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.
This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science Technology and Culture. The scientific responsibility rests with its author.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Y. Genin, “Euclid algorithm, orthogonal polynomials and generalized Routh-Hurwitz algorithm”, to be published in Linear Algebra Appl.
Y. Genin, “On polynomials nonnegative on the unit circle and related questions”, to be published in Linear Algebra Appl.
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.
P. Delsarte, Y. Genin and Y. Kamp, “Pseudo-Carathéodory functions and Hermitian Toeplitz matrices”, Philips J. Res., vol. 41, pp. 1–54, 1986.
M. Marden, Geometry of Polynomials. Providence R.I: American Math. Soc., 1966.
F.R. Gantmacher, The Theory of Matrices, Vol. II. New York: Chelsea, 1959.
V. Belevitch, Classical Network Theory. San Francisco: Holden-Day, 1968.
V.L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of linear differential equations”, Differential Equations, vol. 14, pp. 1483–1485, 1979.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Verlag Basel
About this paper
Cite this paper
Genin, Y.V. (1996). On the Cauchy index of a real rational function and the index theory of pseudo-lossless rational functions. In: Jeltsch, R., Mansour, M. (eds) Stability Theory. ISNM International Series of Numerical Mathematics, vol 121. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9208-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9208-7_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9945-1
Online ISBN: 978-3-0348-9208-7
eBook Packages: Springer Book Archive