## 2 Active Magnetic Bearing Rotor System Model  
The AMB rotor system is an integrated electromechanical system, including an AMB, rotor, controller, power amplifier, displacement sensors and other modules. The system can monitor the state of the rotor through the displacement sensors and perform real-time closed-loop control of the rotor. Affected by various factors, the imbalance of the magnetic bearing rotor is inevitable. The unbalanced mass will cause the rotor center of mass to shift, resulting in a centrifugal force at the same frequency as the rotor speed. Figure 1 is a schematic diagram of AMB rotor structure. The coordinate system is established with the geometric center \(O\), \(G\) is the rotor center of mass, \(l_{a}\) is the distance from the plane center of the AMB at the \(A\) end to the rotor center of mass, \(l_{b}\) is the distance from the plane center of the AMB at the \(B\) end to the rotor center of mass, and \(l\) is the distance from the AMB at both ends. \(\theta_{x}\) is the angle at which the rotor rotates counterclockwise in the \(yz\) plane and \(\theta_{y}\) is the angle at which the rotor rotates counterclockwise in the \(xz\) plane. \(f_{xx},f_{bx},f_{ay}\) and \(f_{by}\) are the levitation forces exerted on the rotor by the AMB at both ends of \(A\) and \(B\) in the \(x\) and \(y\) directions, respectively. The displacements at both ends of the rotor are \(x_{a}\), \(y_{a}\), \(x_{b}\) and \(y_{b}\), respectively. \(x_{c}\) and \(y_{c}\) are the shifts of the center of mass in the \(x\) and \(y\) directions from the geometric center, respectively. The displacement of both ends has the following relationship with the offset of the center of mass. Figure 1: Schematic diagram of AMB rotor structure. \[\left\{ \begin{array}{l}x_{c}=\frac{l_{b}}{l}x_{a}+\frac{l_{f}}{l}x_{b}\\ \theta_{y}=\frac{x_{a}-x_{b}}{l}\\ y_{c}=\frac{l_{b}}{l}y_{a}+\frac{l_{f}}{l}y_{b}\\ \theta_{x}=\frac{y_{b}-y_{c}}{l}\\ \end{array} \right. \tag{1}\]  
We can write this in its matrix form, as shown in Equation (2). \[s_{c}=Tq \tag{2}\]  
\[s_{c}= \begin{bmatrix}x_{c}&\theta_{y}&y_{c}&\theta_{x}\end{bmatrix} ^{T}\text{ is the rotor mass characteristic matrix, }q= \begin{bmatrix}x_{a}&x_{b}&y_{a}&y_{b}\end{bmatrix} ^{T}\]  
is the displacement of the rotor in all directions. \(T= \begin{bmatrix}\frac{l_{b}}{l}&\frac{l_{b}}{l}&0&0\\ \frac{1}{l}&-\frac{1}{l}&0&0\\ 0&0&\frac{l_{b}}{l}&\frac{l_{b}}{l}\\ 0&0&-\frac{1}{l}&\frac{1}{l}\end{bmatrix} \) is the transformation matrix. According to [PERSON]'s second law and the law of rotor rotation at fixed point, the motion equation of the AMB rotor can be obtained, as shown in Equation (3). \[\left\{ \begin{array}{l}m\ddot{x}_{c}=f_{ax}+f_{bx}\\ I_{y}\ddot{\theta}_{y}-\omega I_{z}\dot{\theta}_{x}=-f_{bx}l_{b}+f_{ax}l_{a}\\ m\ddot{y}_{c}=f_{ay}+f_{by}\\ I_{x}\ddot{\theta}_{x}+\omega I_{z}\dot{\theta}_{y}=-f_{ay}l_{a}+f_{by}l_{b} \end{array} \right. \tag{3}\]  
where \(m\) is the mass of the rotor, \(\omega\) is the rotational speed of the rotor and \(I_{x}\), \(I_{y}\) and \(I_{z}\) are about the \(x\)-axis, \(y\)-axis and \(z\)-axis of inertia. We can write this in its matrix form, as shown in Equation (4). \[M\ddot{s}_{c}+G\dot{s}_{c}=BF \tag{4}\]  
\[M=diag(m,I_{y},m,I_{x})\text{ is the rotor mass characteristic matrix, }G= \begin{bmatrix}0&0&0&0\\ 0&0&0&-I_{z}\omega\\ 0&0&0&0\\ 0&I_{z}\omega&0&0\end{bmatrix} \]  
is the gyro matrix and \(B= \begin{bmatrix}1&1&0&0\\ I_{a}&-l_{b}&0&0\\ 0&0&1&1\\ 0&0&-l_{a}&l_{b}\end{bmatrix} \), \(F= \begin{bmatrix}f_{ax}\\ f_{bx}\\ f_{ay}\\ f_{by}\end{bmatrix} \) is the electromagnetic force matrix of the rotor. When the rotor is near the equilibrium point, the linear expression of the AMB electromagnetic force can be obtained as shown in Equation (5) [19]. \[f=k_{i}i+k_{x}x \tag{5}\]  
where \(f\) is the electromagnetic force, \(k_{i}\) is the current stiffness coefficient, \(k_{x}\) is the displacement stiffness coefficient, \(i\) is the coil current and \(x\) is the rotor displacement.