Skip to main content
SpringerLink
Log in

The existence and uniqueness of the solutions of the nonlinear on–off switched systems with switching at variable times

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, by introducing the definition of the solution, we prove the existence and uniqueness of the solution for nonlinear on–off switched systems with switching at variable times (NVTSS). Based on this result, the continuous dependence and differentiability of the solution of NVTSS (4) with respect to the initial state are presented. Besides, the switching phenomenon (the integral curve of NVTSS (4) may hit some surface finite or infinite times causing a rhythmical beating) and periodic solution of NVTSS (4) are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Liberzon, D.: Switching in Systems and Control, pp. 3–9. Bikhäuser, Boston (2003)

    Book  Google Scholar 

  2. Narendra, K.S., Balakrishnan, J.: A common Lyapunov functon for stable LTI systems with commuting A-matrices. IEEE Trans. Autom. Control 39, 2469–2471 (1994)

    Article  Google Scholar 

  3. Liberzon, D., Hespanha, J.P.: Morse AS (1999) Stability of switched systems: a Lie-algebraic condition. Syst. Control Lett. 37, 117–122 (1999)

    Article  Google Scholar 

  4. Sun, Z., Ge, S.S.: Stability Theory of Switched Dynamical Systems. Springer, London (2011)

    Book  Google Scholar 

  5. Morse, A.S.: Supervisory control of families of linear set-point controllers-part 1: exact matching. IEEE Trans. Autom. Control 41, 1413–1431 (1996)

    Article  Google Scholar 

  6. Hespanha, J., Morse, A.S.: Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, pp. 2655–2860 (1999)

  7. Zhai, G., Bo, H., Yasuda, K., Michel, A.N.: Stability analysis of switched with stable and unstanle subsystems: an average dwell timeapproach. Int. J. Syst. Sci. 32, 1055–1061 (2001)

    Article  Google Scholar 

  8. Lee, S.H., Kim, T.H., Lim, J.T.: A new stability analysis of switched systems. Automatica 36, 917–922 (2000)

    Article  MathSciNet  Google Scholar 

  9. Manci, J.L.M., Garca, R.A.: An extension of LaSalle’s invariance principle for switched systems. Syst. Control Lett. 55, 376–384 (2006)

    Article  MathSciNet  Google Scholar 

  10. Gao, F., Yuqiang, W., Zhang, Z.: Global fixed-time stabilization for a class of switched nonlinear systems with general powers and its application. Nonlinear Anal. Hybrid Syst. 31, 56–68 (2019)

    Article  MathSciNet  Google Scholar 

  11. Maniotis, P., Gitzenis, S., Tassiulas, L.: Switched linear systems: control and design. IEEE Trans. Autom. Control 51, 1585–1586 (2006)

    Article  Google Scholar 

  12. Shenyu, D.L.: Global stability and asymptotic gain imply input-to-state stability for state-dependent switched systems. In: 2018 IEEE Conference on Decision and Control, pp. 17–19 (2018)

  13. King, C.K., Griggs, W.M., Shorten, R.N.: A result on the existence of quadratic lyapunov functions for state-dependent switched systems with uncertainty. In: 49th IEEE Conference on Decision and Control, pp. 15–17 (2010)

  14. Heemels, W.P.M.H., Johansson, K.H., Tabuada, P.: An introduction to even-triggered and self-triggered control. In: 51st IEEE Conference on Decision and Control, pp. 3270–3285 (2012)

  15. Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid Dynamical Systems: Modeling, Stability and Robustness. Princeton University Press, Princeton (2012)

    Book  Google Scholar 

  16. Allerhand, L.I., Shaked, U.: Stability of stochastic nonlinear systems with state-dependent switching. IEEE Trans. Autom. Control 58, 994–1001 (2013)

    Article  MathSciNet  Google Scholar 

  17. Cai, C., Teel, A.: Robust input-to-state stability for hybrid systems. SIAM J. Control Optim. 51, 1651–1678 (2013)

    Article  MathSciNet  Google Scholar 

  18. Broucke, M., Arapostathis, A.: Continuous selections of trajectories of hybrid systems. Syst. Control Lett. 47, 149–157 (2002)

    Article  MathSciNet  Google Scholar 

  19. Zhang, G., Shen, Y.: Novel conditions on exponential stability of a class of delayed neural networks with state-dependent switching. Neural Netw. 71, 55–61 (2015)

    Article  Google Scholar 

  20. Brockett, R.W.: Hybrid Models for Motion Control Systems, in Essays on Control: Perspectives in the Theory and Its Applications, pp. 29–53. Bikhäuser, Boston (1993)

    Book  Google Scholar 

  21. Shorten, R., Wirth, F., Mason, O., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49, 545–592 (2007)

    Article  MathSciNet  Google Scholar 

  22. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81, 3088–3092 (1984)

    Article  Google Scholar 

  23. Zhang, X., Li, C., Huang, T.: Hybrid impulsive and switching Hopfield neural networks with state-dependent impulses. Neural Netw. 93, 176–184 (2017)

    Article  Google Scholar 

  24. Liu, C., Yang, Z., Sun, D., Liu, X., Liu, W.: Stability of switched neural networks with time-varying delays. Neural Comput. Appl. 30, 2229–2244 (2018)

    Article  Google Scholar 

  25. Wang, S.J.: Study on Several Predator-Prey Models with Impulsive Control. Shanxi Normal University, Xi’an (2012)

    Google Scholar 

  26. Spero, D.J., Pilutti, T.E., Rupp, M.Y., Joh, P.G.: Autonomous lane control system (2013)

  27. Baek, J., Kang, C., Kim, W.: Practical approach for developing lateral motion control of autonomous lane change system. Appl. Sci. 10, 3143 (2020)

    Article  Google Scholar 

  28. Sun, Z., Ge, S.S.: Analysis and synthesis of switched linear control systems. Automatica 41, 181–195 (2005)

    Article  MathSciNet  Google Scholar 

  29. Linh, V., Morgansen, K.A.: Stability of time-delay feedback switched linear systems. IEEE Trans. Autom. Control 55, 2385–2390 (2010)

    Article  MathSciNet  Google Scholar 

  30. Sun, Z.: A note on marginal stability of switched systems. IEEE Trans. Autom. Control 53, 625–631 (2008)

    Article  MathSciNet  Google Scholar 

  31. Liu, C., Yang, Z., Sun, D., Liu, X.: Stability of variable-time switched systems. Arab. J. Sci. Eng. 42, 2971–2980 (2017)

    Article  MathSciNet  Google Scholar 

  32. Cui, S.: On the general problem of solution and stability for variable-time switched systems. In: 2017 International Workshop on Complex Systems and Networks, pp. 27–33 (2017)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11661020 and 11661017) and the Foundation of Postgraduate of Guizhou Province (No. 2019032), Science and Technology of Guizhou Province (No. [2017]5788).

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Huanting Li, Yunfei Peng & Kuilin Wu

Authors
  1. Huanting Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunfei Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kuilin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunfei Peng.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by NSFC (Nos. 11661020, 11661017), Science and Technology Project of Guizhou Province (No. [2017]5788) and the Foundation of Postgraduate of Guizhou Province (No. 2019032)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Peng, Y. & Wu, K. The existence and uniqueness of the solutions of the nonlinear on–off switched systems with switching at variable times. Nonlinear Dyn 103, 2287–2298 (2021). https://doi.org/10.1007/s11071-021-06214-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06214-8

Keywords

Mathematics Subject Classification

Search

Navigation