Asymptotic Analysis of Multiple Characteristics Roots for Quasi-polynomials of Retarded-Type

  • A. Martínez-González
  • S.-I. Niculescu
  • J. Chen
  • C. F. Méndez-BarriosEmail author
  • J. G. Romero
  • G. Mejía-Rodríguez
Part of the Advances in Delays and Dynamics book series (ADVSDD, volume 10)


In this chapter, the analysis of the behavior of multiple critical roots with respect to the delay parameters for a class of quasi-polynomials is addressed. The analysis is based on the construction of the so-called Weierstrass polynomial. Several numerical examples encountered in the control literature are considered to illustrate the proposed approach.



The research of J. Chen was supported in part by the Hong Kong RGC under Projects F-HK006/11T and CityU 11260016; the work of A. Martínez-González was financially supported by CONACyT, Mexico.


  1. 1.
    Abdallah, C.T., Dorato, P., Benitez-Read, J., Byrne, R.: Delayed positive feedback can stabilize oscillatory systems. In: Proceedings of the American Control Conference, pp. 3106–3107. San Francisco, USA (1993)Google Scholar
  2. 2.
    Boussaada, I., Niculescu, S.-I.: Characterizing the codimension of zero singularities for time-delay systems: a link with Vandermonde and Birkhoff incidence matrices. Acta Appl. Math. 145(1), 47–88 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boussaada, I., Niculescu, S.-I.: Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: a Vandermonde-based approach. IEEE Trans. Autom. Control 61(6), 1601–1606 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cai, T., Zhang, H., Wang, B., Yang, F.: The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux-Newton diagram. Int. J. Syst. Sci. 45(5), 1145–1155 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casas-Alvero, E.: Singularities of Plane Curves. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  6. 6.
    Chen, J., Gu, C., Nett, C.N.: A new method for computing delay margins for stability of linear delay systems. Syst. Control Lett. 26, 107–117 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, J., Fu, P., Niculescu, S.-I., Guan, Z.: An eigenvalue perturbation approach to stability analysis, part I: eigenvalue series of matrix operators. SIAM J. Control Optim. 48(8), 5564–5582 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, J., Fu, P., Niculescu, S.-I., Guan, Z.: An eigenvalue perturbation approach to stability analysis, part II: when will zeros of time-delay systems cross imaginary axis? SIAM J. Control Optim. 48(8), 5583–5605 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, J., Fu, P., Méndez-Barrios, C.-F., Niculescu, S.-I., Zhang, H.: Stability analysis of polynomially dependent systems by eigenvalue perturbation. IEEE Trans. Autom. Control 1–8 (2017)Google Scholar
  10. 10.
    Grigoryan, S.S., Mailybaev, A.A.: On the Weierstrass preparation theorem. Math. Notes 69(2), 170–174 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  12. 12.
    Hale, J.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefGoogle Scholar
  13. 13.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Elsevier Science Publishers, London (1990)zbMATHGoogle Scholar
  14. 14.
    Hryniv, R., Lancaster, P.: On the perturbation of analytic matrix functions. Integr. Equ. Oper. Theory 34, 325–338 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jarlebring, E., Michiels, W.: Invariace properties in the root sensitivity of time-delay systems with double imaginary roots. Automatica 46, 1112–1115 (2010)CrossRefGoogle Scholar
  16. 16.
    Kharitonov, V.L., Niculescu, S.-I., Moreno, J., Michiels, W.: Static output feedback stabilization: necessary conditions for multiple delay controllers. IEEE Trans. Autom. Control 50(1), 82–86 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Krantz, G.S., Parks, R.H.: The Implicit Function Theorem. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  18. 18.
    Langer, H., Najman, B., Veselić, K.: Perturbation of the eigenvalues of quadratic matrix polynomials. SIAM J. Matrix Anal. Appl. 13(2), 474–489 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, X.-G., Niculescu, S.-I., Cela, A., Wang, H.H., Cai, T.-Y.: On computing puiseux series for multiple imaginary characteristic roots of LTI systems with commensurate delays. IEEE Trans. Autom. Control 58(5), 1338–1343 (2013)CrossRefGoogle Scholar
  20. 20.
    Mazenc, F., Mondié, S., Niculescu, S.-I.: Global asymptotic stabilitation for chains of integrators with a delay in the input. IEEE Trans. Autom. Control 48(1), 57–63 (2003)CrossRefGoogle Scholar
  21. 21.
    Méndez-Barrios, C.-F., Niculescu, S.-I., Chen, J., Cárdenas-Galindo, V.M.: On the Weierstrass preparation theorem with applications to the asymptotic analysis of characteristics roots of time-delay systems. In: Proceedings of the 12th IFAC Workshop on Time Delay Systems (TDS), pp. 251–256. Ann Arbor, USA (2015)Google Scholar
  22. 22.
    Michiels, W., Niculescu, S.-I.: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  23. 23.
    Michiels, W., Niculescu, S.I.: Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach. SIAM, Philadelphia (2014)CrossRefGoogle Scholar
  24. 24.
    Niculescu, S.-I., Michiels, W.: Stabilizing a chain of integrators using multiple delays. IEEE Trans. Autom. Control 49(5), 802–807 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ramírez, A., Garrido, R., Mondié, S.: Velocity control of servo systems using an integral retarded algorithm. ISA Trans. Elsevier 58, 357–366 (2015)CrossRefGoogle Scholar
  26. 26.
    Ramírez, A., Mondié, S., Garrido, R., Sipahi, R.: Design of Proportional-Integral-Retarded (PIR) controllers for second-order LTI systems. IEEE Trans. Autom. Control 61, 1688–1693 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ramírez, A., Sipahi, R., Mondié, S., Garrido, R.: An analytical approach to tuning of delay-based controllers for LTI-SISO systems. SIAM J. Control Optim. 55(1), 397–412 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shabat, B.V.: Introduction to Complex Analysis. Part II: Functions of Several Variables. American Mathematical Society, Providence (1992)CrossRefGoogle Scholar
  29. 29.
    Sipahi, R., Niculescu, S.-I., Abdallah, C.-T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. IEEE Control Syst. 31(1), 38–65 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Vainberg, M.M., Trenogin, V.A.: Theory of Branching of Solutions of Non-linear Equations. Noordhoff International Publishing, Leyden (1974)zbMATHGoogle Scholar
  31. 31.
    Wall, C.T.C.: Singular Points of Plane Curves. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • A. Martínez-González
    • 1
  • S.-I. Niculescu
    • 2
  • J. Chen
    • 3
  • C. F. Méndez-Barrios
    • 1
    Email author
  • J. G. Romero
    • 4
  • G. Mejía-Rodríguez
    • 1
  1. 1.Universidad Autónoma de San Luis Potosí (UASLP), Facultad de Ingeniería, Dr. Manuel Nava No. 8, Zona UniversitariaSan Luis Potosí, S.L.P.Mexico
  2. 2.Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS-Supélec 3, rue Joliot CurieGif-sur-YvetteFrance
  3. 3.Department of Electronic EngineeringCity University of Hong KongHong KongChina
  4. 4.Departamento Académico de Sistemas Digitales ITAM, Río Hondo 1Ciudad de MéxicoMexico

Personalised recommendations