International E-publication: Publish Projects, Dissertation, Theses, Books, Souvenir, Conference Proceeding with ISBN.  International E-Bulletin: Information/News regarding: Academics and Research

Time varying delay systems: a survey

Author Affiliations

  • 1Department of ETC, Bhilai Institute of Technology, Durg, CG, India

Res. J. Engineering Sci., Volume 6, Issue (8), Pages 24-28, September,26 (2017)


The development of the hardware systems has incurred various types of delays such as processing and transmission delays. Such delay may be due to the effect of tolerances of electronic components which were used while developing the system. Such time delay parameters must be implemented in the transfer function of the system so as to identify the correct cause of dynamic behavior of the system which in turn affects the stability of the system. For the development of the accurate system it is required to consider the condition for the global robust asymptotic stability. Criteria for verifying robust stability are formulated as feasibility problems over a set of frequency dependent linear matrix inequalities. The criteria can be equivalently formulated as Semi-Definite Programs (SDP) using Kalman-Yakubovich-Popov lemma. Therefore, checking robust stability can be performed in a computationally efficient fashion. The Lyapunov-Krasovskii approach is definitely the most popular method to address this issue and many results have proposed new functionals and enhanced techniques for deriving less conservative stability conditions. The paper surveys the techniques used for developing the stable system from the various literatures published recently and draws the result that which method is best to develop a reliable and stable systems.


  1. Kolmanovskii V.B. and Myshkis A. (1999)., Introduction to the Theory and Applications of Functional Differential Equations., Kluwer Academic Publishers.
  2. Bushnell L.G. (2001)., Networks and control., IEEE Control Systems Magazine, 21(1), 22-23.
  3. He Y., Wang Q.G., Xie L. and Lin C. (2007)., Further improvement of free-weighting matrices technique for systems with time-varying delay., IEEE Trans. on Automat. Control, 52(2), 293-299.
  4. Yue D., Han Q.-L. and Lam J. (2008)., Robust H1 Control and Filtering of Networked Control Systems., Springer London, 121-151.
  5. Fridman E. and Shaked U. (2006)., Input-output approach to stability and l2-gain analysis of systems with time-varying delays., Systems & Control Letters, 55, 1041-1053.
  6. Briat C. (2008)., Robust Control and Observation of LPV Time-Delay Systems., PhD thesis, INP-Grenoble.
  7. Shao H. and Han Q.L. (2012)., Less conservative delay dependent stability criteria for linear systems with interval time-varying delays., International Journal of Systems Science, 43(5), 894-902.
  8. Sun J., Liu G.P., Chen J. and Rees D. (2010)., Improved delay-range-dependent stability criteria for linear systems with time-varying delays., Automatica, 46(2), 466-470.
  9. Sun Yeong-jeu (2007)., Stability criterion for a class of descriptor systems with discrete and distributed time delays., Asian Journal Of Control, 33, 986-993.
  10. Ariba Y. and Gouaisbaut F. (2009)., An augmented model for robust stability analysis of time-varying delay systems., Int. J. Control, 82, 1616-1626.
  11. Du Z., Zhang Q. and Liu L. (2011)., New delay dependent robust stability of discrete singular systems with time-varying delay., Asian Journal of Control, 13, 136-147.
  12. Kao C.Y. and Rantzer A. (2005)., Robust stability analysis of linear systems with time-varying delays., In 16th IFAC World Congress, Prague, Czech Republic.
  13. Ariba Y., Gouaisbaut F. and Peaucelle D. (2008)., Stability analysis of time-varying delay systems in quadratic separation framework., In TheInternational conference on mathematical problems in engineering, aerospace and sciences(ICNPAA’08).
  14. Kao C.-Y. and Rantzer A. (2007)., Stability analysisof systems with uncertain time-varying delays., Automatica, 43(6), 959-970.
  15. Kim J.H. (2001)., Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty., IEEE Trans. On Automat. Control, 46(5), 789-792.
  16. Shao H. (2009)., New delay-dependent stability criteria for systems with interval delay., Automatica, 45(3), 744-749.
  17. Ariba Y. and Gouaisbaut F. (2009)., Input-output framework for robust stability of time-varying delay systems., In the 48th IEEE Conferenceon Decision and Control, Shanghai, China.
  18. Peaucelle D., Arzelier D., Henrion D. and Gouaisbaut F. (2007)., Quadratic separation for feedback connection of an uncertain matrix and an implicitlinear transformation., Automatica, 43(5),795-804.
  19. Iwasaki T. and Hara S. (1998)., Well-posedness offeedback systems: insights into exact robustnessanalysis and approximate computations., IEEE Trans. on Automat. Control, 43(5), 619-630.
  20. Gouaisbaut F. and Peaucelle D. (2007)., Robust stability of time-delay systems with interval delays., In 46th IEEE Conference on Decision and Control, New Orleans, USA.
  21. Fridman E. and Shaked U. (2002)., An improved stabilization method for linear time-delay systems., IEEE Trans. on Automat. Control, 47(11), 1931-1937.
  22. Ariba Y., Gouaisbaut F. and Johansson K.H. (2010)., Stability interval for time-varying delay systems., In the 49th IEEE Conference on Decision and Control (CDC’10), Atlanta, USA, 1017-1022.
  23. Sipahi R., Niculescu S., Abdallah C.T., Michiels W. and Keqin Gu. (2011)., Stability and stabilization of systems with time delay., Control Systems Magazine, IEEE, 31(1), 38-65.
  24. Gu K., Kharitonov V.L. and Chen J. (2003)., Stability of Time-Delay Systems., Birkh¨auser Boston, Control engineering.
  25. Seuret A. (2009)., Lyapunov-Krasovskii functional parameterized with polynomials., In the 6th IFAC Symposium on Robust Control Design, Haifa, Israel, 214-219.
  26. Sun J., Liu G.P., Chen J. and Rees D. (2010)., Improved delay-range-dependent stability criteria for linearsystems with time-varying delays., Automatica, 46(2), 466-470.
  27. Safonov M.G. (1980)., Stability and Robustness of Multivariable Feedback Systems., Signal Processing, Optimization, and Control. MIT Press.
  28. Wu M., He Y., She J.H. and Liu G.P. (2004)., Delay dependent criteria for robust stability of time varying delay systems., Automatica, 40, 1435-1439.
  29. Zhang J., Knopse C.R. and Tsiotras P. (2001)., Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions., IEEE Trans. on Automat. Control, 46(3), 482-486.
  30. Bliman P.-A. (2002)., Lyapunov equation for the stability of linear delay systems of retarded and neutral type., IEEE Trans. on Automat. Control, 47, 327-335.
  31. Ebihara Y., Peaucelle D., Arzelier D. and Hagiwara T. (2005)., Robust performance analysis oflinear time-invariant uncertain systems by taking higher-order time-derivatives of the states., In 44thIEEE Conference on Decision and Control and the European Control Conference, Seville, Spain.
  32. He Y., Wang Q.G., Lin C. and Wu M. (2007)., Delay range-dependent stability for systems with time varying delay., Automatica, 43, 371-376. 30. S. Xu and J. Lam. A survey of linear matrix inequality techniques in stability analysis of delay systems. International Journal of Systems Science, 39(12):1095–1113, December 2008.