STABILIZING UNSTABLE PERIODIC TRAJECTORIES OF CHAOTIC SYSTEMS WITH TIME-VARYING SWITCHING DELAYED FEEDBACK CONTROL
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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.
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关键词:
- 混沌控制 /
- 时滞反馈 /
- 周期轨线 /
- 时变切换 /
- 条件Lyapunov指数
Abstract: In order to improve the effect of the classical delayed feedback control in stabilizing the unstable periodic trajectory and expand the stability region, the time-varying switching strategy is used to modify the classical delayed feedback control, which leads to the method of time-varying switching delayed feedback control. The control signal of the time-varying switching delayed feedback control only exists in specific time intervals, and there is no control signal in other time intervals, which is different from the fixed control signal in the classical delayed feedback control. Through case studies, the specific performance of time-varying switching delayed feedback control in stabilizing unstable periodic trajectory is investigated. The maximum conditional Lyapunov exponent of the controlled periodic trajectory is calculated as a function of the feedback strength. The relationship between the stability region of the controlled periodic trajectory and the switching frequency is obtained. The results show that with the increase of switching frequency, the stable region of the controlled periodic trajectory changes non smoothly. The stability region of the time-varying switching delayed feedback control is significantly larger than that of the classical delayed feedback control when the switching frequency is properly selected. In the engineering practice of chaos control, the control signal is often constrained. To achieve the stable control of the target periodic trajectory, the controlled periodic trajectory needs to have a large enough stable region. Therefore, compared with the classical time-delay feedback control, the time-varying switching time-delay feedback control proposed in this paper has a wider application prospect. -
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TSDFCFigure 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TSDFCFigure 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TSDFCFigure 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TSDFCFigure 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
${{{\boldsymbol{X}}}_k}(t)$ Trajectory Stability region Width of stability region $ {W_s} $ DFC TSDFC DFC TSDFC $ {{{\boldsymbol{X}}}_1}(t) $ (0.22, 2.5) (0.44, 9.9) 2.28 9.46 $ {{{\boldsymbol{X}}}_2}(t) $ (0.11, 0.29) (0.24, 2.0) 0.18 1.76 
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