## 3 Proposed Intelligent Weighting Factors Design Approach
### MOGA-Based NN Weighting Factor Tuning Approach  
As explained, the DRL-based strategy has to be compared to a baseline method to assess its efficiency. The baseline considered in this paper is the neural network presented in Figure 4. This network has two hidden layers, the first one having 15 neurons using the poslin activation function and the second one having 10 neurons using the tansig activation function. The inputs of the network are the \(q\)-axis current error \(\Delta i_{q}(k)=i_{q}^{*}(k)-i_{q}(k)\) at time instant \(k\), the integral of this error, and the number of commutations \(\Xi(k)\) between time instants \(k-1\) and \(k\). The outputs are the weighting factors \(\lambda_{i}\) and \(\lambda_{f}\). **Remark 3**. : _To select the best set of signals as the neural network inputs, several combinations were tried. For each set of input signals, the performance of the control strategy of Figure 3 was evaluated using the neural network resulting from the supervised learning procedure described below. For test cases, the integral square error (ISE), integral time-weighted square error (ITSE), integral absolute error (IAE), and integral time-weighted absolute error (ITAE) of the errors \(\Delta i_{q}\) and \(\Delta i_{d}(k)=i_{d}^{*}(k)-i_{d}(k)\) were evaluated. As can be seen on Table 2 on one example of input signals, the set of inputs we selected always performed better, hence its selection._  
To train this neural network, an optimal value of the weighting factors is first obtained using a MOGA implemented through the Matlab 2024a function gamultiobj. This algorithm uses a population of \(N_{\text{MOGA}}\) individuals. To each individual is assigned a pair of weighting factors \((\lambda_{i},\lambda_{f})\) drawn randomly such that \(\lambda_{i}\in[\underline{\lambda},\overline{\lambda}_{i}]\) and \(\lambda_{f}\in[\underline{\lambda}_{f},\overline{\lambda}_{f}]\). For each individual, a simulation using the structure Figure 3 is run by fixing the weighting factors to those of the individual and using the parameters Table 3 for the different elements  
Figure 3: FCS-MPCC-based control design of SPMSM. in the structure. The speed controller is a PI controller whose parameters are also given in Table 3. The best individuals are selected based on two objectives:  
* The total number of transitions over the simulation, i.e., the sum of the \(\Xi(k)\);
* The ITAE of the \(q\)-axis current \(\sum_{k=0}^{N}k|\Delta i_{q}(k)|\) where \(N\) is the total number of time steps in the simulation. Based on the obtained results, a new generation of individuals is generated using the default methods of the gamultiobj function. This process is then repeated over a maximum of \(N_{\text{gen}}\) generations. Through the use of MOGA, a Pareto front of weighting factors balancing the two objectives is obtained. The structure of Figure 3 is then simulated under different conditions using different pairs from the Pareto front as weighting factors and the parameters of Table 3. The different conditions include varied steady-state and transient conditions, load \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Input Features**} & \multicolumn{4}{c}{\(\Delta i_{q}\) **Error Metrics**} & \multicolumn{4}{c}{\(\Delta i_{d}\) **Error Metrics**} \\ \cline{2-10}  & **ISE** & **ITSE** & **IAE** & **ITAE** & **ISE** & **ITSE** & **IAE** & **ITAE** \\ \hline \([\Delta i_{q}(k),\int\Delta i_{q}(k),\Xi(k),\) & 0.08091 & 0.00500 & 0.02630 & 0.02046 & 0.05033 & 0.00207 & 0.00775 & 0.01292 \\ \(S_{1}(k-1),S_{3}(k-1),S_{5}(k-1)]\) & 0.068 27 & 0.00289 & 0.02044 & 0.01581 & 0.03613 & 0.00083 & 0.00395 & 0.00836 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance metrics for neural network input features. Figure 4: Neural network architecture.