Circuits, Systems, and Signal Processing

, Volume 32, Issue 1, pp 61–81

Stability Analysis and Synthesis of Discrete Impulsive Switched Systems with Time-Varying Delays and Parameter Uncertainty

Article

Abstract

This paper studies the problem of stability and stabilization for discrete impulsive switched systems with time-varying delays and norm-bounded parameter uncertainty. By using the Lyapunov–Krasovskii functional technique and the method of linear matrix inequalities (LMIs), some delay-dependent criteria on asymptotic stability are established. A stabilization condition using feedback control is formulated to stabilize the closed-loop system. Some numerical examples are given to illustrate the main results.

Keywords

Delay-dependent Lyapunov–Krasovskii functional Discrete impulsive switched system 

References

  1. 1.
    M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    E. Fridman, U. Shaked, Delay-dependent stability and H-infinity control: constant and time-varying delays. Int. J. Control 76(1), 48–60 (2003) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    E. Fridman, New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett. 43(4), 309–319 (2001) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Y. He, G. Liu, D. Rees, New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Trans. Neural Netw. 18(1), 310–314 (2007) MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.H. Kim, E.T. Jeung, H.B. Park, Robust control for parameter uncertain delay systems in state and control input. Automatica 32(9), 1337–1339 (1996) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H. Kokame, H. Kobayashi, T. Mori, Robust H∞ performance for linear delay-differential systems with time-varying uncertainties. IEEE Trans. Autom. Control 43(2), 223–226 (1998) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    V. Lakshmikantham, X. Liu, Impulsive hybrid systems and stability theory. Dyn. Syst. Appl. 7, 1–10 (1998) MathSciNetMATHGoogle Scholar
  8. 8.
    V. Lakshmikantham, D. Bauinov, P.S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 (World Scientific, Singapore, 1989) CrossRefMATHGoogle Scholar
  9. 9.
    D. Liberzon, Switching in Systems and Control (Springer, Berlin, 2003) CrossRefMATHGoogle Scholar
  10. 10.
    J. Liu, X. Liu, W.C. Xie, Delay-dependent robust control for uncertain switched systems with time-delay. Nonlinear Anal. Hybrid Syst. 2(1), 81–95 (2008) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    X. Liu, Impulsive stabilization of nonlinear systems. IMA J. Math. Control Inf. 10(1), 11–19 (1993) CrossRefMATHGoogle Scholar
  12. 12.
    X. Liu, K. Rohlf, Impulsive control of a Lotka–Volterra system. IMA J. Math. Control Inf. 15(3), 269–284 (1998) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    X.G. Liu, R.R. Martin, M. Wu, M.L. Tang, Delay-dependent robust stabilisation of discrete-time systems with time-varying delay, in Control Theory and Applications, IEE Proceedings, vol. 153 (IET, Stevenage, 2006), pp. 689–702 Google Scholar
  14. 14.
    M.S. Mahmoud, N.F. Al-Muthairi, Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties. IEEE Trans. Autom. Control 39(10), 2135–2139 (1994) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Y.S. Moon, P. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems. Int. J. Control 74(14), 1447–1455 (2001) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    S.I. Niculescu, E. Verriest, L. Dugard, J.M. Dion, Stability and robust stability of time-delay systems: A guided tour, in Stability and Control of Time-delay Systems (1998), pp. 1–71 CrossRefGoogle Scholar
  17. 17.
    L. Xie, C.E. De Souza, Robust stabilization and disturbance attenuation for uncertain delay systems, in 2nd European Control Conference, Groningen (1993), pp. 667–672 Google Scholar
  18. 18.
    H. Xu, H. Su, L. Xiao, J. Chu, Delay-dependent robust stabilizability for uncertain discrete time-delay systems, in Intelligent Control and Automation 2006 (WCICA 2006), vol. 1 (IEEE Press, New York, 2006), pp. 2239–2243. CrossRefGoogle Scholar
  19. 19.
    H. Xu, X. Liu, K.L. Teo, Delay independent stability criteria of impulsive switched systems with time-invariant delays. Math. Comput. Model. 47(3–4), 372–379 (2008) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    H. Xu, X. Liu, K.L. Teo, A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays. Nonlinear Anal. Hybrid Syst. 2(1), 38–50 (2008) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    C. Yang, W. Zhu, Stability analysis of impulsive switched systems with time delays. Math. Comput. Model. 50(7), 1188–1194 (2009) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Z. Zhang, X. Liu, Robust stability of uncertain discrete impulsive switching systems. Comput. Math. Appl. 58(2), 380–389 (2009) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Statistics and Applied MathematicsHubei University of EconomicsWuhanChina
  2. 2.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations