Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

  • Phan T. NamEmail author
  • Vu N. Phat


This paper proposes new sufficient conditions for the exponential stability and stabilization of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities (LMIs) and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponential stability rate of the solution. Numerical examples illustrating the conditions are given.


Polytopic time-delay system Exponential stability Stabilization Parameter-dependent Lyapunov function Linear matrix inequalities 


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  1. [1]
    E. N. Chukwu. Stability and Time-Optimal Control of Hereditary Systems[M]. Boston, San Diego, New York: Academic Press, Inc., 1992.Google Scholar
  2. [2]
    J. J. DaCunha. Stability for time-varying linear dynamical systems on time scale[J]. Journal of Computer Application Mathematica, 2000, 176(1): 381–410.MathSciNetGoogle Scholar
  3. [3]
    V. L. Kharitonov, D. Hinrichsen. Exponential estimates for time-delay systems[J]. Systems and Control Letters, 2004, 53(2): 395–405.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    V. B. Kolmanovskii, S. Niculescu, J. P. Richard. On the Lyapunov-Krasovskii functionals for stability analysis of linear delay systems[J]. International Journal of Control, 1999, 72(2): 374–384.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    V. N. Phat, S. Pairote. Global stabilization of linear periodically time-varying switched systems via matrix inequalities[J]. Journal of Control Theory and Applications, 2006, 4(1): 24–29.CrossRefGoogle Scholar
  6. [6]
    V. N. Phat, A. V. Savkin. Robust state estimation for a class of linear uncertain time-delay systems[J]. Systems & Control Letters, 2002, 47(2): 237–245.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    V. N. Phat. Weak asymptotic stabilizability of discrete-time inclusions given by set-valued operators[J]. Journal of Mathematical Analysis Application, 1996, 202(2): 353–369.MathSciNetGoogle Scholar
  8. [8]
    V. N. Phat, P. T. Nam. Exponential stability criteria of linear non-autonomous systems with multiple delays[J]. Electronic Journal of Differencial Equations, 2005, 2005(58): 1–9.Google Scholar
  9. [9]
    T. Yoshizawa. Stability Theory by Lyapunov’s Second Method[M]. Tokyo: Publication of the Mathematic Society, 1966.Google Scholar
  10. [10]
    S. Boyd, L. El Ghaou, E. Feron, et al. Linear matrix inequalities in system and control theory[M]//Studies in Applied Mathematics, Philadenphia: SIAM, 1994.Google Scholar
  11. [11]
    P. Colaneri, J. C. Geromel. Parameter dependent Lyapunov function for time-varying polytopic systems[C]// Proceedings of the American Control Conference. Portland OR: IEEE, 2005: 604–608.CrossRefGoogle Scholar
  12. [12]
    J. Daafouz, J. Bernusson. Pararmeter dependent Lyapunov functions for discrete-time systems with time varying parametric uncertaintiey[J]. Systems & Control Letters, 2001, 43(2): 355–359.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    E. Feron, P. Apkarian, P. Gahinet. Analysis and synthesis of robust control systems via parameter dependent Lyapunov functions[J]. IEEE Transactions on Automatic Control, 1996, 41(7): 1041–1046.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    T. Mori, H. Kokame. A parameter-dependent Lyapunov function for a polytope of matrices[J]. IEEE Transactions on Automatic Control, 2000, 45(8): 1516–1519.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    A. G. Spark. Analysis of affinely parameter-varying systems using parameter dependent Lyapunov functions[C]//Proceedings Conference on Decision and Control. California, USA: IEEE, 997: 990–991.Google Scholar
  16. [16]
    Y. Jia. Alternative proofs for inproved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty: a predective approach[J]. IEEE Transactions on Automatic Control, 2003, 48(10): 1413–1416.Google Scholar
  17. [17]
    J. Bernusson, P. L. Peres, J. C. Geromel. A linear programming oriented procedure for quadratic stabilization of uncertain systems[J]. Systems & Control Letters, 1989, 13(1): 65–72.CrossRefMathSciNetGoogle Scholar
  18. [18]
    D. C. Ramos, P. L. D. Peres. An LMI condition for the robust stability of uncertain continuous time linear systems[J]. IEEE Transactions on Automatic Control, 2002, 47(4): 675–678.CrossRefMathSciNetGoogle Scholar
  19. [19]
    K. Tanaka, T. Hori, H. O. Wang. A multiple Lyapunov function approach to stabilization of fuzzy control systems[J]. IEEE Transactions on Automatic Control, 2003, 11(4): 582–589.Google Scholar
  20. [20]
    Y. He, M. Wu, J. She, et al. Parameter-dependent Lyapunov functional for stability fo time-delay systems with polytopic-type uncertainties[J]. IEEE Transactions on Automatic Control, 2004, 49(5): 828–832.CrossRefMathSciNetGoogle Scholar
  21. [21]
    Q. Xia, M. Jia. Robust stability functional of state delayed systems with polytopic type uncertainties via parameter dependent Lyapunov functions[J]. International Journal of Control, 2002, 75(16): 1427–1434.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    S. Mondie, V. L. Kharitonov. Exponential estimates for retarded time-delay systems: an LMI approach[J]. IEEE Transactions on Automatic Control, 2005, 50(2): 268–273.CrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Board of Control Theory and Applications, South China University of Technology and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsQuynhon UniversityQuynhonVietnam
  2. 2.Institute of MathematicsHanoiVietnam

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