Mathematics of Control, Signals, and Systems

, Volume 19, Issue 3, pp 183–205

Sufficient conditions for robustness of \(\mathcal{K}\mathcal{L}\) -stability for difference inclusions

Original Article

Abstract

Difference inclusions arise naturally in the study of discrete-time or sampled-data systems. We develop two novel sufficient conditions for robustness of a stability property referred to as \(\mathcal{K}\mathcal{L}\) -stability with respect to an arbitrary measure; i.e., where a continuous positive definite function of the solutions satisfies a class-\(\mathcal{K}\mathcal{L}\) estimate of time and the continuous positive definite function of the initial condition.

Keywords

Difference inclusions Stability with respect to two measures Robust stability Compact attractors 

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References

  1. 1.
    Agarwal RP (2000) Difference equations and inequalities: theory, methods, and applications, 2 edn. Marcel DekkerGoogle Scholar
  2. 2.
    Clarke FH, Ledyaev YuS and Stern RJ (1998). Asymptotic stability and smooth Lyapunov functions. J Differ Equ 149: 69–114 MATHCrossRefGoogle Scholar
  3. 3.
    Hoppensteadt FC (1966). Singular perturbations on the infinite interval. Trans Am Math Soc 123: 521–535 MATHCrossRefGoogle Scholar
  4. 4.
    Jiang ZP and Wang Y (2002). A converse Lyapunov theorem for discrete-time systems with disturbances. Systems Control Lett 45: 49–58 MATHCrossRefGoogle Scholar
  5. 5.
    Kellett CM (2002) Advances in converse and control Lyapunov functions. PhD thesis, University of California, Santa BarbaraGoogle Scholar
  6. 6.
    Kellett CM and Teel AR (2004). Smooth Lyapunov functions and robustness of stability for difference inclusions. Systems Control Lett 52(5): 395–405 MATHCrossRefGoogle Scholar
  7. 7.
    Kellett CM and Teel AR (2004). Weak converse Lyapunov theorems and control Lyapunov functions. SIAM J Control Optim 42(6): 1934–1959 MATHCrossRefGoogle Scholar
  8. 8.
    Kellett CM and Teel AR (2005). On the robustness of \(\mathcal{K}\mathcal{L}\)-stability for difference inclusions: smooth discrete-time Lyapunov functions. SIAM J Control Optim 44(3): 777–800 MATHCrossRefGoogle Scholar
  9. 9.
    Kurzweil J (1956). On the inversion of Ljapunov’s second theorem on stability of motion. Am Math Soc Trans 24(2): 19–77 Google Scholar
  10. 10.
    Lakshmikantham V and Liu XZ (1993). Stability analysis in terms of two measures. World Scientific, Singapore MATHGoogle Scholar
  11. 11.
    Lakshmikantham V and Salvadori L (1976). On Massera type converse theorem in terms of two different measures. Bollettino U.M.I. 13: 293–301 MATHGoogle Scholar
  12. 12.
    Lin Y, Sontag ED and Wang Y (1996). A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim 34: 124–160 MATHCrossRefGoogle Scholar
  13. 13.
    Lyapunov AM (1892) The general problem of the stability of motion. Math. Soc. of Kharkov (Russian). (English Translation, Int J Control, 55:531–773, 1992)Google Scholar
  14. 14.
    Massera JL (1956) Contributions to stability theory. Ann Math 64:182–206. (Erratum: Ann Math 68:202, 1958)Google Scholar
  15. 15.
    Movchan AA (1960) Stability of processes with respect to two measures. Prikl Mat Mekh, 988–1001. English translation in J Appl Mathe MechGoogle Scholar
  16. 16.
    Nešić D, Teel AR and Kokotović PV (1999). Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. Systems Control Lett 38: 259–270 CrossRefGoogle Scholar
  17. 17.
    Stuart AM and Humphries AR (1996). Dynamical systems and numerical analysis. Cambridge University Press, Cambridge MATHGoogle Scholar
  18. 18.
    Sontag ED (1998). Comments on integral variants of ISS. Systems Control Lett 34: 93–100 MATHCrossRefGoogle Scholar
  19. 19.
    Teel AR and Praly L (2000). A smooth Lyapunov function from a class-\(\mathcal{K}\mathcal{L}\) estimate involving two positive semidefinite functions. ESAIM-Control Optim Calc Var 5: 313–367 MATHCrossRefGoogle Scholar
  20. 20.
    Wilson FW (1969). Smoothing derivatives of functions and applications. Trans Am Math Soc 139: 413–428 MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2007

Authors and Affiliations

  1. 1.School of Electrical Engineering and Computer ScienceUniversity of NewcastleCallaghanAustralia
  2. 2.Department of Electrical and Computer EngineeringUniversity of CaliforniaSanta BarbaraUSA

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