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Limit cycle oscillator in nonlinear systems with multiple time delays

Author

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  • Hakimi, A.R.
  • Azhdari, M.
  • Binazadeh, T.

Abstract

This paper concerns with the problem of generation stable limit cycles in a class of nonlinear time-delay systems with multiple time delays. With the help of the Lyapunov–Krasovskii functional approach, a state-feedback controller is constructed based on the backstepping technique. According to the recursive procedure of the backstepping technique, at first, the desired limit cycle is generated in the second-order subsystem by utilizing the Lyapunov theorem analysis of the positive limit sets. Then, this method is expanded for higher-order systems. By introducing an appropriate Lyapunov-Krasovskii functional, the complexities raised by unknown time-delays are solved in the control design process. Based on this scheme, a delay-independent controller is explicitly designed which does not need the precise knowledge of time delays. The proposed method rigorously guarantees the practical stability of the closed-loop system and also ensures the convergence of the phase trajectories of the closed-loop system to the target limit cycle. Finally, to prove the theoretical achievement and also to illustrate the effective performance of the proposed approach, the simulation results are provided for several examples.

Suggested Citation

  • Hakimi, A.R. & Azhdari, M. & Binazadeh, T., 2021. "Limit cycle oscillator in nonlinear systems with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    • Handle: RePEc:eee:chsofr:v:153:y:2021:i:p2:s0960077921008080
    DOI: 10.1016/j.chaos.2021.111454
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    References listed on IDEAS

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    1. Meysam Azhdari & Tahereh Binazadeh, 2021. "Output tracker design for uncertain nonlinear sandwich systems with sandwiched dead-zone nonlinearity based on adaptive finite-time control," International Journal of Systems Science, Taylor & Francis Journals, vol. 52(3), pages 596-611, February.
    2. Ali Reza Hakimi & Tahereh Binazadeh, 2018. "Robust limit cycle control in a class of nonlinear discrete-time systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 49(15), pages 3108-3116, November.
    3. T. Binazadeh & A. R. Hakimi, 2020. "Adaptive generation of limit cycles in a class of nonlinear systems with unknown parameters and dead-zone nonlinearity," International Journal of Systems Science, Taylor & Francis Journals, vol. 51(15), pages 3134-3145, November.
    4. H. Arismendi-Valle & D. Melchor-Aguilar, 2019. "On the Lyapunov matrices for integral delay systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 50(6), pages 1190-1201, April.
    5. Shen, Yongjun & Yang, Shaopu & Sui, Chuanyi, 2014. "Analysis on limit cycle of fractional-order van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 94-102.
    6. Lingrong Xue & Zhenguo Liu & Zongyao Sun & Wei Sun, 2019. "New results on robust tracking control for a class of high-order nonlinear time-delay systems," International Journal of Systems Science, Taylor & Francis Journals, vol. 50(10), pages 2002-2014, July.
    7. Shu-Min Lu & Dong-Juan Li, 2017. "Adaptive Neural Network Control for Nonlinear Hydraulic Servo-System with Time-Varying State Constraints," Complexity, Hindawi, vol. 2017, pages 1-11, October.
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    Cited by:

    1. Azhdari, Meysam & Binazadeh, Tahereh, 2022. "A novel adaptive SMC strategy for sustained oscillations in nonlinear sandwich systems based on stable limit cycle approach," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).

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