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时变切换时滞反馈镇定混沌系统不稳定周期轨线

曾尖尖 郑远广

曾尖尖, 郑远广. 时变切换时滞反馈镇定混沌系统不稳定周期轨线. 力学学报, 2022, 54(12): 3477-3485 doi: 10.6052/0459-1879-22-361
引用本文: 曾尖尖, 郑远广. 时变切换时滞反馈镇定混沌系统不稳定周期轨线. 力学学报, 2022, 54(12): 3477-3485 doi: 10.6052/0459-1879-22-361
Zeng Jianjian, Zheng Yuanguang. Stabilizing unstable periodic trajectories of chaotic systems with time-varying switching delayed feedback control. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3477-3485 doi: 10.6052/0459-1879-22-361
Citation: Zeng Jianjian, Zheng Yuanguang. Stabilizing unstable periodic trajectories of chaotic systems with time-varying switching delayed feedback control. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3477-3485 doi: 10.6052/0459-1879-22-361

时变切换时滞反馈镇定混沌系统不稳定周期轨线

doi: 10.6052/0459-1879-22-361
基金项目: 国家自然科学基金资助项目(11872197)
详细信息
    作者简介:

    郑远广, 教授, 主要研究方向: 时滞系统动力学与控制. E-mail: zheng-yg@nchu.edu.cn

  • 中图分类号: O323

STABILIZING UNSTABLE PERIODIC TRAJECTORIES OF CHAOTIC SYSTEMS WITH TIME-VARYING SWITCHING DELAYED FEEDBACK CONTROL

  • 摘要: 为提高经典时滞反馈控制镇定不稳定周期轨线的效果, 扩大受控周期轨线的稳定区域, 本文基于时变切换策略对经典时滞反馈控制进行改进, 提出了时变切换时滞反馈控制. 时变切换时滞反馈控制的控制信号仅在特定的时段中存在, 而在其他时段上不存在控制信号, 这与经典时滞反馈控制中具有固定的控制信号是不同的. 通过实例分析, 研究了时变切换时滞反馈控制在镇定不稳定周期轨线中的具体性能. 以反馈增益系数为变量, 计算受控周期轨线的最大条件Lyapunov指数, 得到了受控周期轨线的稳定区域随切换频率变化的关系曲线. 结果表明, 随着切换频率增大, 受控周期轨线的稳定区域呈现非平滑地变化. 当选取恰当的切换频率时, 时变切换时滞反馈控制的稳定区域显著大于经典时滞反馈控制的稳定区域. 在混沌控制的工程实践中, 控制信号常常受到一定的限制. 要实现对目标周期轨线的稳定控制, 就需要受控周期轨线具有足够大的稳定区域. 因此, 与经典时滞反馈控制相比, 本文提出的时变切换时滞反馈控制具有更广泛的应用前景.

     

  • 图  1  $ {\rm{R\ddot ossler}} $系统的混沌吸引子

    Figure  1.  The chaotic attractor of $ {\rm{R\ddot ossler}} $system

    图  2  ${\rm{ R\ddot ossler}} $系统的不稳定周期轨线在$ xoy $平面上的投影

    Figure  2.  The projection of the unstable periodic trajectory of $ {\rm{R\ddot ossler}} $system on the$ xoy $plane

    图  3  受控周期轨线$ {{{\boldsymbol{X}}}_k}(t) $最大条件Lyapunov指数随反馈增益系数$ g $的变化曲线对比图, 红色虚线和蓝色实线分别对应于DFC和TSDFC

    Figure  3.  The comparison of the maximum Lyapunov exponent of periodic trajectory$ {{{\boldsymbol{X}}}_k}(t) $ as a function of the feedback strength, the red dotted line and the blue solid line correspond to DFC and TSDFC, respectively

    图  4  系统(9)的时间序列和TSDFC控制信号在不同时段上切换示意图, 在灰色时段中控制信号存在, 而在灰色时段外控制信号消失

    Figure  4.  The time series of system (9) and the schematic diagram of the switching of TSDFC control signals in different time intervals, where the control signal exists in the gray time intervals and disappears outside the gray time intervals

    图  5  受控周期轨线$ {{{\boldsymbol{X}}}_k}(t) $的稳定区域宽度$ {W_s} $随切换频率$ \omega $的变化曲线

    Figure  5.  The stable region width $ {W_s} $of controlled periodic trajectory$ {{{\boldsymbol{X}}}_k}(t) $ as a function of the switching frequency$ \omega $

    图  6  受控周期轨线$ {{{\boldsymbol{X}}}_k}(t) $的误差指数随反馈增益系数$ g $的变化曲线对比图, 红色虚线和蓝色实线分别对应于DFC和TSDFC

    Figure  6.  The comparison of the error index of the controlled periodic trajectory $ {{{\boldsymbol{X}}}_k}(t) $ as a function of the feedback strength$ g $, the red dotted line and the blue solid line correspond to DFC and TSDFC, respectively

    图  7  Van der Pol系统的混沌吸引子和不稳定周期轨线在$ xoy $平面上的投影

    Figure  7.  The projection of the chaotic attractors and unstable periodic trajectory of Van der Pol systems on the$ xoy $plane

    图  8  受控周期轨线$ {{{\boldsymbol{X}}}_3}(t) $最大条件Lyapunov指数随反馈增益系数$ g $的变化曲线对比图, 红色虚线和蓝色实线分别对应于DFC和TSDFC

    Figure  8.  The comparison of the maximum conditional Lyapunov exponent of the controlled periodic trajectory$ {{{\boldsymbol{X}}}_3}(t) $ as a function of the feedback strength$ g $, the red dotted line and the blue solid line correspond to DFC and TSDFC, respectively

    图  9  受控周期轨线$ {{{\boldsymbol{X}}}_3}(t) $的稳定区域宽度$ {W_s} $随切换频率$ \omega $的变化曲线

    Figure  9.  The width of the stable region of the controlled periodic trajectory $ {{{\boldsymbol{X}}}_3}(t) $ as a function of the switching frequency $ \omega $

    图  10  受控周期轨线$ {{{\boldsymbol{X}}}_3}(t) $的误差指数随反馈增益系数$ g $的变化曲线对比图, 红色虚线和蓝色实线分别对应于DFC和TSDFC

    Figure  10.  The comparison of the error index of the controlled periodic trajectory$ {{{\boldsymbol{X}}}_3}(t) $ as a function of the feedback strength $ g $, the red dotted line and the blue solid line correspond to DFC and TSDFC, respectively

    表  1  受控周期轨线${{{\boldsymbol{X}}}_k}(t)$的稳定区域

    Table  1.   The stability regions of the controlled periodic trajectory${{{\boldsymbol{X}}}_k}(t)$

    TrajectoryStability regionWidth of stability region $ {W_s} $
    DFCTSDFCDFCTSDFC
    $ {{{\boldsymbol{X}}}_1}(t) $(0.22, 2.5)(0.44, 9.9)2.289.46
    $ {{{\boldsymbol{X}}}_2}(t) $(0.11, 0.29)(0.24, 2.0)0.181.76
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-08-08
  • 录用日期:  2022-10-08
  • 网络出版日期:  2022-10-09
  • 刊出日期:  2022-12-15

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