# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 2 Modeling of Vehicle-Bridge Interaction
### Modeling of Levitation System  
Suppose the turns of a single electromagnet is \(N\), the pole area is \(A\), and the magnetic permeability of vacuum is \(\mu_{0}\). Then, for a single electromagnet [3], the balance equations of electromagnetic force \(F_{\mathrm{E}}(t)\) and voltage \(u(t)\) are  
Figure 1: The minimum model of Maglev vehicle-bridge system. Red arrows: \(F_{\mathrm{E}}\) is electromagnetic force; blue arrows: \(y_{\mathrm{B}}\) and \(y_{\mathrm{E}}\) are the vertical displacements of bridge and electromagnet. \[\left\{ \begin{array}{l}F_{\rm E}(t)=\frac{\mu_{0}\Lambda N^{2}}{2}\Big{(}\frac{i( t)}{\delta(t)}\Big{)}^{2}\\ 2 Ri(t)+\frac{\mu_{0}\Lambda N^{2}}{\delta(t)}i(t)-\frac{\mu_{0}\Lambda N^{2}i(t) }{\delta^{2}(t)}\dot{\delta}(t)=u(t)\end{array} \right. \tag{8}\]  
where \(R\) is the resistance, \(i(t)\) is the current of electromagnet. In light of Equation (8), the balance equation of voltage is related with the four variables, control voltage \(u(t)\), current \(i(t)\), levitation gap \(\delta(t)\) and its derivative \(\dot{\delta}(t)\). Besides, the value of \(F_{\rm E}(t)\) refers to the two variables, current \(i(t)\) and levitation gap \(\delta(t)\). To simplify the above relationship, the magnetic flux \(B(t)\) instead of the current \(i(t)\) may be adopted when developing the dynamic equations. In this way, Equation (8) is updated as  
\[\left\{ \begin{array}{l}F_{\rm E}(t)=\frac{2A}{\mu_{0}}B^{2}(t)\\ 2 NA\dot{B}(t)+\frac{4R}{\mu_{0}N}\delta(t)B(t)=u(t)\end{array} \right. \tag{9}\]  
In light of Equation (9), it can be seen that the balance equation of voltage is related with three variables and the electromagnetic force is solely determined by the magnetic flux. Compared with Equation (8), the dynamic equation is much simpler and clearer, which may be favorable adopted for the synthesis of the vehicle-bridge interaction system. Generally, the natural frequency of air springs is far less than the bandwidth of the levitation control system and the modal frequencies of bridges, so the dynamics of sprung mass is neglected. Combining Equation (7), the movement of electromagnet is  
\[m_{\rm E}\ddot{y}_{\rm E}(t)=-F_{\rm E}(t)+(m_{\rm C}+m_{\rm E}){\rm g} \tag{10}\]  
where the variable \(y_{\rm E}(t)\) is the vertical displacement of electromagnet, g is the acceleration of gravity, \(m_{\rm C}\) is the sprung mass, and \(m_{\rm E}\) is the mass of electromagnet. According to Equations (9) and (10), it can be seen that the steady voltage of electromagnet is  
\[u_{0}=2R\delta_{0}\sqrt{2(m_{\rm C}+m_{\rm E}){\rm g}/(\mu_{0}N^{2}A)} \tag{11}\]  
where the variable \(\delta_{0}\) is the desired clearance between the upper surface of the electromagnet and the lower surface of the guideway. Traditionally, the cascade control, the inner-loop with current feedback, the outer-loop with states feedback (proportion, damping and acceleration feedback) is widely adopted in maglev engineering. It gives that  
\[u(t)=k_{\rm C}\big{[}i_{\rm exp}(t)-i(t)\big{]}+u_{0} \tag{12}\]  
\[i_{\rm exp}(t)=k_{\rm p}e(t)+k_{\rm d}\dot{y}_{\rm E}(t)+k_{\rm a}\ddot{y}_{ \rm E}(t) \tag{13}\]  
where the variable \(i_{\rm exp}(t)\) is the desired current of electromagnet and the error \(e(t)\) of levitation gap is set as  
\[e(t)=y_{E}(t)-y_{\rm B}(t)-\delta_{\rm set} \tag{14}\]  
where \(\delta_{\rm set}\) is the expected levitation gap. To stabilize the maglev vehicle-bridge interaction system with magnetic flux feedback, a similar cascade control scheme, the inner-loop with magnetic flux feedback, the outer-loop with states feedback, is adapted, which gives  
\[u(t)=k_{\rm B}\big{[}B_{\rm exp}(t)-B(t)\big{]}+u_{0} \tag{15}\]  
\[B_{\rm exp}(t)=k_{\rm p}e(t)+k_{\rm d}\dot{y}_{\rm E}(t)+k_{\rm a}\ddot{y}_{\rm E }(t) \tag{16}\]  
where \(B_{\rm exp}(t)\) is the desired magnetic flux of levitation gap. To draw the main innovation of this work, combining with Equation (14), Equations (13) and (16) may be rewritten as Equation (17) when the parameter \(\overline{k}_{\rm p}\) is equal to parameter \(k_{\rm p}\). \[\left\{