Internally Positive Representations and Stability Analysis of Linear Delay Systems with Multiple Time-Varying Delays

  • Francesco Conte
  • Vittorio De IuliisEmail author
  • Costanzo Manes
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)


This chapter introduces the Internally Positive Representation of linear systems with multiple time-varying state delays. The technique, previously introduced for the undelayed case, aims at building a positive representation of systems whose dynamics is, in general, indefinite in sign. As a consequence, stability criteria for positive time-delay systems can be exploited to analyze the stability of the original system. As a result, an easy-to-check sufficient condition for the delay-independent stability is provided, that is compared with some well known conditions available in the literature.


Positive delay systems Time-varying delays Internally positive representation (IPR) Stability analysis 



We would like to thank Alfredo Germani and Filippo Cacace for their encouragement and helpful suggestions in doing this work.


  1. 1.
    Ait Rami, M.: Positive Systems: Proceedings of the third Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009) Valencia, Spain, September 2-4, 2009, pp. 205–215. Springer, Berlin (2009)Google Scholar
  2. 2.
    Cacace, F., Farina, L., Germani, A., Manes, C.: Internally positive representation of a class of continuous time systems. IEEE Trans. Autom. Control 57(12), 3158–3163 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cacace, F., Germani, A., Manes, C.: Stable internally positive representations of continuous time systems. IEEE Trans. Autom. Control 59(4), 1048–1053 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cacace, F., Germani, A., Manes, C., Setola, R.: A new approach to the internally positive representation of linear MIMO systems. IEEE Trans. Autom. Control 57(1), 119–134 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, J., Latchman, H.A.: Frequency sweeping tests for stability independent of delay. IEEE Trans. Autom. Control 40(9), 1640–1645 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications, vol. 50. Wiley (2011)Google Scholar
  7. 7.
    Fridman, E.: Introduction to Time-Delay Systems. Birkhuser Basel (2014)Google Scholar
  8. 8.
    Germani, A., Manes, C., Palumbo, P.: State space representation of a class of MIMO systems via positive systems. In: 2007 46th IEEE Conference on Decision and Control, pp. 476–481. IEEE (2007)Google Scholar
  9. 9.
    Germani, A., Manes, C., Palumbo, P.: Representation of a class of MIMO systems via internally positive realization. Eur. J. Control 16(3), 291–304 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhuser Basel (2003)Google Scholar
  11. 11.
    Gu, K., Niculescu, S.I.: Survey on recent results in the stability and control of time-delay systems. J. Dyn. Syst. Meas. Control 125(2), 158–165 (2003)CrossRefGoogle Scholar
  12. 12.
    Haddad, W.M., Chellaboina, V.: Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst. Control Lett. 51(5), 355–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hale, J., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kaczorek, T.: Positive 1D and 2D Systems. Springer, London, UK (2001)zbMATHGoogle Scholar
  15. 15.
    Kaczorek, T.: Stability of positive continuous-time linear systems with delays. In: 2009 European Control Conference (ECC), pp. 1610–1613. IEEE (2009)Google Scholar
  16. 16.
    Liu, X., Lam, J.: Relationships between asymptotic stability and exponential stability of positive delay systems. Int. J. Gen. Syst. 42(2), 224–238 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, X., Yu, W., Wang, L.: Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans. Autom. Control 55(4), 1024–1028 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Luenberger, D.: Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley (1979)Google Scholar
  19. 19.
    Mazenc, F., Malisoff, M.: Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach. IEEE Trans. Autom. Control 61(3), 771–776 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Michiels, W., Niculescu, S.I.: Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach, vol. 27. Siam (2014)Google Scholar
  21. 21.
    Mori, T., Fukuma, N., Kuwahara, M.: Simple stability criteria for single and composite linear systems with time delays. Int. J. Control 34(6), 1175–1184 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mori, T., Fukuma, N., Kuwahara, M.: On an estimate of the decay rate for stable linear delay systems. Int. J. Control 36(1), 95–97 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ngoc, P.H.A.: Stability of positive differential systems with delay. IEEE Trans. Autom. Control 58(1), 203–209 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schoen, G.M.: Stability and stabilization of time-delay systems. Ph.D. thesis, Swiss Federal Institute of Technology, Zurich, Switzerland (1995).
  25. 25.
    Schoen, G.M., Geering, H.P.: On stability of time delay systems. In: Proceedings of the 31st Annual Allerton Conference on Communications Control and Computing, pp. 1058–1060. Monticello, Il (1993)Google Scholar
  26. 26.
    Wang, S.S., Lee, C.H., Hung, T.H.: New stability anlysis of system with multiple time delays. In: American Control Conference (ACC 1991), pp. 1703–1704 (1991)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Francesco Conte
    • 1
  • Vittorio De Iuliis
    • 2
    Email author
  • Costanzo Manes
    • 2
  1. 1.Dipartimento di Ingegneria Navale, Elettrica, Elettronica e delle TelecomunicazioniUniversità degli Studi di GenovaGenovaItaly
  2. 2.Dipartimento di Ingegneria e Scienze Dell’Informazione, e MatematicaUniversità degli Studi dell’AquilaCoppitoItaly

Personalised recommendations