American Journal of Applied Mathematics
Volume 7, Issue 1, February 2019, Pages: 1-4
Received: Jan. 3, 2019;
Accepted: Jan. 24, 2019;
Published: Feb. 25, 2019
Views 287 Downloads 120
Ziad Zahreddine, Mathematics Division, College of Science & Information Systems, Rafik Hariri University, Mechref, Damour, Lebanon
The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.
New Versions of the Hermite Bieler Theorem in Stability Contexts, American Journal of Applied Mathematics.
Vol. 7, No. 1,
2019, pp. 1-4.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
S. Elmadssia, K. Saadaoui, M. Benrejeb, PI controller design for time delay systems using an extension of the Hermite-Biehler theorem, Journal of Industrial Engineering Volume 2013, Article ID 813037, 6 pages.
G. Heinig, U. Jungnickel, On the Routh-Hurwitz and Schur-Cohn Problems for Matrix Polynomials and Generalized Bezoutians, Math. Nachr. 11G (1984), 185-196.
R. Hovstad, A short proof of continued fraction test for the stability of polynomials, Pro. Amer. Math. Soc., 105, No. 1 (1989), 76-79.
V. L. Kharitonov, The asymptotic stability of the equilibrium state of a family of systems of linear differential equations, Differentsial’nye Uravneniya 14 (1978), no. 11, 2086–2088.
V. Pivovarchik, H. Woracek, The square-transform of hermite-biehler functions. a geometric approach, Methods of Functional Analysis and Topology, Vol. 13 (2007), no. 2, pp. 187–200.
M. M. Postnikov, Stable polynomials, Moscow, Nauka, 1981 (in Russian).
B. Senol, C. Yeroglu, N. Tan, Analysis of fractional order polynomials using Hermite-Biehler theorem, International Conference on Fractional Differentiation and Its Applications 2014, 1-5, 2014.
L. B. Yang, P. B. Zhang, The real-rootedness of Eulerian polynomials via the Hermite–Biehler Theorem, Discrete Mathematics and Theoretical Computer Science proc., FPSAC’15, 2015, 465–474 Nancy, France.
Z. Zahreddine, On Positive Para-odd and Complex Discrete Reactance Functions, Journal of Interdisciplinary Mathematics, Taylor & Francis, Vol. 21, 2018 – Issue 1, 243-251.
Z. Zahreddine, Parallel Properties of Poles of Positive Functions and those of Discrete Reactance Functions, International Journal of Mathematical Analysis. Vol 11, 2017, no. 24, 1141 – 1150.
Z. Zahreddine, Alternative Approaches to the General Stability Problem, Conference in Mathematical Analysis & Applications, American University of Sharjah, 2003.
Z. Zahreddine, On the -stability of systems of differential equations in the Routh-Hurwitz and Schur-Cohn cases, Bulletin of the Belgian Mathematical Society, 3 (1996), 363-368.
Z. Zahreddine, An extension of the Routh array for the asymptotic stability of a system of differential equations with complex coefficients, Applicable Analysis, 49 (1993), 61-72.
Z. Zahreddine, Explicit relationships between Routh-Hurwitz and Schur-Cohn types of stability, Bulletin of the Irish Mathematical Society, 29 (1992), 49-54.