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 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.
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.
 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.

## 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.

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.

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