Circuits, Systems, and Signal Processing

, Volume 35, Issue 5, pp 1545–1561 | Cite as

Exponential Stability Criteria for Positive Systems with Time-Varying Delay: A Delay Decomposition Technique

  • W. ElloumiEmail author
  • D. Mehdi
  • M. Chaabane
  • G. Hashim


This paper is concerned with exponential stability analysis for linear continuous positive systems with bounded time-varying delay. Our approach is based on Lyapunov–Krasovskii functional with delay decomposition. The main idea is to divide uniformly the discrete delay interval into multiple segments and then to choose proper functionals with different weighted matrices corresponding to different segments. The synthesis of our delay-dependent sufficient stability criteria is formulated in terms of linear matrix inequalities. The proposed results are shown to be less conservative compared to other results from the literature. Some illustrative examples are given to show the effectiveness of the obtained results.


Linear systems Positive systems Time delay Exponential stability 


  1. 1.
    M. Araki, Application of M-matrices to the stability problems of composite dynamical systems. J. Math. Anal. Appl. 52(2), 309–321 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J.J. Batzel, F. Kappel, Time delay in physiological systems: analyzing and modeling its impact. Math. Biosci. 234(2), 61–74 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    V. Chellaboina, W.M. Haddad, J. Ramakrishnan, J. Bailey, On monotonocity of solutions of non negative and compartmental dynamical systems with time delays, in Conference on Decision and Control, vol. 4, ed. by In Proc (USA, Hawaii, 2003), pp. 4008–4013Google Scholar
  4. 4.
    H.R. Feyzmahdavian, T. Charalambous, M. Johansson, On the rate of convergence of continuous-time linear positive systems with heterogeneous time-varying delays, in European Control Conference (ECC) (Switzerland, Zurich, 2013), pp. 3372–3377Google Scholar
  5. 5.
    K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings of the 39th IEEE Conference on Decision and Control, (Sydney, Australia, 2000), pp. 2805–2810Google Scholar
  6. 6.
    Q. Han, A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica 45(2), 517–524 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Hmamed, M.A. Rami, A. Benzaouia, F. Tadeo, Stabilization under constrained states and controls of positive systems with time delays. Mech. Syst. Signal Process. 18(2), 182–190 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Ilchmann, P.H.A. Ngoc, Stability and robust stability of positive Volterra systems. Int. J. Robust Nonlinear Control 22(6), 604–629 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S.T. Kaczorek, Stability of positive continuous-time linear systems with delays. Bull. Pol. Acad. Sci. Tech. Sci. 57(4), 395–398 (2009)MathSciNetGoogle Scholar
  10. 10.
    T. Kaczorek, Realization problem for positive linear systems with time delay. Math. Probl. Eng. 4, 455 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.H. Kim, Note on stability of linear systems with time-varying delay. Automatica 47(9), 2118–2121 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    P. Kokil, V.K.R. Kandanvli, H. Kar, Delay-partitioning approach to stability of linear discrete-time systems with interval-like time-varying delay. Int. J. Eng. Math. 2013, 291976 (2013). doi: 10.1155/2013/291976
  13. 13.
    X. Liu, W. Yu, L. Wang, Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans. Autom. Control 55(4), 1024–1028 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Y. Mao, H. Zhang, C. Dang, Stability analysis and constrained control of a class of fuzzy positive systems with delays using linear copositive Lyapunov functional. Circuits Syst. Signal Process. 31(5), 1863–1875 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    K. Mathiyalagan, R. Sakthivel, S.M. Anthoni, New stability criteria for stochastic Takagi–Sugeno fuzzy systems with time-varying delays. J. Dyn. Sys. Meas. Control. 136(2), 021013-1–021013-9 (2013)Google Scholar
  16. 16.
    W. Mitkowski, Dynamical properties of Metzler systems. Bull. Pol. Acad. Sci. Tech. Sci. 56(4), 309–312 (2008)Google Scholar
  17. 17.
    J.D. Murray, Murray, Mathematical Biology Part I: An Introduction, 3rd edn. (Springer, Berlin, 2002)Google Scholar
  18. 18.
    P. Ngoc, A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices. Appl. Math. Lett. 19(8), 747–751 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Ramachandran, Y.M. Ram, Stability boundaries of mechanical controlled system with time delay. Mech. Syst. Signal Process. 27(2), 523–533 (2012)CrossRefGoogle Scholar
  20. 20.
    M.A. Rami, U. Helmke, F. Tadeo, Positive observation problem for linear time-delay positive systems, in Control and Automation, 2007. MED’07. Mediterranean Conference, (Athens, 2007) pp. 1–6Google Scholar
  21. 21.
    R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results. IEEE Trans. Netw. 14(2), 616–629 (2006)CrossRefGoogle Scholar
  22. 22.
    H.P. Wang, X. Gu, L. Xie, L.S. Shieh, J.S.H. Tsai, Y. Zhang, Digital controller design for analog systems represented by multiple input–output time-delay transfer function matrices with long time delays. Circuits Syst. Signal Process. 31(5), 1653–1676 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. Xiang, Z. Xiang, Observer design of switched positive systems with time-varying delays. Circuits Syst. Signal Process. 32(5), 2171–2184 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Z. Zhang, H. Yang, Stability and Hopf bifurcation in a three-species food chain system with harvesting and two delays. J. Comput. Nonlinear Dyn. 9(2), 024501 (2013). doi: 10.1115/1.4025670
  25. 25.
    S. Zhu, Z. Li, C. Zhang, Exponential stability analysis for positive systems with delays, in Proceedings of the 30th Chinese Control Conference, (Yantai, China, 2011), pp. 1234–1239Google Scholar
  26. 26.
    S. Zhu, M. Meng, C. Zhang, Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization. J. Frankl. Inst. 350(3), 617–636 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    X. Zhu, G.H. Yang, New results of stability analysis for systems with time-varying delay. Int. J. Robust Nonlinear Control 20(5), 596–606 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Lab-STA, National School of Engineering of SfaxUniversity of SfaxSfaxTunisia
  2. 2.Université de PoitiersPoitiers Cedex 9France
  3. 3.Computer Science DepartmentCihan UniversityErbilIraq

Personalised recommendations