Skip to main content
SpringerLink
Log in

A new stability result for nonlinear cascade time-delay system and its application in chaos control

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

Abstract

Based on the converse Lyapunov stability theorem and invariant set theory, this paper presents a new theorem for the nonlinear cascade delay system. With this new proposed method, a lot of coupled items can be taken as zero items. So the whole system can be converted to a very simple form. Also, a simple chaos control technique is proposed for the uncertain time-delay Lorenz chaotic system via this new method. The controller designed is linear and easy to be implemented. Simulation results for uncertain chaotic systems are provided to illustrate the effectiveness of the proposed scheme.

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.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Mazenc, F., Praly, L., Dayawansa, W.P.: Global stabilization by output feedback: examples and counter examples. Syst. Control Lett. 23, 119–125 (1994)

  2. Qian, C., Lin, W.: Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Autom. Control 47(10), 1710–1715 (2002)

  3. Sun, M., Tian, T., Xu, J.: Time-delayed feedback control of the energy resource chaotic system. Int. J. Nonlinear Sci. 1(3), 172–177 (2006)

    MathSciNet  Google Scholar 

  4. Li, X., Song, S.: Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3892–3900 (2014)

    Article  MathSciNet  Google Scholar 

  5. Guo, R.: Finite-time stabilization of a class of chaotic systems via adaptive control method. Commun. Nonlinear Sci. Numer. Simul. 17(1), 255–262 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wang, H., Zhang, X., Wang, X., Zhu, X.: Finite time chaos control for a class of chaotic systems with input nonlinearities via TSM scheme. Nonlinear Dyn. 69, 1941–1947 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, X., Liu, C.: Passive control on a unified chaotic system. Nonlinear Anal. Real World Appl. 11, 683–687 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wang, Y., Ma, J., Xu, Y.: Finite-time chaos control of the chaotic financial system based on control Lyapunov function. Appl. Mech. Mater. 55–57, 203–208 (2011)

    Article  Google Scholar 

  9. Fatih, O., Sirma, Y.: Designing chaotic S-boxes based on time-delay chaotic system. Nonlinear Dyn. 74(3), 551–557 (2013)

    Article  Google Scholar 

  10. Cho, S., Jin, M., Kuc, T., Lee, J.: Control and synchronization of chaos systems using time-delay estimation and supervising switching control. Nonlinear Dyn. 75(3), 549–560 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jack, H.: Theory of Functional Differential Equations. Springer, Berlin (1977)

    MATH  Google Scholar 

  12. Krasovskii, N.: Stability of Motion. Stanford University Press, Stanford (1963)

    MATH  Google Scholar 

  13. Burton, T.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, New York (1985)

    MATH  Google Scholar 

  14. Cao, Y., Sun, Y., Cheng, C.: Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans. Automat. Contr. 43, 1608–1612 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Vegas, J.: Reduced stability of parameter-dependent matrices. Linear Algebra Appl. 268, 289–321 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would thank Professor Stefan Siegmund for his useful suggestions in the proof of Theorem 1.

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, 200072, China

    Hua Wang, Jiu-Peng Wu, Xiao-Shu Sheng, Xin Wang & Peng Zan

Authors
  1. Hua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiu-Peng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Shu Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peng Zan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Zan.

Additional information

Supported by National Natural Science Foundation of China (11202121, 61104006, 31370998).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Wu, JP., Sheng, XS. et al. A new stability result for nonlinear cascade time-delay system and its application in chaos control. Nonlinear Dyn 80, 221–226 (2015). https://doi.org/10.1007/s11071-014-1862-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1862-7

Keywords

Search

Navigation