## 3 Multi-Objective Optimization Design
### Optimization Objectives and Parameters  
In order to improve the output stability and smoothness of the built-in permanent magnet synchronous motor, as well as the overall efficiency of the motor under the condition of satisfying the output, a multi-objective optimization strategy is proposed here. Since reducing the torque pulsation can effectively reduce noise and vibration, and appropriately increasing the reverse electromotive force can effectively improve the motor efficiency, the torque ripple (\(T_{r}\)), output torque (\(T_{a}\)), and back EMF amplitude (\(E\)) were taken as the optimization objectives, and seven structural parameters were selected as the optimization parameters. Table 3 shows the names and value ranges of the optimization parameters, and Figure 4 shows the optimization variable diagram. Usually, when the torque ripple of the permanent magnet synchronous motor is less than 5%, the motor output is smoother; thus, considering that the output torque of the initial motor model is 9.7 Nm, and its back EMF amplitude is 245 V, in order to better improve the performance of the motor and satisfy the optimization requirements with the set of solutions obtained through algorithm optimization, the multi-objective optimization problem, as well as its constraints, can be expressed as follows:  
\[\min\left\{\begin{array}{l}f_{1}(x)=-T_{a}\\ f_{2}(x)=T_{r}\\ f_{3}  
Improved Latin hypercube sampling is mainly divided into three steps: stratification, sampling, and disorder. The specific steps are as follows:  
1. Determine the number of samples \(N\) and the dimension \(D\) to be sampled, and divide each dimension into N equal parts to obtain N values to be taken, denoted as \(V_{1}\), \(V_{2}\), \(\ldots\), \(V_{N}\);
2. For the first dimension, randomly take a number from \(V_{1}\), \(V_{2}\), \(\ldots\), \(V_{N}\) as the first sample value, remove it from the list, and then repeat for further sample values until the value of the Dth dimension is removed;
3. Finally, the obtained samples are discharged in columns to form a matrix of \(N\) rows and \(D\) columns, which is noted as the Latin hypercube sample matrix.  
Considering that the relationship between optimization objectives and design variables in the motor is more complex, the sampling points were selected with non-normal distribution; as such, it would be more rough to judge the sensitivity using a linear relationship, so the Spearman coefficient was used to describe the convergence between the objectives and the variables, as it does not require the sampling data to be normally distributed. Therefore, the correlation coefficient sensitivity analysis formula is as follows [27]:  
\[S(x)=\frac{\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})(Y_{i}-\overline{Y})}}{ \sqrt{\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})^{2}}}\sqrt{\sum\limits_{i=1}^ {N}{(Y_{i}-\overline{Y})^{2}}}} \tag{3}\]  
where \(x_{i}\) and \(Y_{i}\) are the design parameter values and design objective values of the ith sampling point; \(\overline{x}\), and \(\overline{Y}\) are the design parameter average and design objective average; and \(N\) is the number of sampling points. Since it was necessary to study the influence of a single parameter on multiple optimization objectives, we introduced a comprehensive sensitivity formula. Considering that permanent magnet synchronous motors have high requirements on torque ripple, and, at the same time, torque ripple serves as an important measure of motor output performance, weighting coefficients were introduced to reflect the importance of different optimization objectives, so as to ensure that the parameters that have greater impacts on torque ripple can be retained.