# Levitating Control System of Maglev Ruler Based on Active Disturbance Rejection Controller
## 2 Analysis Model of Levitating Force
### Dynamic Model of Levitating System  
According to the analysis of electromagnetic suction force in Figure 1, the MC has two degrees of freedom in the levitating system: (1) vertical motion in the z-direction; (2) rotation around the y-axis with the center of mass of the MC as the center of rotation, denoted by \(\psi\). This paper uses [PERSON]'s second law of motion to describe the vertical motion of the MC and uses the torque equation to describe the rotational motion of the MC [18]. \[F=m\cdot\frac{\partial^{2}z_{a}}{\partial t^{2}} \tag{39}\]  
\[M_{a}=J_{a}\cdot\frac{\partial^{2}\alpha}{\partial t^{2}} \tag{40}\]  
In Equation (39), \(F\) is the sum of all forces on the MC in the z-direction, \(m\) is the mass of the MC, and \(z_{a}\) is the displacement of the mass center of the MC in the z-direction. In Equation (40), \(M_{a}\) is the torque of the resultant force of \(F\), \(J_{a}\) is the moment of inertia, and \(\alpha\) is the rotation angle. Figure 7 shows the position of the mass center, the force points of the levitating forces on the left and right ends of the MC, and the length of the arm of force. \[J_{a}=m_{h}r^{2} \tag{41}\]  
In Equation (41), \(m_{h}\) is the mass of the mass center of the MC to the end, and \(r\) is the distance from the mass center to the point of action of the levitating force. From this, \(J_{a}\) can be calculated: \(J_{a}=1.49\ \times 10^{-2}\) kg\(\cdot\)m\({}^{2}\). Ignoring other disturbances, such as base vibration and horizontal thrust, the sum of all forces on the MC in the z-direction is:  
\[F=-F_{g}+F_{left}+F_{right} \tag{42}\]  
Figure 6: (a) The calculation relationship between \(z\) and \(F_{left}\). (b) The calculation relationship between \(i_{a}\) and \(F_{left}\). Figure 7: Force analysis of the MC. In Equation (42), \(F_{g}=mg=17.93\) N is the gravity force on the MC. From the dimensions indicated in Figure 7, the moment equation around the y-axis is as follows:  
\[M_{\alpha}=-0.1275\times F_{left}+0.1275\times F_{right} \tag{43}\]  
According to Equation (40), a new equation can be obtained:  
\[-0.1275\times F_{left}+0.1275\times F_{right}=J_{\alpha}\cdot\frac{\partial^{2 }\psi}{\partial t^{2}} \tag{44}\]  
From Equations (39) and (42), a new equation can be obtained:  
\[F_{left}+F_{right}-F_{g}=m\cdot\frac{\partial^{2}z_{a}}{\partial t^{2}} \tag{45}\]  
Thus, the rigid body dynamics equation of the MC is obtained:  
\[ \begin{bmatrix}1&1\\ -0.1275&0.1275\end{bmatrix} \cdot \begin{bmatrix}F_{left}\\ F_{right}\end{bmatrix} + \begin{bmatrix}17.93\\ 0\end{bmatrix} = \begin{bmatrix}1.83&0\\ 0&0.0273\end{bmatrix}\begin{bmatrix}\frac{\partial^{2}z_{a}}{\partial t^{2}} \\ \frac{\partial^{2}\psi}{\partial t^{2}}\end{bmatrix}  \tag{46}\]  
From Equations (33) and (34), it can be seen that \(F_{left}\) and \(F_{right}\) are related to \(z_{l}\) and \(z_{r}\), which are in the \([z_{l},\,z_{r}]\) coordinate system with the air gap length as the research point. However, Equation (45) is in the \([z_{a},\psi]\) coordinate system with the mass center as the research point. A coordinate transformation is needed to convert the air gap length coordinate system to the mass center coordinate system. As shown in Figure 1b, when the MC rotates around the y-axis, there is a relationship between \(z_{l}\) and \(z_{r}\), the displacement of the mass center \(z_{a}\), and the rotation angle \(\psi\) as follows:  
\[z_{l}=z_{a}-0.1275\times\tan\psi \tag{47}\]  
\[z_{r}=z_{a}+0.1275\times\tan\psi \tag{48}\]  
Due to the very small length of the air gap, the rotation angle \(\psi\) is also very small, so tan\(\psi\)\(\approx\)\(\psi\). Therefore, Equations (47) and (48) can be written as:  
\[ \begin{bmatrix}z_{l}\\ z_{r}\end{bmatrix} = \begin{bmatrix}1&-0.1275\\ 1&0.1275\end{bmatrix}\begin{bmatrix}z_{a}\\ \psi\end{bmatrix}  \tag{49}\]  
The above equation can be used to convert the air gap length coordinate system into the centroid coordinate system.