# Optimization Design and Control of Six-Phase Switched
## 3 Surrogate Modeling Method
### Kriging Model  
The ranges of optimization variables are listed in Table 2. It requires a large number of samples if directly applying FEMs to reflect the relationship between optimization objectives and variables. Kriging model, one of the effective surrogate modeling methods, can be used to represent the relationship between the optimization parameters and objectives. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Initial Value** & **Scope** \\ \hline Yoke width ratio (\(\lambda_{\text{hcs}}\)) & \(x_{1}\) & 0.8 & [0.8, 1] \\ A-phase turn-on angle (\(\alpha_{\text{Aon}}/^{\circ}\)) & \(x_{2}\) & 13 & [13; 14] \\ B-phase turn-on angle (\(\alpha_{\text{Bon}}/^{\circ}\)) & \(x_{3}\) & 19 & [18; 19] \\ A-phase conduction width (\(P_{\text{dA}}/^{\circ}\)) & \(x_{4}\) & 16 & [15; 16] \\ B-phase conduction width (\(P_{\text{dB}}/^{\circ}\)) & \(x_{5}\) & 16 & [15; 16] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization parameters and their range. Figure 4: Schematic of parameters in six-phase SRM. The relationship between the variables and the outputs can be obtained by the Kriging model based on the existing data achieved from the FEM. Given n sample points \([x_{1},x_{2},\ldots,x_{n}]\) and their responses \([y(x_{1}),y(x_{2}),\ldots,y(x_{n})]\), for a set of input parameters \(x\), the response of the Kriging model can be expressed as [29; 30; 31]  
\[y(x)=f(x)^{T}\beta+z(x) \tag{10}\]  
where \(f(x)^{T}\beta\) is the regression model, and \(z(x)\) is a random error term used for the modeling of local deviation, which is usually assumed to be a vector with a mean of zero, covariance \(\sigma^{2}\), and covariance matrix \(\text{cov}_{ij}\) as  
\[\text{cov}_{ij}=\sigma^{2}\mathbf{R}[R(\mathbf{x_{i}},\mathbf{x_{j}})] \tag{11}\]  
where \(\mathbf{R}\) is a correlation matrix that can be represented by the Gaussian correlation function matrix, and \(R\) is the user-specified correlation function  
\[\mathbf{R}= \begin{bmatrix}r(\mathbf{x_{1}},\mathbf{x_{1}})&r(\mathbf{x_{1}}, \mathbf{x_{2}})&\cdots&r(\mathbf{x_{1}},\mathbf{x_{n}})\\ r(\mathbf{x_{2}},\mathbf{x_{1}})&r(\mathbf{x_{2}},\mathbf{x_{2}})&\cdots&r( \mathbf{x_{2}},\mathbf{x_{n}})\\ \vdots&\vdots&\ddots&\vdots\\ r(\mathbf{x_{1}},\mathbf{x_{1}})&r(\mathbf{x_{1}},\mathbf{x_{1}})&\cdots&r( \mathbf{x_{1}},\mathbf{x_{1}})\end{bmatrix}  \tag{12}\]  
where \(r(\mathbf{x}_{i},\mathbf{x}_{j})=\exp\biggl{\{}-\sum\limits_{k=1}^{D}\alpha_{ k}\Big{|}\mathbf{x}_{ik}-\mathbf{x}_{jk}\Big{|}^{2}\biggr{\}}\), \(D\) is the dimension of the design values, and \(\alpha_{k}\) is a flexible parameter that varies randomly. The outcomes \(y(\mathbf{x})\) and \(\beta\) can be estimated by using the best linear unbiased estimator in statistics  
\[\left\{ \begin{array}{l}\hat{y}(x)=f(x)^{T}\hat{\beta}+g(x)^{T}R^{-1}(\mathbf{y}- \mathbf{F}\hat{\beta})\\ \hat{\beta}=\left(\mathbf{F}^{T}R^{-1}\mathbf{F}\right)^{-1}\mathbf{F}^{T}R^{-1}\mathbf{y} \end{array} \right. \tag{13}\]  
where \(\mathbf{F}\), \(r(\mathbf{x})\), and \(\mathbf{y}\) can be represented by  
\[\left\{ \begin{array}{l}\mathbf{F}=\left[\begin{array}{cccc}f_{1}(\mathbf{x}_ {1})&f_{2}(\mathbf{x}_{1})&\cdots&f_{q}(\mathbf{x}_{1})\\ \vdots&\vdots&\vdots\\ f_{1}(\mathbf{x}_{n})&f_{2}(\mathbf{x}_{n})&\cdots&f_{q}(\mathbf{x}_{n})\end{array} \right]\\ \\ g(x)=\left[\begin{array}{c}R(\mathbf{x},\mathbf{x}_{1})\\ \vdots\\ R(\mathbf{x},\mathbf{x}_{n})\end{array}\right]\\ \\ \mathbf{y}=[y(\mathbf{x}_{1}),y(\mathbf{x}_{2}),\ldots,y(\mathbf{x}_{n})]^{T} \end{array} \right. \tag{14}\]  
By using maximum likelihood estimation, \(\sigma^{2}\) can be estimated by:  
\[\hat{\sigma}^{2}=\frac{1}{n}(\mathbf{y}-\mathbf{F}\hat{\beta})^{T}\mathbf{R}^{-1}(\mathbf{y}- \mathbf{F}\hat{\beta}) \tag{15}\]  
Through the above analysis, the Kriging model of the \(12/10\) SRM can be established based on a certain number of sample points.