## 3 Parameter Assessment for Optimization
### Levitation Performance  
As shown in Figure 3a, the axial levitation force (\(F_{z}\)) of the BPMSM increases linearly with axial offsets (\(z\)) ranging from \(-1\) to \(1\) mm, evenly, and the rotor tilting levitation torque (\(T_{b}\)) increases linearly with the tilting angle (\(\alpha\)) ranging from \(-1^{\circ}\) to \(1^{\circ}\), evenly. Therefore, the axial levitation force (\(F_{z}\)) and the tilting levitation torque (\(T_{b}\)) can be linearized as:  
\[F_{z}=-k_{z}\cdot z \tag{1}\]  
\[T_{b}=-k_{\alpha}\cdot\alpha \tag{2}\]  
Figure 2: Optimization flowchart.  
By using least-square (LS) fitting, the linear expression of \(F_{z}\) is: \(y=-2.12x-0.5\) with \(k_{z}=2.12\) N/mm, and the linear fitting expression of \(T_{b}\) is: \(y=-5.39x+0.02\) with \(k_{\rm{g}}=0.309\) N\(\cdot\)m/rad.  
The levitation windings are energized to generate a stable radial levitation force. The radial levitation force (\(F_{r}\)) increases linearly with the winding current (\(i\)), as shown in Figure 3a. The LS fitting is utilized to linearly fit the rotor radial levitation force (\(F_{r}\)) evenly at different currents, ranging from 0 to 2 A. The linear fitting expression is \(y=2.65x+0.145\) with \(k=2.65\) N/A.  
To analyze the error introduced in the linear fitting process, the fitted levitation performance and the experimental results based on the virtual prototype with the rotor axial offsets, tilting angles, and currents evenly ranging from \(-1\) to 1 mm, from \(-1^{\circ}\) to \(1^{\circ}\), and from 0 to 2 A, are compared. In total, 11 groups of points are derived from the fitting model and the virtual model in Ansoft Maxwell. To demonstrate the fitting accuracy, the relative fitting error (\(\epsilon\)) of \(F_{z}\), \(T_{b}\), and \(F_{r}\) are presented in Figure 3b. It can be seen in Figure 3a that, for each linear fitting line, there is one sample of which the value is close to zero, specifically, sample No. 5 for the axial levitation force (\(F_{z}\)), sample No. 6 for the tilting levitation torque (\(T_{b}\)), and sample No. 1 for the rotor radial levitation force (\(F_{r}\)). Considering that distortion of \(\epsilon\) will occur (\(\rightarrow\infty\)), these three sample points are excluded in computing the relative fitting error (\(\epsilon\)) presented in Figure 3b. It can be observed in Figure 3b that:  
* For the fitted value near the excluded samples close to zero, involving sample Nos. 4 and 6 for the axial levitation force (\(F_{z}\)), sample Nos. 5 and 7 for the tilting levitation torque (\(T_{b}\)) and sample Nos. 2 and 3 for the rotor radial levitation force (\(F_{r}\)), their relative fitting errors are more significant (maximum 16%).
* For other samples, the relative fitting error is comparatively small (most within 4%), indicating the linear fitting accuracy of BPMSM performance.  
Since the passive axial and tilting stiffness represent the levitation capability of the BPMSM and the force-current coefficient represents the radial levitation stability of the BPMSM, \(k_{z}\), \(k_{\rm{g}}\), and \(k\) are selected as the optimization objects for BPMSM performance improvement.