# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
## 3 Wavelet-GMDH Based Inductance Estimation Method
### Group Method of Data Handling (GMDH)  
Model-based inductance estimation utilizes the same motor dynamics variables with extended EMF sensorless control. This leads to a partial contradiction in the response speed of the estimated rotor position and inductance, resulting in a time delay. However, the machine learning approach employing the GMDH algorithm predicts a system equation by selecting input variables, segmenting input-output variables, and defining partial expressions. As a result, despite the nonlinear relationship between input-output variables or the unspecified function type of the model, hierarchically combining partial expressions enables obtaining the estimation equation accurately without time delays. Thus, machine learning based on the GMDH algorithm can be seamlessly integrated into an embedded system. The GMDH algorithm is comprised of a layered network with a specific structure honed during training. It not only captures nonlinear dynamics via a mathematical model but also encompasses higher-order polynomials, thus ensuring stability. The sophisticated relationship between input and output variables can be illustrated through a complex polynomial series reminiscent of the Kolmogorov-Gabor polynomial [15]:  
\[y=a_{0}+\sum_{i=1}^{m}a_{i}x_{i}+\sum_{i=1}^{m}\sum_{j=1}^{m}a_{ij}x_{i}x_{j}+ \sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{m}a_{ijk}x_{i}x_{j}x_{k}+\cdots, \tag{16}\]  
where \(x\), \(y\), \(m\), and \(a\) represent the input into the system, the output of each regression layer, the number of inputs, and a coefficient, respectively.  
A GMDH algorithm can serve as a predictor for estimating the output of nonlinear complex systems, applicable to most scenarios represented by quadratic forms of input variables. The output of each quadratic neuron is calculated as  
\[y(x_{1},x_{2})=a_{0}+a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{1}^{2}+a_{4}x_{2}^{2}+a_{5} x_{1}x_{2}, \tag{17}\]  
where \(a_{i}(i=0,1,\ldots,5)\) are the weight coefficients of the quadratic neuron to be learned. The coefficients \(a_{i}\) are derived from linear regression analysis and a recursive algorithm, which iteratively minimizes errors when validating data at each layer.  
To train a GMDH network with inputs, it is essential to consider the combinations of all possible input pairs. A system of Gaussian normal equations must be solved to determine the coefficients for each model. The coefficient of nodes in each layer can be represented by [15; 16; 17]  
\[\mathbf{A}=\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{Y}\]  
where  
\[\mathbf{X}=\begin{bmatrix}\mathbf{Y}=\left[y_{1},y_{2},\cdots,y_{m}\right]^{T},\\ \mathbf{A}=\left[a_{0},a_{1},a_{2},a_{3},a_{4},a_{5}\right],\\ \end{bmatrix} \tag{18}\]  
Therefore, the output of each polynomial can be calculated as  
\[\mathbf{Y}=\mathbf{A}\times\mathbf{X}. \tag{19}\]  
To assess the adequacy of the partial description of the validation dataset, the linear regression index \(R_{k}\) is obtained as  
\[R_{k}=\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-g_{i})^{2}}{\sum_{i=1}^{n}y_{i}^{2}}},k =1,2,\cdots\frac{1}{2}m(m-1). \tag{20}\]  
In addition, to evaluate how well a regression model fits a dataset, it is necessary to compute the RMSE (root mean square error). This metric indicates the average deviation between the predicted values from the model and the actual values in the dataset.  
The RMSE can be calculated as  
\[RMSE=\sqrt{\frac{\sum(y_{i}-\hat{y}_{i})^{2}}{N}}, \tag{21}\]  
where \(\hat{y}_{i}\) is the predicted value for the \(i\)th observation, and \(N\) is the number of observations.  
The main function of GMDH involves the forward propagation of a signal through the network nodes, similar to the principle utilized in typical neural networks. Every layer comprises simple nodes, each executing its own polynomial transfer function and transmitting its output to the nodes in the subsequent layer.