# Identification Approach for Nonlinear MIMO Dynamics of Closed-Loop Active Magnetic Bearing System
## 2 System Identification on Controllers for Radial Magnetic Bearings
### System Identification on Controllers for URAMB  
There are two controllers applied to regulate the position deviations of a rotor by upper radial AMB (URAMB) along the X-axis and Y-axis, respectively. The controller for URAMB along the X-axis, \(G_{\text{UX\_C}}\), is composed of a notch filter and a regulator, i.e., \(G_{\text{UX\_Notch}}\) and \(G_{\text{UX\_Reg}}\), respectively, shown in Figure 4. That is, \(G_{\text{UX\_C}}=G_{\text{UX\_Reg}}G_{\text{UX\_Notch}}\). By direction of the signal flow, the notch filter is cascaded ahead of the regulator. To identify the controller for URAMB along the X-axis, the chirp signal is exerted to _Point S3_, shown in Figure 4, i.e., the output of the upper gap sensor along the X-axis and the corresponding response is recorded at _Point CP4_, i.e., the output of the regulator for URAMB along the X-axis. The resulting Bode plot of the controller for URAMB along the X-axis is depicted in Figure 7. It is observed that the frequency of the notch filter is around 550 Hz.  
Figure 6: Test rig of system identification on the controller.  
Figure 7: Bode plot of controller for URAMB along the X-axis.  
The order of the dynamic model of controller for URAMB along the X-axis is 4. The corresponding state space representation and the discrete-time transfer function are summarized as:  
\[\text{A}_{\text{UX\_C}}=\begin{bmatrix}-3072.915&-1593.095&0&0\\ 1593.095&-3072.915&0&0\\ 0&0&-635.947&-3357.384\\ 0&0&3357.384&-635.947\end{bmatrix} \tag{1}\]  
\[\text{B}_{\text{UX\_C}}=\begin{bmatrix}-147.249\\ 182.816\\ 13.242\\ -19.935\end{bmatrix} \tag{2}\]  
\[\text{C}_{\text{UX\_C}}=\begin{bmatrix}-14.930&-43.428&24.018&-37.681\end{bmatrix} \tag{3}\]  
\[\text{D}_{\text{UX\_C}}=0.06795 \tag{4}\]  
\[\text{G}_{\text{UX\_C}}=(0.06795\;x^{4}-0.4431\;x^{3}+0.9146\;x^{2}-0.7769\;x+ 0.2373)/\]  
\[\text{(}x^{4}-3.62\;x^{3}+4.94\;x^{2}-3.009\;x+0.6901), \tag{5}\]  
where \(T_{\text{S}}=0.05\) ms  
Since the purpose of the controller for URAMB along the X-axis is the same as the controller for URAMB along the Y-axis, i.e., to levitate the rotor stably and to suppress the position deviation of the rotor at the position of URAMB at a certain frequency, the Bode plot of the controller for URAMB along the Y-axis is almost the same as Figure 7. On the other hand, the state space representation and the discrete-time transfer function of the controller for URAMB along the Y-axis are listed as:  
\[\text{A}_{\text{UY\_C}}=\begin{bmatrix}-3117.165&-1570.445&0&0\\ 1570.445&-3117.165&0&0\\ 0&0&-615.177&-3332.377\\ 0&0&3332.377&-615.177\end{bmatrix} \tag{6}\]  
\[\text{B}_{\text{UY\_C}}=\begin{bmatrix}152.878\\ -187.223\\ 13.747\\ -19.596\end{bmatrix} \tag{7}\]  
\[\text{C}_{\text{UY\_C}}=\begin{bmatrix}15.508&43.751&24.061&-37.467\end{bmatrix} \tag{8}\]  
\[\text{D}_{\text{UY\_C}}=0.06957 \tag{9}\]  
\[\text{G}_{\text{UY\_C}}=(0.06957\;x^{4}-0.452\;x^{3}+0.9319\;x^{2}-0.7913\;x+ 0.2417)/\]  
\[\text{(}x^{4}-3.619\;x^{3}+4.936\;x^{2}-3.005\;x+0.6885), \tag{10}\]  
where \(T_{\text{S}}=0.05\) ms