## 2 Modeling and Blade Modal Identification of AMB-Rotor System
### Dynamic Modeling of AMB-Rotor System
\begin{array}{l}m\ddot{x}=(k_{ia}i_{\rm AX}+k_{ha}m_{\rm AX})+(k_{ib }i_{\rm BX}+k_{hb}m_{\rm BX})\\ J_{r}\ddot{\beta}-J_{z}\Omega\dot{u}=(k_{ia}i_{\rm AX}+k_{ha}m_{\rm AX})a_{m} -(k_{ib}i_{\rm BX}+k_{hb}m_{\rm BX})b_{m}\end{array} \right.. \tag{4}\]  
The rotor displacement in the magnetic bearing coordinate system (\(m_{\rm AX},m_{\rm AX},m_{\rm BX},m_{\rm BY}\), and in the sensor coordinate system (\(d_{\rm AX},d_{\rm AY},d_{\rm BX},d_{\rm BY}\)) satisfy the following transformation relationships, respectively  
\[\left\{ \begin{array}{l}\alpha=\frac{m_{\rm BY}-m_{\rm AX}}{a_{a}+b_{\rm X}} \\ \beta=\frac{m_{\rm AX}-m_{\rm AX}}{a_{a}+b_{\rm X}}\\ x=\frac{b_{\rm x}m_{\rm AX}+a_{\rm m}m_{\rm BX}}{a_{a}+b_{\rm X}}\\ y=\frac{b_{\rm x}m_{\rm AX}+a_{\rm m}m_{\rm BX}}{a_{a}+b_{\rm X}}\end{array}  \right., \tag{5}\]  
\[\left\{ \begin{array}{l}\alpha=\frac{d_{\rm BY}-d_{\rm AX}}{a_{a}+b_{\rm X}} \\ \beta=\frac{d_{\rm AX}-d_{\rm BX}}{a_{a}+b_{\rm X}}\\ x=\frac{b_{\rm x}d_{\rm AY}+a_{\rm x}d_{\rm BX}}{a_{a}+b_{\rm X}}\\ y=\frac{b_{\rm x}d_{\rm AY}+a_{\rm x}d_{\rm BY}}{a_{a}+b_{\rm X}}\end{array}  \right.. \tag{6}\]  
By putting Equation (5) into (4), the following formula can be obtained  
\[\left\{ \begin{array}{l}m\ddot{x}=(k_{ha}+k_{hb})x+(k_{ha}a_{m}-k_{hb}b_{m}) \beta+k_{ia}i_{\rm AX}+k_{ib}i_{\rm BX}\\ J_{r}\ddot{\beta}-J_{z}\Omega\dot{u}=(k_{ha}a_{m}-k_{hb}b_{m})x+(k_{ha}a_{m}^{2}+ k_{hb}b_{m}^{2})\beta+k_{ia}a_{m}i_{\rm AX}-k_{ib}b_{m}i_{\rm BX}\end{array} \right.. \tag{7}\]  
By putting Equation (6) into (7), and by Laplace transformation, the following formula can be obtained  
\[\left\{ \begin{array}{l}\frac{m\dot{x}}{l_{s}}d_{\rm AX}(s)s^{2}+\frac{m_{a} }{l_{s}}d_{\rm BX}(s)s^{2}-\frac{k_{ha}}{l_{s}}ed_{\rm AX}(s)-\frac{k_{ha}}{l_{ s}}fd_{\rm BX}(s)=k_{ia}i_{\rm AX}(s)+k_{ib}i_{\rm BX}(s)\\ \frac{l_{s}}{l_{s}}d_{\rm AX}(s)s^{2}-\frac{l_{s}}{l_{s}}d_{\rm BX}(s)s^{2}- \frac{l_{s}}{l_{s}}[d_{\rm BY}(s)s-d_{\rm AX}(s)s]-\frac{k_{ha}}{l_{s}}gd_{ \rm AX}(s)-\frac{k_{ha}}{l_{s}}hd_{\rm BX}(s)=k_{ia}a_{m}i_{\rm AX}(s)-k_{ib}b_ {m}i_{\rm BX}(s)\end{array} \right. \tag{8}\]where \(l_{s}=a_{s}+b_{s}\), \(e=2b_{s}+a_{m}-b_{m}\), \(f=2a_{s}+b_{m}-a_{m}\), \(g=a_{m}b_{s}-b_{m}b_{s}+a_{m}^{2}+b_{m}^{2}\), \(h=a_{m}a_{s}-b_{m}a_{s}-a_{m}^{2}-b_{m}^{2}\). A block diagram of the AMB-rotor control system with single input and single output (SISO) is shown in Figure 2. According to (8), the model of the AMB-rotor system \(G_{r}(s)\) is  
\[G_{r}(s)=\frac{d_{\text{AX}}(s)}{i_{\text{AX}}(s)}=\frac{k_{ii}A}{s^{2}-k_{ha}B} \tag{9}\]  
where  
\[ \begin{array}{l}A=\frac{1}{m}+\frac{a_{m}a_{s}}{l_{s}},\\ B=\frac{a_{s}\left(a_{m}^{2}+b_{m}^{2}+a_{m}b_{s}-b_{m}b_{s}\right)}{l_{s}(a_{ s}+b_{s})}+\frac{2b_{s}+a_{m}-b_{m}}{m(a_{s}+b_{s})}.\end{array} \]  
For radial magnetic bearings, \(R\) is the coil resistance, and \(L\) is the coil inductance. The AMB-rotor system is driven by the switching power amplifier, \(k_{v}\) is a quantity related to the mains voltage and the PWM duty cycle, \(k_{ad}\) is the transfer factor of the analog-to-digital (A/D) converter, \(k_{ic}\) is the enlargement factor of the current sensor, and \(k_{amp}\) and \(k_{ico}\) are the preamplification factor and feedback factor of the current controlled circuit, respectively. The model of the amplifier \(G_{w}(s)\) is  
\[G_{w}(s)=\frac{k_{amp}k_{v}\frac{1}{L_{S}+R}}{1+k_{amp}k_{v}\frac{1}{L_{S}+R} \cdot k_{ic}k_{ad}k_{ico}}=K_{w}\frac{W_{w}}{s+W_{w}} \tag{10}\]  
where \(K_{w}=k_{amp}k_{v}/(R+k_{amp}k_{v}k_{ic}k_{ad}k_{ico})\), and \(W_{w}=(R+k_{amp}k_{v}k_{ic}k_{ad}k_{ico})/L\). The incomplete differential PID controller is used as the base controller, and the model of the controller \(G_{c}(s)\) is  
\[G_{c}(s)=k_{p}\left(1+\frac{1}{T_{iS}}+\frac{T_{d}s}{T_{f}s+1}\right) \tag{11}\]  
where \(k_{p}\) is the scale factor, \(T_{i}\) is the integration factor, \(T_{d}\) is the incomplete differential factor, and \(T_{f}\) is the time constant.