## 2 Optimal Design Strategy for a BLAC Motor in Integrated Brake Systems
### Optimal Design
#### 2.3.3 DOE with the Moving Least Square Method  
These types of experimental designs are frequently used together with second-order response models. The design consists of three types of points, as shown in Figure 10.  
- Axial points: \(2\times 3\) axial points are created by a screening analysis. - Cube points: \(2^{3}\) cube points are obtained from a full factorial design. - Center point: A single point in the center is created by a nominal design. Figure 8: Current vector control strategy by using the experimental load profile. Figure 9: Load pressure: (**a**) experimental load pressure profile (\(PI_{LOAD}\)); (**b**) concept of the motor and pump gear system of the integrated brake system. The objective functions Y1-Y6 for the optimal design of the motor in the integrated brake system were constructed using the MLSM [18]. The following matrix expresses the relationship between the responses and the variables:  
\[y^{\prime}=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}+\sum_{i=1,j\leq i}^{k}\beta_{i} x_{i}x_{j}+\varepsilon=X\beta+\varepsilon, \tag{2}\]  
where \(y^{\prime}\) denotes the vector of the fitted values from the true function \(y\), k is the number of design variables, X denotes the matrix of the levels of independent variables \(x_{i}\) and \(x_{j}\), \(\beta\) denotes the vector of the regression coefficients, and \(\varepsilon\) denotes the vector of random errors. The least-square function Ly(x) is defined by the following equation, which expresses the sum of weighted errors as follows:  
\[L_{y}(x)=\sum_{i=1}^{k}W(x)\big{(}y^{\prime}-y\big{)}^{2}=W(x)(X\beta(x)-y)^{2}, \tag{3}\]  
where W(x) denotes a weighting matrix as a function of the location, which is obtained from the weighting function, \(y\) is a known value obtained from the simulation results, and N is the number of CCDs. The weighting function W(x) uses the Gaussian function expressed in (4) and depicted in Figure 11.  
\[w(x)=exp\bigg{(}-\frac{x^{2}}{h^{2}}\bigg{)} \tag{4}\]  
Figure 11: Gaussian weighting function. Figure 10: Example points of a central composite design with three input parameters. The weighting matrix W(x) is constructed using the weighting function W(x) in the diagonal terms as follows:  
\[W(x)=\left[ \begin{array}{cccc}w(x-x_{1})&0&\ldots&0\\ 0&w(-x_{1})&0\\ \vdots&&\ddots&\vdots\\ 0&0&\ldots&w(x-x_{N})\\ \end{array} \right], \tag{5}\]  
where x denotes the vector of the approximate location, and x\({}_{\rm i}\) denotes the vector of the \(i\)th sampling point. To minimize Ly(x), the least square estimators must satisfy the following expressions:  
\[\frac{\partial L_{y}(x)}{\partial\beta(x)}=2X^{T}(W(x)(X\beta(x)-y))=0 \tag{6}\]  
\[2X^{T}(W(x)(X\beta(x)))=2X^{T}(W(x)y) \tag{7}\]  
The coefficients of the regressive equation \(\beta\)(x) are obtained using the matrix operation as follows:  
\[\beta(x)=\left[X^{T}W(x)X\right]^{-1}X^{T}W(x)y, \tag{8}\]  
where the coefficient \(\beta\)(x) denotes a function of position x. It should be noted that the procedure for calculating \(\beta\)(x) denotes a local approximation, and the moving processes perform a global approximation throughout the design domain. The simulation results of the CCD are listed in Table 2.  
The objective functions were modeled using the MLSM, such as meta modeling [18]. Figure 12 shows the response surfaces of the objective functions according to the change in the design variables x1 and x2 at x3 = 0.