Research on Time Lag Control of Levitation System of Two-Degree-of-Freedom Magnetic Levitation Train
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摘要:
为研究控制器时滞对磁浮列车悬浮系统稳定性的影响，首先，以位移−速度为反馈控制参数，建立考虑控制器时滞的二自由度磁浮列车悬浮系统模型；其次，通过Routh-Hurwitz稳定性判据得到无时滞系统的稳定性区域，同时，依据特征根穿越虚轴边界条件，获得系统发生Hopf分岔的控制器时滞临界值；最后，分析反馈控制参数及系统参数与控制器时滞临界值的关系. 研究结果表明：当系统参数一定时，控制器时滞临界值随位移控制增益的增大而减小，随速度控制增益的增大先增大后减小；当反馈控制参数一定时，控制器时滞临界值随二系刚度的增大而减小，随二系阻尼的增大而增大；当系统时滞以10−6数量级在时滞临界值附近渐渐增大，系统会从稳定−周期运动−不稳定逐渐变化，期间发生超临界Hopf分岔.
Abstract:In order to study the influence of controller time lag on the stability of the levitation system of magnetic levitation train, this paper firstly established a levitation system model of two-degree-of-freedom magnetic levitation train by considering controller time lag and taking the displacement-velocity as the feedback control parameter. Secondly, the stability region of the system without time lag was obtained through the Routh-Hurwitz stability criterion. At the same time, according to the boundary condition of the characteristic root crossing the imaginary axis, the critical value of the controller time lag for the occurrence of the Hopf bifurcation of the system was obtained. Finally, the relationship between the feedback control and system parameters and the critical value of controller time lag was analyzed. The results show that when the system parameters are certain, the critical value of the controller time lag decreases with the increase in the displacement control gain, and it increases and then decreases with the increase in the velocity control gain. When the feedback control parameters are certain, the critical value of the controller time lag decreases with the increase in the diaphragm stiffness and increases with the increase in the diaphragm damping. As the time lag of the system increases asymptotically by an order of magnitude 10−6 around the critical value of time lag, the system changes gradually from stable-periodic motion to unstable state, during which a supercritical Hopf bifurcation occurs.
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Key words:
- magnetic levitation system /
- stability /
- time lag /
- feedback control /
- Hopf bifurcation
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图 4 控制器时滞临界值与反馈控制参数的关系
Figure 4. Relationship between critical value of controller time lag and feedback control parameters
图 5 固有频率与反馈控制参数的关系
Figure 5. Relationship between natural frequency and feedback control parameters
图 6 控制器时滞临界值与系统参数的关系
Figure 6. Relationship between critical value of controller time lag and system parameters
图 7 固有频率与系统控制参数的关系
Figure 7. Relationship between natural frequency and system parameters
图 8 τ=0.000 900 s，τ<τ0时的系统响应
Figure 8. System response with time lag τ=0.000 900 s, τ<τ0
图 10 τ=0.000 986 s，τ>τ0时的系统响应
Figure 10. System response with time lag τ=0.000 986 s, τ>τ0
图 13 τ=0.000 900 s，τ <τ0时的系统响应
Figure 13. System response with time lag τ=0.000 900s, τ<τ0
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