## 4 Multi-Structural Optimization of BPMSM
### Dimensionless Multi-Structural Optimization Model  
To compare the different optimized performances of different units, it is necessary to standardize the raw data and transform them into dimensionless values. Normalization is a typical process of data standardization. From the selection of the BPMSM performance in Section 3, we can determine that \(k_{z}\), \(k_{a}\), \(k\), and \(T\) are benefit-based performances, but \(P_{Cu}\) and \(P_{Fe}\) are cost-based performances. Thus, the performance min-max normalizations can be defined as follows:  
\[\text{Benefit-based performance}:\ x_{ij}^{\prime}=\frac{x_{ij}-min(x_{1j}, \cdots x_{nj})}{max(x_{1j},\cdots,x_{nj})-min(x_{1j},\cdots x_{nj})} \tag{9}\]  
\[\text{Cost-based performance}:\ x_{ij}^{\prime}=\frac{max(x_{1j},\cdots x_{nj})- x_{ij}}{max(x_{1j},\cdots,x_{nj})-min(x_{1j},\cdots x_{nj})} \tag{10}\]  
where \(x_{ij}\) is the parameter of the \(j^{th}\) optimized performance of the \(j^{th}\) test scheme in the sample dataset. Figure 7: The error percentage of the optimized performance of the test samples. By using the linear weighted method to combine the regression model, a dimensionless multi-objective optimization function can be expressed as follows:  
\[f_{opt}=max(\omega_{1}x_{i1}^{\prime}+\omega_{2}x_{i2}^{\prime}+\omega_{3}x_{i3} ^{\prime}+\omega_{4}x_{i4}^{\prime}+\omega_{5}x_{i5}^{\prime}+\omega_{6}x_{i6}^ {\prime}) \tag{11}\]  
where \(\omega_{i}\) (\(i=1,\,\ldots,6\)) is the weight coefficient of the \(i^{\text{th}}\) regression model. In this paper, the analytic hierarchy process (AHP) is utilized to determine the weight of each optimized performance in the multi-structural optimization function. The process of AHP is shown in Figure 8.  
When using AHP to determine the weights, the importance degrees of the optimized performance need to be derived using the pairwise comparison method, and then a judgment matrix is constructed; \(P_{Cu}\) is large, but \(P_{Fe}\) is small. To improve the efficiency of the motor, \(P_{Cu}\) and \(T\) are selected as the most important optimized performances. To achieve better controllability, \(k\) is selected as the second most important optimized performance. For better passive levitation, \(k_{z}\) and \(k_{a}\) are selected as the third most important optimized performances. Lastly, \(P_{Fe}\) is selected as the fourth most important optimized performance. The corresponding weights, \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\), \(\omega_{4}\), \(\omega_{5}\), and \(\omega_{6}\) of \(k_{a}\), \(k_{z}\), \(P_{Fe}\), \(P_{Cu}\), \(k\), and \(T\) add up to 1. The calculation results of the weights are obtained using MATLAB, as shown in Table 4.  
For multi-structural optimization of the BPMSM, certain constraints must be satisfied:  
1. The rotor diameter-to-height ratio is an important parameter that influences the axial levitation force and tilting levitation torque. In order to achieve a larger axial levitation force and tilting levitation torque, the optimal size of the BPMSM rotor diameter-to-height ratio satisfies \(\frac{d}{h}\in[2.5,4]\). 2. Due to the power design requirement of the BPMSM, \(P_{e}=120\) W, the electromagnetic torque needs to meet \(T\geq 9550\frac{P_{e}}{n_{N}}\) \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Performance Rank** & \(\mathbf{\omega_{1}}\) & \(\mathbf{\omega_{2}}\) & \(\mathbf{\omega_{3}}\) & \(\mathbf{\omega_{4}}\) & \(\mathbf{\omega_{5}}\) & \(\mathbf{\omega_{6}}\) \\ \hline \(P_{Cu}>T=k>k_{z}=k_{a}>P_{Fe}\) & 0.107 & 0.107 & 0.064 & 0.332 & 0.195 & 0.195 \\ \(P_{Cu}=T>k_{z}=k_{a}>P_{Fe}\) & 0.098 & 0.098 & 0.059 & 0.288 & 0.169 & 0.288 \\ \(P_{Cu}>T>k>k_{z}=k_{a}>P_{Fe}\) & 0.088 & 0.088 & 0.054 & 0.375 & 0.151 & 0.244 \\ \(P_{Cu}=T=k>k_{z}=k_{a}>P_{Fe}\) & 0.122 & 0.122 & 0.069 & 0.229 & 0.229 & 0.229 \\ \(T>P_{Cu}>k>k_{z}=k_{a}>P_{Fe}\) & 0.088 & 0.088 & 0.054 & 0.244 & 0.151 & 0.375 \\ \(k>T=P_{Cu}>k_{z}=k_{a}>P_{Fe}\) & 0.107 & 0.107 & 0.064 & 0.195 & 0.332 & 0.195 \\ \hline \end{tabular}
\end{table}