Advertisement

On the Cauchy index of a real rational function and the index theory of pseudo-lossless rational functions

  • Yves V. Genin
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)

Abstract

The index theory relative to rational pseudo-lossless functions has been shown to be an interesting substitute for the Cauchy index theory and the argument principle theorem to discuss polynomial zero location problems. The reasons underlying this fact are put into light by working out the algebraic relations between these two equivalent approaches. A new simple proof of Kharitonov’s theorem is proposed to further illustrate this issue.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Y. Genin, “Euclid algorithm, orthogonal polynomials and generalized Routh-Hurwitz algorithm”, to be published in Linear Algebra Appl.Google Scholar
  2. [2]
    Y. Genin, “On polynomials nonnegative on the unit circle and related questions”, to be published in Linear Algebra Appl.Google Scholar
  3. [3]
    P. Delsarte, Y. Genin and Y. Kamp, “Pseudo-lossless functions with application to the problem of locating the zeros of a polynomial”, IEEE Trans. Circuits and Systems, vol. CAS-32, pp. 373–381, 1985.Google Scholar
  4. [4]
    P. Delsarte, Y. Genin and Y. Kamp, “Pseudo-Carathéodory functions and Hermitian Toeplitz matrices”, Philips J. Res., vol. 41, pp. 1–54, 1986.MathSciNetzbMATHGoogle Scholar
  5. [5]
    M. Marden, Geometry of Polynomials. Providence R.I: American Math. Soc., 1966.zbMATHGoogle Scholar
  6. [6]
    F.R. Gantmacher, The Theory of Matrices, Vol. II. New York: Chelsea, 1959.Google Scholar
  7. [7]
    V. Belevitch, Classical Network Theory. San Francisco: Holden-Day, 1968.zbMATHGoogle Scholar
  8. [8]
    V.L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of linear differential equations”, Differential Equations, vol. 14, pp. 1483–1485, 1979.zbMATHGoogle Scholar
  9. [9]
    R.J. Minnichelli, J.J. Anagnost and C.A. Desoer, “An elementary proof of Kharitonov’s stability theorem with extensions”, IEEE Trans. Automatic Control, vol. AC-34, pp. 995–998, 1989.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Yves V. Genin
    • 1
  1. 1.CESAMEUniversité Catholique de Louvain, Bâtiment EulerLouvain-La-NeuveBelgium

Personalised recommendations