# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 4 Suppression Strategy of Self-Excited Vibration
### The Estimation of Electromagnet's Displacement \(y_{\mathrm{E}}\)  
In an actual magnetic levitation system, there are two real-time signals available. The first signal is the levitation clearance \(\delta(t)=y_{\mathrm{E}}(t)-y_{\mathrm{B}}(t)\), measured by the eddy-current sensor, and the other is the acceleration signal of the electromagnet \(a_{\mathrm{E}}(t)=\dot{y}_{\mathrm{E}}(t)\), detected by the accelerometer. Traditionally, the signal of levitation gap is adopted for the outer-loop of the levitation controller. Separately speaking, the feedback gain of the electromagnet's displacement \(y_{\mathrm{E}}\) is \(k_{\mathrm{P}}\), and the feedback gain of the bridge's displacement are in pairs is \(\overline{k}_{\mathrm{P}}=-k_{\mathrm{P}}\).  
When the amplitude of the feedback gains of the electromagnet and bridge's displacement is different, i.e., \(\overline{k}_{\mathrm{P}}\
eq-k_{\mathrm{P}}\), we should measure the signal of electromagnet's displacement \(y_{\mathrm{E}}\) and the bridge's displacement \(y_{\mathrm{B}}\) separately. For a special case, \(\overline{k}_{\mathrm{P}}=0\). We still should measure the signal of the electromagnet's displacement \(y_{\mathrm{E}}\).  
Considering the signal of the electromagnet's displacement \(y_{\mathrm{E}}\) is immeasurable directly, we should develop a method to estimate it. Theoretically, the displacement of electromagnet may be obtained by the double integration of acceleration of the electromagnet:  
\[y_{\mathrm{E}}(s)=\frac{1}{s^{2}}a_{\mathrm{E}}(s) \tag{38}\]  
However, in a real maglev system, measurements of acceleration \(a_{\mathrm{E}}(t)\) are polluted by its inexact direct bias and ultra low frequency disturbance, which will lead to integral saturation.  
To prevent the integral saturation, the estimator \(1/(s+\tau)^{2}\) is adopted to instead of the double-integrator \(1/s^{2}\). This is to say, the estimated value \(\hat{y}_{\mathrm{E}}\) of the electromagnet's displacement is  
\[\hat{y}_{\mathrm{E}}(s)=\frac{1}{(s+\tau)^{2}}a_{\mathrm{E}}(s) \tag{39}\]  
where \(\tau\) is the time constant of the estimator. The comparison between the double-integrator and the estimator is shown in Figure 8.  
Considering Figure 8, when the frequency is less than 10 rad/s, the amplitude of double-integrator is oversized, which may result in excessive transient response when the ultra-low frequency component of \(a_{\mathrm{E}}(t)\) is shifted. Hence, the double-integrator is unsuitable for the engineering application.  
Figure 8: The comparison between the double-integrator and estimator.  
Luckily, compared with the double-integrator, the amplitude of the estimator is much smaller. Furthermore, when the frequency is larger than 10 rad/s, the amplitude and phase of estimator are highly consistent with the double-integrator, which provides enough phase advances over the upper frequency range. Therefore, compared with the double-integrator, the estimator is more suitable for the engineering applications. When the electromagnet's displacement \(y_{\mathrm{E}}\) is replaced by the estimation one, the control Equation (17) is updated to Equation (40).  
\[B_{\mathrm{exp}}(t)=k_{\mathrm{p}}\overline{y}_{\mathrm{E}}(t)+k_{\mathrm{d}} \dot{y}_{\mathrm{E}}(t)+k_{\mathrm{a}}\ddot{y}_{\mathrm{E}}(t) \tag{40}\]  
However, the phase distortion of the estimator at the low frequency range, which may degrade the stability of the levitation system, should be considered. Generally, the larger the time constant \(\tau\) is, the less the amplitude of the estimator will be at the ultra low frequency range, and the more seriously its phase distortion will be. Hence, the time constant \(\tau\) should be selected comprehensively according to the amplitude elimination and phase distortion. When the estimator \(1/\left(s+\tau\right)^{2}\) is adopted to replace the double-integrator \(1/s^{2}\), the block diagram of maglev vehicle-bridge system is updated as Figure 9.