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, 61:092203 | Cite as

Achievable delay margin using LTI control for plants with unstable complex poles

  • Peijun Ju
  • Huanshui ZhangEmail author
Research Paper


We consider the achievable delay margin of a real rational and strictly proper plant, with unstable complex poles, by a linear time-invariant (LTI) controller. The delay margin is defined as the largest time delay such that, for any delay less than this value, the closed-loop stability is maintained. Drawing upon a frequency domain method, particularly a bilinear transform technique, we provide an upper bound of the delay margin, which requires computing the maximum of a one-variable function. Finally, the effectiveness of the theoretical results is demonstrated through a numerical example.


delay margin systems with time-delay time-invariant systems unstable complex poles frequency domain method 



This work was partially supported by Taishan Scholar Construction Engineering by Shandong Government and National Natural Science Foundation of China (Grant Nos. 61573220, 61573221, 61633014).


  1. 1.
    Krstic M. Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Boston: Birkhauser, 2009CrossRefzbMATHGoogle Scholar
  2. 2.
    Michiels W, Niculescu S I. Stability, Control and Computation of Time-delay Systems: An Eigenvalue Based Approach. 2nd ed. Philadelphia: SIAM Publications, 2014CrossRefzbMATHGoogle Scholar
  3. 3.
    Fridman E. Introdution to Time-delay Systems: Analysis and Control. London: Springer, 2014CrossRefGoogle Scholar
  4. 4.
    Richard J-P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 39: 1667–1694MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gu K Q, Niculescu S I. Survey on recent results in the stability and control of time-delay systems. J Dyn Syst Control, 2003, 125: 158–165CrossRefGoogle Scholar
  6. 6.
    Sipahi R, Niculescu S I, Abdallah C T, et al. Stability and stabilization of systems with time delay: limitations and opportunities. IEEE Contr Syst Mag, 2011, 31: 38–65MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davison D E, Miller D E. Determining the least upper bound on the achievable delay margin. In: Open Problems in Mathematical Systems and Control Theory. Blondel V D, Megretski A, eds. Princeton: Princeton University Press, 2004. 276–279Google Scholar
  8. 8.
    Middleton R H, Miller D E. On the achievable delay margin using LTI control for unstable plants. IEEE Trans Automat Control, 2007, 52: 1194–1207MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ju P J, Zhang H S. Further results on the achievable delay margin using LTI control. IEEE Trans Automat Control, 2016, 61: 3134–3139MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Qi T, Zhu J, Chen J. On delay radii and bounds of MIMO systems. Automatica, 2017, 77: 214–218MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Qi T, Zhu J, Chen J. Fundamental limits on uncertain delays: when is a delay system stabilizable by LTI controllers? IEEE Trans Automat Control, 2017, 62: 1314–1328MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Michiels W, Engelborghs K, Vansevenant P, et al. Continuous pole placement for delay equations. Automatica, 2002, 38: 747–761MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhu Y, Krstic M, Su H Y. Adaptive output feedback control for uncertain linear time-delay systems. IEEE Trans Automat Control, 2017, 62: 545–560MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gaudette D L, Miller D E. Adaptive stabilization of a class of time-varying systems with an uncertain delay. Math Control Signal, 2016, 28: 1–39MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gaudette D L, Miller D E. Stabilizing a SISO LTI plant with gain and delay margins as large as desired. IEEE Trans Automat Control, 2014, 59: 2324–2339MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mazenc F, Malisoff M, Niculescu S I. Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach. SIAM J Contr Optim, 2017, 55: 533–556MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vidyasagar M. Control System Synthesis: A Factorization Approach. Cambridge: MIT Press, 1985zbMATHGoogle Scholar
  18. 18.
    Rekasius Z V. A stability test for systems with delays. In: Proceedings of Joint Automatic Control Conference, San Francisco, 1980Google Scholar
  19. 19.
    Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Trans Automat Control, 2002, 47: 793–797MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gu K Q, Kharitonov V L, Chen J. Stability of Time-delay Systems. Boston: Birkhäuser, 2003CrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Control Science and EngineeringShandong UniversityJi’nan, ShandongChina
  2. 2.School of Mathematics and StatisticsTaishan UniversityTai’an, ShandongChina

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