar
  29. 29.
    Uteshev, A.Y., Cherkasov, T.M.: The search for the maximum of a polynomial. J. Symb. Comput. 25(5), 87–618 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Uteshev, A.Y., Borovoi, I.I.: Solution of a rational interpolation problem using Hankel polynomials. Vestn. St.-Peterbg. Univ. Ser. 10 Prikl. Mat. Inform. Protsessy Upr. 4, 31–43 (2016). (in Russian)MathSciNetGoogle Scholar
  31. 31.
    Uteshev, A.Yu., Goncharova, M.V.: Metric problems for algebraic manifolds: analytical approach. In: Constructive Nonsmooth Analysis and Related Topics – CNSA 2017 Proceedings, 7974027 (2017)Google Scholar
  32. 32.
    Uteshev, A.Y., Yashina, M.V.: Metric problems for quadrics in multidimensional space. J. Symb. Comput. 68, 287–315 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wayland, H.: Expansion of determinantal equations into polynomial form. Quart. Appl. Math. 2(4), 277–305 (1945)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yli-Kaakinen, J., Saramaki, T.: Efficient recursive digital filters with variable magnitude characteristics. In: Proceedings of 7th Nordic Signal Processing Symposium – NORSIG 2006, pp. 30–33 (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Applied MathematicsSt. Petersburg State UniversitySt. PetersburgRussia

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