New Versions of the Hermite Bieler Theorem in Stability Contexts

Ziad Zahreddine

doi:doi:10.11648/j.ajam.20190701.11

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New Versions of the Hermite Bieler Theorem in Stability Contexts

Ziad Zahreddine

Published in American Journal of Applied Mathematics (Volume 7, Issue 1)

Received: 3 January 2019 Accepted: 24 January 2019 Published: 25 February 2019

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.ajam.20190701.11
Page(s)	1-4
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Hermite-Bieler Theorem, Routh-Hurwitz Criterion, Schur-Cohn Criterion, Stability Analysis, Positive Para-Odd Functions, Conformal Mappings

References

[1]	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.
[2]	G. Heinig, U. Jungnickel, On the Routh-Hurwitz and Schur-Cohn Problems for Matrix Polynomials and Generalized Bezoutians, Math. Nachr. 11G (1984), 185-196.
[3]	R. Hovstad, A short proof of continued fraction test for the stability of polynomials, Pro. Amer. Math. Soc., 105, No. 1 (1989), 76-79.
[4]	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.
[5]	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.
[6]	M. M. Postnikov, Stable polynomials, Moscow, Nauka, 1981 (in Russian).
[7]	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.
[8]	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.
[9]	Z. Zahreddine, On Positive Para-odd and Complex Discrete Reactance Functions, Journal of Interdisciplinary Mathematics, Taylor & Francis, Vol. 21, 2018 – Issue 1, 243-251.
[10]	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.
[11]	Z. Zahreddine, Alternative Approaches to the General Stability Problem, Conference in Mathematical Analysis & Applications, American University of Sharjah, 2003.
[12]	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.
[13]	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.
[14]	Z. Zahreddine, Explicit relationships between Routh-Hurwitz and Schur-Cohn types of stability, Bulletin of the Irish Mathematical Society, 29 (1992), 49-54.

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APA Style

Ziad Zahreddine. (2019). New Versions of the Hermite Bieler Theorem in Stability Contexts. American Journal of Applied Mathematics, 7(1), 1-4. https://doi.org/10.11648/j.ajam.20190701.11

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ACS Style

Ziad Zahreddine. New Versions of the Hermite Bieler Theorem in Stability Contexts. Am. J. Appl. Math. 2019, 7(1), 1-4. doi: 10.11648/j.ajam.20190701.11

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AMA Style

Ziad Zahreddine. New Versions of the Hermite Bieler Theorem in Stability Contexts. Am J Appl Math. 2019;7(1):1-4. doi: 10.11648/j.ajam.20190701.11

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@article{10.11648/j.ajam.20190701.11,
  author = {Ziad Zahreddine},
  title = {New Versions of the Hermite Bieler Theorem in Stability Contexts},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {1-4},
  doi = {10.11648/j.ajam.20190701.11},
  url = {https://doi.org/10.11648/j.ajam.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.11},
  abstract = {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.},
 year = {2019}
}

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TY  - JOUR
T1  - New Versions of the Hermite Bieler Theorem in Stability Contexts
AU  - Ziad Zahreddine
Y1  - 2019/02/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190701.11
DO  - 10.11648/j.ajam.20190701.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
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EP  - 4
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190701.11
AB  - 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.
VL  - 7
IS  - 1
ER  -

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Author Information

Ziad Zahreddine

Mathematics Division, College of Science & Information Systems, Rafik Hariri University, Mechref, Damour, Lebanon

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APA Style

Ziad Zahreddine. (2019). New Versions of the Hermite Bieler Theorem in Stability Contexts. American Journal of Applied Mathematics, 7(1), 1-4. https://doi.org/10.11648/j.ajam.20190701.11

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ACS Style

Ziad Zahreddine. New Versions of the Hermite Bieler Theorem in Stability Contexts. Am. J. Appl. Math. 2019, 7(1), 1-4. doi: 10.11648/j.ajam.20190701.11

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AMA Style

Ziad Zahreddine. New Versions of the Hermite Bieler Theorem in Stability Contexts. Am J Appl Math. 2019;7(1):1-4. doi: 10.11648/j.ajam.20190701.11

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@article{10.11648/j.ajam.20190701.11,
  author = {Ziad Zahreddine},
  title = {New Versions of the Hermite Bieler Theorem in Stability Contexts},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {1-4},
  doi = {10.11648/j.ajam.20190701.11},
  url = {https://doi.org/10.11648/j.ajam.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.11},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - New Versions of the Hermite Bieler Theorem in Stability Contexts
AU  - Ziad Zahreddine
Y1  - 2019/02/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190701.11
DO  - 10.11648/j.ajam.20190701.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 1
EP  - 4
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190701.11
AB  - 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.
VL  - 7
IS  - 1
ER  -

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