# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 2 Vertical Dynamic Modeling of the Electromagnetic Levitation System
### Linear Model in the Frequency Domain  
Under the levitation control, the levitation electromagnet module strictly tracks the space trajectory of the track, but in this process, the levitation gaps fluctuate around the set gap \(\delta_{0}\). The linear model can be used to simplify the analysis process without introducing obvious errors. By using the Laplace transform function, the linear model is changed into the frequency domain. Define \(k_{u1}\left(s\right)=e^{-\frac{4i\delta  
\[B_{s}=\left[{0\quad b_{2}\quad 0\quad b_{4}\quad b_{5}\quad b_{6}}\right]^{T} \tag{28}\]  
\[C_{s}=\left[{ \begin{matrix}1&0&0&0&0&0\\ 0&0&1&0&0&0\end{matrix} }\right] \tag{29}\]  
\[D_{s}=\left[{ \begin{matrix}-k_{u1}&-1\end{matrix} }\right]^{T} \tag{30}\]  
\[ \begin{cases}a_{21}=3.5k_{2}m_{e}^{-1}\\ a_{23}=0.5k_{2}m_{e}^{-1}\\ a_{25}=-5k_{i}m_{e}^{-1}\\ a_{26}=k_{i}m_{e}^{-1}\\ a_{51}=\left({k_{c}k_{P}+k_{c}k_{s}}^{-1}+3.5k_{2}k_{c}k_{A}m_{e}^{-1}\right)2 ^{-1}L_{0}^{-1}s^{-1}\\ a_{52}=0.5k_{c}k_{D}L_{0}^{-1}+0.75k_{i}L_{0}^{-1}\\ a_{53}=0.25k_{2}k_{c}k_{A}m_{e}^{-1}L_{0}^{-1}\\ a_{54}=0.25k_{i}L_{0}^{-1}\\ a_{55}=-RL_{0}^{-1}-0.5k_{c}L_{0}^{-1}-2.5k_{c}k_{A}k_{i}m_{e}^{-1}L_{0}^{-1}\\ a_{56}=0.5k_{c}k_{A}k_{i}m_{e}^{-1}L_{0}^{-1}\end{cases}  \tag{31}\]  
\[ \begin{cases}b_{2}=\left({-5k_{u2}+k_{u3}}\right)k_{2}m_{e}^{-1}\\ b_{4}=\left({-5k_{u3}+k_{u2}}\right)k_{2}m_{e}^{-1}\\ b_{5}=-\frac{\left({k_{c}k_{P}+k_{c}k_{s}}^{-1}\right)k_{41}+k_{c}k_{A}k_{s}m_{ e}^{-1}\left({5k_{u2}-k_{u3}}\right)+2k_{i}k_{u2}s}{2\ell_{0}}\\ b_{6}=-\frac{\left({k_{c}k_{P}+k_{c}k_{s}}^{-1}\right)k_{41}+k_{c}k_{A}k_{s}m_{ e}^{-1}\left({5k_{u3}-k_{u2}}\right)+2k_{i}k_{u3}s}{2\ell_{0}}\end{cases}  \tag{32}\]  
The transfer functions from \(u\left(s\right)\) to \(Y\left(s\right)\):  
\[G_{s}\left(s\right)=C_{s}\left({sI_{6\times 6}-A_{s}}\right)^{-1}B_{s}+D_{s} \tag{33}\]  
Then, the dynamic response of the levitation gap is:  
\[Y\left(s\right)=G_{s}\left(s\right)u\left(s\right) \tag{34}\]  
The track irregularity is generally considered as a stationary random process [4], which is generally analyzed by a PSD function. For each frequency in the PSD function, there is no phase correlation with other frequencies, so the discrete frequency analysis method can be used to analyze each frequency separately. Under the excitation of the track irregularity, the levitation gap signal is also a stationary random signal. Therefore, the variance of the time-domain levitation gap signal can be obtained by the spectral integration of the PSD functions according to [PERSON]'s theorem. To simplify the analysis, we transform the PSD function into the amplitude spectrum function as the main function. The space trajectory of the track in a constant wavelength \(\lambda\) at a constant speed \(v\) is modeled as a sinusoid function:  
\[y_{t2}\left(t\right)=A_{r}\sin\left(wt\right)+A_{r0} \tag{35}\]  
where \(w=2\pi v/\lambda\), \(A_{r}\) is the amplitude of the track, and \(A_{r0}\) is the constant value. To simplify the analysis, we set \(A_{r0}=0\). Then, the output levitation gaps are deduced as:  
\[Y\left(t\right)=A_{r}\left|G_{s}\left(s=jw\right)\right|\sin\left(wt+\angle G_ {s}\left(s=jw\right)\right) \tag{36}\]  
Therefore, the amplitude of levitation gaps are deduced as:  
\[\left|\delta\left(s=jw\right)\right|=A_{r}\left|G_{s}\left(s=jw\right)\right| \tag{37}\]The amplitude of the levitation gap is composed of two parts, the amplitude of the track \(A_{r}\) and the amplitude of transfer function \(\left|G_{s}\right|\). To simulate the real track situation, we choose the German low interference track PSD function as the input source in this paper. The PSD function \(S_{v}\left(\Omega\right)\) is:  
\[S_{v}\left(\Omega\right)=\frac{A_{v}\Omega_{c}^{2}}{\left(\Omega^{2}+\Omega_{r }^{2}\right)\left(\Omega^{2}+\Omega_{c}^{2}\right)} \tag{38}\]  
where \(\Omega=2\pi\lambda^{-1}\), \(A_{v}=4.032\times 10^{-7}\) m\({}^{2}\cdot\)rad/m, \(\Omega_{c}=0.8246\) rad/m, and \(\Omega_{r}=0.0206\) rad/m.