## 4 Parameter Identification  
In order to accurately estimate the motor temperature, it is necessary to identify the parameters in the proposed thermal model. Although the motor model was simplified to reduce the number of parameters, the calculation of these parameters is still very complicated. These parameters can be roughly estimated based on heat transfer theory, but the main disadvantage of this method is the high dependence on the motor geometry model and the impossibility of the real-time execution. Therefore, an offline parameter identification algorithm based on the multiple linear regression model is proposed in this paper under the consideration of the good linear relationship of the state equation for the simplified thermal model. This method does not depend on any motor geometry parameters and material characteristics. Thus, the proposed parameter identification scheme is performed with better robustness, which improves the accuracy of temperature estimation. After discretizing the state equation of the proposed thermal model, it can be discovered that the time derivative of the state variable (the node temperature) is a linear function related to the state variable and the input (the heat generation), which makes the multiple linear regression algorithm possible to perform offline parameter identification. The general form of the regression model is:  
\[ \begin{array}{l}Y=\beta X\\ y_{i}=\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{N}x_{N}+\varepsilon_{i}(i=1,2,3,\ldots,M)\end{array} , \tag{11}\]  
where the dependent variable, \(Y\), is a vector consisting of \(y_{i}\); the independent variable, \(X\), is a vector consisting of \(x_{i}\). \(y_{i}\) can be approximately expressed as a linear function of the independent variable, \(x_{1},x_{2},x_{3},\ldots,x_{N}\). \(\beta_{1},\beta_{2},\ldots,\beta_{N}\) represent the regression coefficient, and \(\varepsilon_{i}\) is the random error. In order to identify the parameters in the state equation, it is necessary to sample the stator temperature, rotor temperature, end cap temperature, and cooling water temperature of the motor according to Equation (3). The acquired stator temperature, rotor temperature, end cap temperature, and cooling water temperature and the calculated loss are taken as independent variables of the multiple linear regression function, as shown in Equation (12). \(T_{si}\), \(T_{Ri}\), \(T_{Ei}\), \(T_{Wi}\), \(P_{Si}\), and \(P_{Ri}\) (\(i=1,2,3\ldots M\)) represent the sampled stator temperature, rotor temperature, end cap temperature, and cooling water temperature and the calculated loss, respectively. The derivative of the sampled node temperature is regarded as the dependent variable of the multiple linear regression function, as shown in Equation (13). \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) are the stator temperature, rotor temperature, and endcap temperature, respectively:  
\[X_{snm}=\left[ \begin{array}{ccccc}T_{S1}&T_{S2}&T_{S3}&\ldots&T_{SM}\\ T_{R1}&T_{R2}&T_{R3}&\ldots&T_{RM}\\ T_{E1}&T_{E2}&T_{E3}&\ldots&T_{EM}\\ T_{W1}&T_{W2}&T_{W3}&\ldots&T_{WM}\\ P_{S1}&P_{S2}&P_{S3}&\ldots&P_{SM}\\ P_{R1}&P_{R2}&P_{R3}&\ldots&P_{RM}\\ \end{array} \right]=\left[ \begin{array}{ccccc}X_{1}&X_{2}&X_{3}&\ldots&X_{M} \\ \hline\end{array} \right], \tag{12}\]  
\[Y_{snm}=\left[ \begin{array}{c}Y_{1}\\ Y_{2}\\ Y_{3}\\ \end{array} \right]=\left[ \begin{array}{ccccc}\frac{\Delta T_{S1}}{\Delta t}& \frac{\Delta T_{S2}}{\Delta t}&\frac{\Delta T_{S3}}{\Delta t}&\ldots&\frac{ \Delta T_{SM}}{\Delta t}\\ \frac{\Delta T_{R1}}{\Delta t}&\frac{\Delta T_{R2}}{\Delta t}&\frac{\Delta T_ {R3}}{\Delta t}&\ldots&\frac{\Delta T_{EM}}{\Delta t}\\ \frac{\Delta T_{S1}}{\Delta t}&\frac{\Delta T_{E2}}{\Delta t}&\frac{\Delta T_ {E3}}{\Delta t}&\ldots&\frac{\Delta T_{EM}}{\Delta t}\\ \end{array} \right]. \tag{13}\]  
According to Equations (3) and (11), the relationship between the regression coefficient, \(\beta\), and the parameters of the motor state equation can be expressed as:  
\[\beta=\left[ \begin{array}{c}\beta_{1}\\ \beta_{2}\\ \beta_{3}\\ \end{array}