Advertisement

On the Problem of Nonlinear Stabilization of Switched Systems

  • A. V. PlatonovEmail author
Article
  • 18 Downloads

Abstract

The asymptotic stability and ultimate boundedness of the solutions for nonlinear switched systems is investigated. It is assumed that every mode of the system is unstable. The required behaviour of the switched system is achieved by using a special nonstationary coefficient, which can be considered as a multiplicative control. It is assumed that this coefficient is a piecewise constant function, and the value of the coefficient changes when the mode changes. Sufficient conditions on the coefficient are found to guarantee the asymptotic stability of the zero solution of the switched system or the ultimate boundedness of solutions with a given bound. These results are applied to the stability analysis of some classes of mechanical systems. Numerical examples are presented to demonstrate the effectiveness of the proposed approach.

Keywords

Nonlinear switched systems Stability Stabilization Multiple Lyapunov functions 

Notes

Acknowledgements

This work was supported by the Russian Foundation of Basic Researches, Grant No. 16-01-00587.

References

  1. 1.
    A.Yu. Aleksandrov, E.B. Aleksandrova, A.V. Platonov, Stability analysis of equilibrium positions of nonlinear mechanical systems with nonstationary leading parameter at the potential forces, Vestnik of St. Petersburg State University. Ser. 10. Appl. Math. Comput. Sci. Control Process. (1), 107–119 (2015). (in Russian) Google Scholar
  2. 2.
    A.Yu. Aleksandrov, A.A. Kosov, A.V. Platonov, On the asymptotic stability of switched homogeneous systems. Syst. Control Lett. 61(1), 127–133 (2012)Google Scholar
  3. 3.
    C.F. Beards, Engineering Vibration Analysis with Application to Control Systems (Edward Arnold, London, 1995)Google Scholar
  4. 4.
    M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43, 475–482 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. Cruz-Zavala, J.A. Moreno, Homogeneous high order sliding mode design: a Lyapunov approach. Automatica 80, 232–238 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88(7), 1069–1082 (2000)CrossRefGoogle Scholar
  7. 7.
    X. Ding, X. Liu, On stabilizability of switched positive linear systems under state-dependent switching. Appl. Math. Comput. 307, 92–101 (2017)MathSciNetGoogle Scholar
  8. 8.
    O.V. Gendelman, C.H. Lamarque, Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium. Chaos Solitons Fract. 24, 501–509 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lj.T. Grujic, A.A. Martynyuk, M. Ribbens-Pavella, Large Scale Systems Stability under Structural and Singular Perturbations (Springer, Berlin, 1987)Google Scholar
  10. 10.
    P.O. Gutman, Stabilizing controllers for bilinear systems. IEEE Trans. Autom. Control 26(4), 917–922 (1981)Google Scholar
  11. 11.
    J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in Proceedings of the 38th IEEE Conference on Decision and Control (1999), pp. 2655–2660Google Scholar
  12. 12.
    E. Kazkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation (Birkhauser, Boston, 1999)Google Scholar
  13. 13.
    H.K. Khalil, Nonlinear Systems (Prentice-Hall, Upper Saddle River, 2002)zbMATHGoogle Scholar
  14. 14.
    J. La Salle, S. Lefschetz, Stability by Liapunov’s Direct Method (Academic Press, New York, 1961)zbMATHGoogle Scholar
  15. 15.
    Y. Li, S. Tong, Adaptive fuzzy output-feedback stabilization control for a class of switched nonstrict-feedback nonlinear systems. IEEE Trans. Cybern. 47(4), 1007–1016 (2017)CrossRefGoogle Scholar
  16. 16.
    Y. Li, S. Tong, L. Liu, G. Feng, Adaptive output-feedback control design with prescribed performance for switched nonlinear systems. Automatica 80, 225–231 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Z.G. Li, C.Y. Wen, Y.C. Soh, Stabilization of a class of switched systems via designing switching laws. IEEE Trans. Autom. Control 46, 665–670 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. Liberzon, Switching in Systems and Control (Birkhauser, Boston, 2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    D. Liberzon, Lie algebras and stability of switched nonlinear systems, in Unsolved Problems in Mathematical Systems and Control Theory (Princeton University Press, ed. by V.D. Blondel, A. Megretski (Princeton, Oxford, 2004), pp. 90–92Google Scholar
  20. 20.
    D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19(15), 59–70 (1999)zbMATHGoogle Scholar
  21. 21.
    L. Liu, Q. Zhou, H. Liang, L. Wang, Stability and stabilization of nonlinear switched systems under average dwell time. Appl. Math. Comput. 298, 77–94 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    N. Rouche, P. Habets, M. Laloy, Stability Theory by Liapunov’s Direct Method (Springer, New York, 1977)CrossRefzbMATHGoogle Scholar
  23. 23.
    Y. Shen, Y. Huang, J. Gu, Global finite-time observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 56(2), 418–424 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. Shorten, F. Wirth, O. Mason, K. Wulf, G. King, Stability criteria for switched and hybrid systems. SIAM Rev. 49(4), 545–592 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    S. Sui, Y. Li, S. Tong, Observer-based adaptive fuzzy control for switched stochastic nonlinear systems with partial tracking errors constrained. IEEE Trans. Syst. Man Cybern. Syst. 46(12), 1605–1617 (2016)CrossRefGoogle Scholar
  26. 26.
    S. Sui, S. Tong, Fuzzy adaptive quantized output feedback tracking control for switched nonlinear systems with input quantization. Fuzzy Sets Syst. 290, 56–78 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    R. Wang, J. Xing, Z. Xiang, Finite-time stability and stabilization of switched nonlinear systems with asynchronous switching. Appl. Math. Comput. 316, 229–244 (2018)MathSciNetGoogle Scholar
  28. 28.
    F. Wang, X. Zhang, B. Chen, C. Lin, X. Li, J. Zhang, Adaptive finite-time tracking control of switched nonlinear systems. Inf. Sci. 421, 126–135 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    W. Xiang, J. Xiao, Stabilization 0f switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50, 940–945 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    D. Xie, H. Zhang, H. Zhang, B. Wang, Exponential stability of switched systems with unstable subsystems: a mode-dependent average dwell time approach. Circuits Syst. Signal Process. 32(6), 3093–3105 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    H. Yang, B. Jiang, V. Cocquempot, A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes. Nonlinear Anal. Hybrid Syst. 13, 45–60 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    B. Yang, Z. Zhao, R. Ma, Y. Liu, F. Wang, Stabilization of sector-bounded switched nonlinear systems with all unstable modes, in Proceedings of the 29th Chinese Control and Decision Conference (CCDC) (2017), pp. 2465–2470Google Scholar
  33. 33.
    Y. Yin, X. Zhao, X. Zheng, New stability and stabilization conditions of switched systems with mode-dependent average dwell time. Circuits Syst. Signal Process. 36(1), 82–98 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attention properties of time-controlled switched systems. J. Frankl. Inst. 338, 765–779 (2001)CrossRefzbMATHGoogle Scholar
  35. 35.
    B. Zhang, On finite-time stability of switched systems with hybrid homogeneous degrees. Math. Probl. Eng. 2018, Article ID 3096986 (2018)Google Scholar
  36. 36.
    X. Zhao, Y. Yin, H. Yang, R. Li, Adaptive control for a class of switched linear systems using state-dependent switching. Circuits Syst. Signal Process. 34(11), 3681–3695 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Q. Zheng, H. Zhang, Robust stabilization of continuous-time nonlinear switched systems without stable subsystems via maximum average dwell time. Circuits Syst. Signal Process. 36(4), 1654–1670 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    V.I. Zubov, Methods of A.M. Lyapunov and Their Applications. P. Noordhoff Ltd., Groningen (1964)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations