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New Input-to-State Stability Condition for Continuous-Time Switched Nonlinear Systems

Abstract

This paper investigates the input-to-state stability (ISS) for the switched nonlinear system (SNS) under edge-dependent switching signals, which consists of both ISS and non-ISS subsystems. Sufficient conditions ensuring the ISS properties of the SNS are proposed, which indicate that the switched system is ISS if the edge-dependent average dwell time is large enough and the activation time of non-ISS subsystems is comparatively small. The established criterion is quite general and makes an improvement compared with the existing related results. A numerical example and a chemical process are provided to illustrate the advantage and effectiveness of the theoretical result.

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Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

  1. 1.

    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, pp. 2655–2660 (1999). https://doi.org/10.1109/cdc.1999.831330

  2. 2.

    A. Kundu, D. Chatterjee, A graph theoretic approach to input-to-state stability of switched systems. Eur. J. Control 29, 44–50 (2016). https://doi.org/10.1016/j.ejcon.2016.03.003

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    A. Kundu, D. Chatterjee, D. Liberzon, Generalized switching signals for input-to-state stability of switched systems. Automatica 64, 270–277 (2016). https://doi.org/10.1016/j.automatica.2015.11.027

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    S. Li, J. Guo, Z. Xiang, Global stabilization of a class of switched nonlinear systems under sampled-data control. IEEE Trans. Syst., Man, Cybern.: Syst. 49(9), 1912–1919 (2019). https://doi.org/10.1109/tsmc.2018.2836930

    Article  Google Scholar 

  5. 5.

    D. Liberzon, Switching in Systems and Control (Birkhäuser, Boston, 2003). https://doi.org/10.1007/978-1-4612-0017-8

    Book  MATH  Google Scholar 

  6. 6.

    T. Liu, Z.P. Jiang, Event-triggered control of nonlinear systems with state quantization. IEEE Trans. Autom. Control 64(2), 797–803 (2019). https://doi.org/10.1109/tac.2018.2837129

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    H. Meng, Z. Chen, Input-to-state stability of switched systems with explicit gain functions. Syst. Control Lett. 122, 39–47 (2018). https://doi.org/10.1016/j.sysconle.2018.09.010

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    M.A. Müller, D. Liberzon, Input/output-to-state stability and state-norm estimators for switched nonlinear systems. Automatica 48(9), 2029–2039 (2012). https://doi.org/10.1016/j.automatica.2012.06.026

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Y. Qi, P. Zeng, W. Bao, Event-triggered and self-triggered \(H_\infty \) control of uncertain switched linear systems. IEEE Trans. Syst., Man, Cybern.: Syst. 50(4), 1442–1454 (2020). https://doi.org/10.1109/tsmc.2018.2801284

    Article  Google Scholar 

  10. 10.

    E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34(4), 435–443 (1989). https://doi.org/10.1109/9.28018

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    E.D. Sontag, Y. Wang, On characterizations of the input-to-state stability property. Syst. Control Lett. 24(5), 351–359 (1995). https://doi.org/10.1016/0167-6911(94)00050-6

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    E.D. Sontag, Y. Wang, Output-to-state stability and detectability of nonlinear systems. Syst. Control Lett. 29(5), 279–290 (1997). https://doi.org/10.1016/s0167-6911(97)90013-x

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. Autom. Control 52(9), 1680–1685 (2007). https://doi.org/10.1109/TAC.2007.904277

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    L. Vu, D. Chatterjee, D. Liberzon, Input-to-state stability of switched systems and switching adaptive control. Automatica 43(4), 639–646 (2007). https://doi.org/10.1016/j.automatica.2006.10.007

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Y. Wang, Y. Chang, A.F. Alkhateeb, N.D. Alotaibi, Adaptive fuzzy output-feedback tracking control for switched nonstrict-feedback nonlinear systems with prescribed performance. Circuits, Syst., Signal Process. 40(1), 88–113 (2020). https://doi.org/10.1007/s00034-020-01466-y

    Article  Google Scholar 

  16. 16.

    X. Xiao, J.H. Park, L. Zhou, G. Lu, Event-triggered control of discrete-time switched linear systems with network transmission delays. Automatica (2020). https://doi.org/10.1016/j.automatica.2019.108585

    Article  MATH  Google Scholar 

  17. 17.

    X. Xiao, L. Zhou, D.W.C. Ho, G. Lu, Event-triggered control of continuous-time switched linear systems. IEEE Trans. Autom. Control 64(4), 1710–1717 (2019). https://doi.org/10.1109/tac.2018.2853569

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    W. Xie, C. Wen, Z. Li, Input-to-state stabilization of switched nonlinear systems. IEEE Trans. Autom. Control 46(7), 1111–1116 (2001). https://doi.org/10.1109/9.935066

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    J. Yang, X. Zhao, X. Bu, W. Qian, Stabilization of switched linear systems via admissible edge-dependent switching signals. Nonlinear Anal.: Hybrid Syst. 29, 100–109 (2018). https://doi.org/10.1016/j.nahs.2018.01.003

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    M.B. Yazdi, M.R. Jahed-Motlagh, Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chem. Eng. J. 155(3), 838–843 (2009). https://doi.org/10.1016/j.cej.2009.09.008

    Article  Google Scholar 

  21. 21.

    L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010). https://doi.org/10.1016/j.automatica.2010.02.021

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    X. Zhao, P. Shi, Y. Yin, S.K. Nguang, New results on stability of slowly switched systems: A multiple discontinuous Lyapunov function approach. IEEE Trans. Autom. Control 62(7), 3502–3509 (2017). https://doi.org/10.1109/tac.2016.2614911

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    X. Zhao, Y. Yin, L. Liu, X. Sun, Stability analysis and delay control for switched positive linear systems. IEEE Trans. Autom. Control 63(7), 2184–2190 (2018). https://doi.org/10.1109/tac.2017.2757460

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012). https://doi.org/10.1109/tac.2011.2178629

    MathSciNet  Article  MATH  Google Scholar 

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Funding

This work was supported by the Project 62073181 from National Natural Science Foundation of China, Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents and Nantong 226 High-level Talents Project. The work of Xiaoqing Xiao was also supported by China Scholarship Council (CSC NO. 201908320096).

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Correspondence to Xiaoqing Xiao.

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Zhou, L., Xiao, X. New Input-to-State Stability Condition for Continuous-Time Switched Nonlinear Systems. Circuits Syst Signal Process (2021). https://doi.org/10.1007/s00034-021-01845-z

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Keywords

  • Switched nonlinear system
  • Input-to-state stability
  • Edge-dependent average dwell time