Kybernetika 54 no. 3, 496-521, 2018

Converse theorem for practical stability of nonlinear impulsive systems and applications

Boulbaba Ghanmi, Mohsen Dlala and Mohamed Ali HammamiDOI: 10.14736/kyb-2018-3-0496

Abstract:

The Lyapunov's second method is one of the most famous techniques for studying the stability properties of dynamic systems. This technique uses an auxiliary function, called Lyapunov function, which checks the stability properties of a specific system without the need to generate system solutions. An important question is about the reversibility or converse of Lyapunov's second method; i. e., given a specific stability property does there exist an appropriate Lyapunov function? The main result of this paper is a converse Lyapunov Theorem for practical asymptotic stable impulsive systems. Applying our converse Theorem, several criteria on practical asymptotic stability of the solution of perturbed impulsive systems and cascade impulsive systems are established. Finally, some examples are given to show the effectiveness of the derived results.

Keywords:

converse Lyapunov theorem, practical asymptotic stability, impulsive systems, cascade systems, perturbed systems

Classification:

34A37, 34D20

paper.pdf

References:

  1. A. Bacciotti and L. Rosier: Lyapunov and Lagrange stability: Inverse theorems for discontinuous systems. Math. Control Signals Systems 11 (1998), 101-128.   DOI:10.1007/bf02741887
  2. D. D. Bainov and P. S. Simeonov: Systems with Impulse Effect: Stability, Theory, and Applications. Ellis Horwood, Chichester 1989.   CrossRef
  3. A. Benabdallah, I. Ellouze and M. A. Hammami: Practical stability of nonlinear time-varying cascade systems. J. Dynamical Control Systems 15 (2009), 45-62.   DOI:10.1007/s10883-008-9057-5
  4. A. Benabdallah, M. Dlala and M. A. Hammami: A new Lyapunov function for stability of time-varying nonlinear perturbed systems. Systems Control Lett. 56 (2007), 179-187.   DOI:10.1016/j.sysconle.2006.08.009
  5. B. Ben Hamed, I. Ellouze and M. A. Hammami: Practical uniform stability of nonlinear differential delay equations. Mediterranean J. Math. 8 (2011), 603-616.   DOI:10.1007/s00009-010-0083-7
  6. B. Ben Hamed and M. .A Hammami: Practical stabilization of a class of uncertain time-varying nonlinear delay systems. J. Control Theory Appl. 7 (2009), 175-180.   DOI:10.1007/s11768-009-8017-2
  7. C. Cai, A. Teel and R. Goebel: Smooth Lyapunov functions for hybrid systems, Part I: Existence is equivalent to robustness. IEEE Trans. Automat. Control 52 (2007), 7, 1264-1277.   DOI:10.1109/tac.2007.900829
  8. M. Corless: Guaranteed rates of exponential convergence for uncertain systems. J. Optim. Theory Appl. 64 (1990), 481-494.   DOI:10.1007/bf00939420
  9. M. Dlala, B. Ghanmi and M. A Hammami: Exponential practical stability of nonlinear impulsive systems: converse theorem and applications. Dynamics Continuous Discrete Impulsive Systems 21 (2014), 37-64.   CrossRef
  10. M. Dlala and M. A. Hammami: Uniform exponential practical stability of impulsive perturbed systems. J. Dynamical Control Systems 13 (2007), 373-386.   DOI:10.1007/s10883-007-9020-x
  11. P. Giesl and S. Hafstein: Review on computational methods for Lyapunov functions. Discrete Continuous Dynamical Systems: Series B 20 (2015), 2291-2331.   DOI:10.3934/dcdsb.2015.20.2291
  12. S. P. Gordon: On converse to the stability theorems for difference equations. SIAM J. Control Optim. 10 (1972), 76-81.   DOI:10.1137/0310007
  13. W. Hahn: Stability of Motion. Springer-Verlag, 1967.   DOI:10.1007/978-3-642-50085-5
  14. A. Isidori: Nonlinear Control Systems. Second edition. Springer-Verlag, 1989.   DOI:10.1007/978-3-662-02581-9
  15. C. Kellett: Converse Theorems in lyapunov's second method. Discrete Continuous Dynamical Systems: Series B 20 (2015), 2333-2360.   DOI:10.3934/dcdsb.2015.20.2333
  16. Z. P. Jiang, A. R. Teel and L. Praly: Small gain theorem for ISS systems and applications. Math. Control, Signals Systems 7 (1995), 95-120.   DOI:10.1007/bf01211469
  17. and Y. Wang: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems Control Lett, 45 (2002), 49-58.   DOI:10.1016/s0167-6911(01)00164-5
  18. H. K. Khalil: Nonlinear Systems. Third edition. Macmillan Publishing Company, 2002.   CrossRef
  19. V. Lakshmikantham, S. Leela and A. A. Martynyuk: Practical Stability of Nonlinear Systems. World Scientific, Singapore 1990.   DOI:10.1142/1192
  20. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics. Singapore and Teaneck, World Scientific, NJ 1989.   DOI:10.1142/0906
  21. J. P. LaSalle and S. Lefschetz: Stability by Lyapunov's Direct Method with Applications. Academic Press, New York 1961.   DOI:10.1002/zamm.19620421022
  22. Y. Lin, E. D. Sontag and Y. Wang: A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34 (1996), 124-160.   DOI:10.1137/s0363012993259981
  23. J. L. Mancilla-Aguilar and R. A. Garcia: A converse Lyapunov theorem for nonlinear switch systems. Systems Control Lett. 41 (2000), 67-71.   DOI:10.1016/s0167-6911(00)00040-2
  24. E. Panteley and A. Loria: On global uniform asymptotic stability of non linear time-varying non autonomous systems in cascade. Systems Control Lett. 33 (1998), 131-138.   DOI:10.1016/s0167-6911(97)00119-9
  25. C. Pradalier, R. Siegwart and G. Hirzinger: Robotics Research. Springer-Verlag, Berlin 2011.   DOI:10.1007/978-3-642-19457-3
  26. M. W. Spong and M. Vidyasagar: Robot Dynamics and Control. John Wiley and Sons, Inc, New York 1989.   CrossRef
  27. M. W. Spong: The control of underactuated mechanical systems. In: First International Conference on Mecatronics, Mexico City 1994.   CrossRef
  28. J. Tsinias: A converse Lyapunov theorem for nonuniform in time, global exponential robust stability. Systems Control Lett. 44 (2001), 373-384.   DOI:10.1007/978-3-642-19457-3
  29. X.-S. Yang: Existence of unbounded solutions of time varying systems and failure of global asymptotic stability in discrete-time cascade systems. IMA J. Math. Control Inform. 22 (2005), 80-87.   DOI:10.1093/imamci/dni006
  30. T. Yang: Impulsive Control Theory. Springer, 2001.   DOI:10.1007/3-540-47710-1
  31. T. Yoshizawa: Stability Theory by Lyapunov's Second Method. Mathematical Society of Japan, 1966.   CrossRef
  32. V. I. Zubov: Methods of A. M. Lyapunov and their Application. P. Noordhoff Ltd, Groningen 1964; translated from the Russian edition of 1957.   CrossRef