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Finite-Time Stability Analysis of Impulsive Switched Discrete-Time Linear Systems: The Average Dwell Time Approach

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Abstract

This paper investigates the finite-time stability problem for a class of discrete-time switched linear systems with impulse effects. Based on the average dwell time approach, a sufficient condition is established which ensures that the state trajectory of the system remains in a bounded region of the state space over a pre-specified finite time interval. Different from the traditional condition for asymptotic stability of switched systems, it is shown that the total activation time of unstable subsystems can be greater than that of stable subsystems. Moreover, the finite-time stability degree can also be greater than one. Two examples are given to illustrate the merit of the proposed method.

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References

  1. F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)

    Article  MATH  Google Scholar 

  2. F. Amato, M. Ariola, Finite-time control of discrete-time linear systems. IEEE Trans. Autom. Control 50(5), 724–729 (2005)

    Article  MathSciNet  Google Scholar 

  3. F. Amato, R. Ambrosino, C. Cosentino, G. De Tommasi, Finite-time stabilization of impulsive dynamical linear systems. Nonlinear Anal. Hybrid Syst. 5(1), 89–101 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Amato, R. Ambrosino, M. Ariola, G. De Tommasi, Robust finite-time stability of impulsive dynamical linear systems subject to norm-bounded uncertainties. Int. J. Robust Nonlinear Control 21(10), 1080–1082 (2011)

    Article  MATH  Google Scholar 

  5. R. Ambrosino, F. Calabrese, C. Cosentino, G. De Tommasi, Sufficient conditions for finite-time stability of impulsive dynamical systems. IEEE Trans. Autom. Control 54(4), 861–865 (2009)

    Article  Google Scholar 

  6. S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. M.S. Branicky, Stability of switched and hybrid systems, in Proceedings of the 33rd IEEE Conference on Design and Control (1999), pp. 3498–3503

    Google Scholar 

  9. D. Cheng, Controllability of switched bilinear systems. IEEE Trans. Autom. Control 40(4), 511–515 (2001)

    Google Scholar 

  10. J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002)

    Article  MathSciNet  Google Scholar 

  11. R.A. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88(7), 1069–1082 (2000)

    Article  Google Scholar 

  12. H. Du, X. Lin, S. Li, Finite-time boundedness and stabilization of switched linear systems. Kybernetika 46(5), 870–889 (2010)

    MathSciNet  MATH  Google Scholar 

  13. J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in Proceedings of the 38th Conference on Decision and Control (1999), pp. 2655–2660

    Google Scholar 

  14. M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43(4), 555–559 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulse Differential Equations (World Scientific, Singapore, 1989)

    Google Scholar 

  16. S.H. Lee, J.T. Lim, Stability analysis of switched systems with impulse effects, in Proceedings of the 1999 IEEE International Symposium on Intelligent (1999), pp. 79–83

    Google Scholar 

  17. D. Liberzon, Switching in Systems and Control (Brikhauser, Boston, 2003)

    Book  MATH  Google Scholar 

  18. H. Lin, P.J. Anstaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 2351–2355 (2009)

    Google Scholar 

  19. X. Lin, H. Du, S. Li, Finite-time boundedness and L 2-gain analysis for switched delay systems with norm-bounded disturbance. Appl. Math. Comput. 217(12), 5982–5993 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Liu, J.T. Sun, Finite-time stabilization of linear systems via impulsive control. Int. J. Control 81(6), 905–909 (2008)

    Article  MATH  Google Scholar 

  21. L. Lv, Z. Lin, H. Fang, \(\mathcal{L}_{2}\) gain analysis for a class of switched systems. Automatica 45(4), 965–972 (2009)

    Article  MathSciNet  Google Scholar 

  22. S. Pettersson, Analysis of switched linear systems, in Proceedings of the 42nd IEEE Conference on Decision and Control (2003), pp. 5283–5288

    Google Scholar 

  23. S. Pettersson, Controller design of switched linear systems, in Proceedings of the 2004 American Control Conference (2004), pp. 3869–3874

    Google Scholar 

  24. S. Pettersson, Synthesis of switched linear systems handling sliding motions, in Proceedings of the 2005 International Symposium on Intelligent Control (2005), pp. 18–23

    Google Scholar 

  25. Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design (Springer, New York, 2005)

    MATH  Google Scholar 

  26. J. Xu, J.T. Sun, Finite-time stability of linear time-varying singular impulsive systems. IET Control Theory Appl. 4(10), 2239–2244 (2010)

    Article  MathSciNet  Google Scholar 

  27. X. Xu, G. Zhai, Practical stability and stabilization of hybrid and switched systems. IEEE Trans. Autom. Control 50(11), 1897–1903 (2005)

    Article  MathSciNet  Google Scholar 

  28. H. Ye, A.N. Michael, L. Hou, Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43(4), 461–474 (1998)

    Article  MATH  Google Scholar 

  29. G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. Int. J. Syst. Sci. 32(8), 1055–1061 (2001)

    MathSciNet  MATH  Google Scholar 

  30. G. Zong, S. Xu, Y. Wu, Robust H ∞ stabilization for uncertain switched impulsive control systems with state delay: an LMI approach. Nonlinear Anal. Hybrid Syst. 2(4), 1287–1300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Zong, L. Hou, J. Li, A descriptor system approach to l 2−l ∞ filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Int. J. Innov. Comput. Inf. Control 7(5A), 2213–2224 (2011)

    Google Scholar 

  32. G. Zong, L. Hou, J. Li, Robust l 2−l ∞ guaranteed cost filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Circuits Syst. Signal Process. 30(1), 17–33 (2011)

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China No. 60774039, No. 60974024, No. 61074089, No. 61174129, Program for New Century Excellent Talents in University No. NCET-11-0379, and the Independent Innovation Foundation of Tianjin University.

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Authors and Affiliations

  1. Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, P.R. China

    Yijing Wang, Gai Wang, Xiaomeng Shi & Zhiqiang Zuo

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  1. Yijing Wang
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  2. Gai Wang
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  3. Xiaomeng Shi
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  4. Zhiqiang Zuo
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Correspondence to Yijing Wang.

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Wang, Y., Wang, G., Shi, X. et al. Finite-Time Stability Analysis of Impulsive Switched Discrete-Time Linear Systems: The Average Dwell Time Approach. Circuits Syst Signal Process 31, 1877–1886 (2012). https://doi.org/10.1007/s00034-012-9403-4

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  • DOI: https://doi.org/10.1007/s00034-012-9403-4

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