# Effects of Imperfect Assembly and Magnetic Properties on the Three-Pole AMB System
## 4 Simulation and Experimental Results  
For numerical simulations and experimental validation, the system is shown in Figure 3, with parameters given by  
\[m=2.45\times 10^{-2}\text{ kg},\ g=9.81\text{ m/s}^{2},\ \mu=4\pi\times 10^{-7}\text{ H/m},\ A=3.6\times 10^{-5}\text{ m}^{2},\ N=350,\ l_{0}=3\times 10^{-4}\text{ m}\] From calibration, we have \(k_{1}=0.763\), \(k_{2}=0.468\), \(k_{3}=0.905\), indicating the strong non-uniformity of the magnetic flux distributions. For simulations, it is assumed that \(\Delta x=\Delta y=0.1l_{0}\). Under such an assumption, it can be shown that \(\left\|\Delta\right\|_{\infty}\leq 0.75\left\|\tilde{i}\right\|_{\infty}\). The parameters of the ISMC controller are: \(\rho=60\), \(k=0.8\), \(\alpha=20\), \(\varepsilon=0.01\), \(b_{1}=1,\ b_{2}=5\). The rotor is initially at rest on the back-up bearing, i.e., the initial condition is \( \begin{bmatrix}x_{r}(0)&y_{r}(0)\end{bmatrix} = \begin{bmatrix}0&-1.528\times 10^{-4} \text{ m}\end{bmatrix} \) and \( \begin{bmatrix}\dot{x}_{r}(0)&\dot{y}_{r}(0)\end{bmatrix} = \begin{bmatrix}0&0 \end{bmatrix} \). Figures 4 and 5 show the simulation results, where the rotor trajectory is shown in Figure 4 and the rotor displacement responses are shown in Figure 5. As one can see, the rotor is first deviated a little bit to the right and overshoots to the upper position before settling down to the center eventually. This is due to the fact that we have assumed the assembly errors to be positive. Hence, the magnetic forces to the right and to the upper surface are larger since the air gaps are smaller. Figure 4: Simulation result: rotor trajectory. Figure 3: The 3-pole AMB system under study. For the experimental validation, the rotor displacements are measured by the laser displacement sensor of Keyence, LK-G35 with a resolution of 0.05 \(\upmu\)m. The power amplifier is made by Advanced Motion Control with Model 25A20. The controller is implemented using dSPACE's DS 1103 motion-control card. The same control parameters as the simulation ones have been taken. To de-noise the measured signal, a Butterworth filter with bandwidth 600 Hz is taken:  
\[H(z)=\frac{0.0008+0.0032z^{-1}+0.0048z^{-2}+0.0032z^{-3}+0.0008z^{-4}}{1-3.0176z ^{-1}+3.5072z^{-2}-1.8476z^{-3}+0.3708z^{-4}}\]  
For the experimental results, the rotor trajectory is shown in Figure 6, and the steady-state rotor trajectory is shown in Figure 7. The rotor displacements are shown in Figure 8, and the control currents are shown in Figure 9. It should be note that the rotor is initially at rest on the back-up bearing and the levitation controller is activated at \(t=2\) s. It is clear that the performance is not as good as the simulation one, with longer settling time and larger steady state error of around 20 um. Recall that it is assumed that the assembly error is \(\Delta x=\Delta y=0.1l_{0}\) for numerical simulations. However, the actual assembly error is unknown. Also, for the non-uniformity of the magnetic flux distributions, the parameters \(k_{1}=0.763\), \(k_{2}=0.468\), \(k_{3}=0.905\) are taken in the simulations. These parameters were obtained by calibrations. Again, there must be some deviations from the actual parameters. Hence, Figure 8 (experimental response) cannot correspond to Figure 5 (simulation response). Figure 5: Simulation result: rotor displacement responses. Figure 6: Experimental result: rotor trajectory. Figure 7: Steady-state rotor trajectory. Figure 8: Experimental result: rotor-displacement responses. Figure 4:0 Experimental result: the control currents. In Figure 8, one can observe high frequency vibrations and some \"irregularity\" in the rotor response. The \"irregularity\" shown in Figure 8 is mostly in the transient period (before 15 s), which could be due to the un-modeled higher-order dynamics. As shown in Figure 7, the high-frequency vibrations at the steady state period (after 15 s) are confined within about 20 um, which are due to sensor noise.