# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 4 Suppression Strategy of Self-Excited Vibration  
### Influence on Stability with Regard to \(\overline{k}_{p}\)  
According to Equation (17), the expected magnetic flux \(B_{\rm exp}\) consists of the states (displacement \(y_{\rm E}\), velocity \(\dot{y}_{\rm E}\) and acceleration \(\ddot{y}_{\rm E}\)) of levitation subsystem and the state (displacement \(y_{\rm B}\)) of bridge subsystem.  
In maglev engineering, the levitation eddy-current sensor can detect the relative displacement (\(y_{\rm E}-y_{\rm B}\)) between the upper surface of the electromagnet and the lower surface of the guideway.  
Figure 4: The averaged powers.  
However, the electromagnet's displacement \(y_{\rm E}\) and bridge's displacement \(y_{\rm B}\) can be measured independently. Traditionally, there is no choice but to feed back the displacement of electromagnet and bridge is in pairs (\(y_{\rm E}-y_{\rm B}\)).  
Up to now, no literature indicates that the displacement feedback in pairs is optimal for the levitation stability and the suppression of the self-excited vibration. Taking the stability condition of the levitation subsystem for example, according to the Equation (21), the stability of the levitation subsystem is uncorrelated with the feedback gain of the bridge's displacement \(\overline{k}_{\rm P}\).  
Similarly, according to Figure 2, to some extent, the feedback of the bridge's displacement increases the complexity of the block diagram. Furthermore, its influence on the occurrence of the self-excited vibration is unclear and should be explored.  
To study the influence on the stability of the interaction system with the feedback of the bridge's displacement \(\overline{k}_{\rm P}\), for an extreme case, we suppose that \(\overline{k}_{\rm P}=0\). This is to say, the feedback path of the bridge's displacement is removed from traditional control framework. In this case, the block diagram of maglev vehicle-bridge system is updated as Figure 5.  
In light of Figure 5, the transfer function from disturbance \(F_{\rm d}\) to the displacement \(y_{\rm B}\) of bridge is degraded to  
\[T_{3}(s)=\frac{G_{\rm B}(s)}{1-G_{\rm E}(s)H_{\rm E}(s)} \tag{34}\]  
In this case, the characteristic equation of the transfer function \(T_{3}(s)\) is  
\[\Delta_{3}=\Delta_{1}\cdot\left(s^{2}+2\xi_{\rm B}\omega_{\rm B}s+\omega_{\rm B }^{2}\right) \tag{35}\]  
In light of Equation (35), the characteristic equation of the interaction system is the product of the characteristic polynomial of levitation system itself \(\Delta_{1}\) and \(s^{2}+2\xi_{\rm B}\omega_{\rm B}s+\omega_{\rm B}^{2}\).  
Obviously, the characteristic polynomial \(s^{2}+2\xi_{\rm B}\omega_{\rm B}s+\omega_{\rm B}^{2}\) is stable. Hence, we can conclude that the interaction system is stable provided that the levitation subsystem is stable. This is to say, the stability of the interaction system is degenerated into the stability of the levitation subsystem. In this case, the self-excited vibration will be avoided if Equation (23) is satisfied.  
To illustrate the conclusion quantitatively, the gain \(\overline{k}_{\rm P}\) is set as zero, and the other parameters are kept the same as the above section. The real parts of characteristic roots of the maglev vehicle-bridge interaction system are shown in Figure 6 when the modal frequency \(\omega_{\rm B}\) is varying.  
In light of Figure 6, it can be seen that the real parts \(R_{1}\), \(R_{2}\) and \(R_{3}\) are all negative when \(\overline{k}_{\rm P}=0\). Hence, from the perspective of the characteristic roots, removing the bridge's displacement feedback is beneficial for the stability of the interaction system.  
Figure 5: The block diagram of maglev vehicle-bridge system.  
passive if the feedback of the bridge's displacement \(y_{\mathrm{B}}\) is removed. Furthermore, considering the passivity and the dissipation of bridge due to its modal damping, the vibration energy of bridge is delay to zero no matter how large the initial states are. In this case, the self-excited vibration is avoided.