## 3 Proposed MPTC with Multi-Objective Ranking
### Multi-Objective Ranking-Based Strategy  
The multi-objective sorting-based MPTC, aiming to eliminate the dimensional and magnitude differences between motor torque and flux, independently evaluates the control effectiveness of each control variable after obtaining the predicted values of the control variables. These variables are then ranked using dimensionless sorting results to represent the control effectiveness for the control objectives [38]. The optimal solution is obtained by minimizing the sorting results. The complete sorting algorithm mainly consists of the following four steps: (1) Prediction: Utilize the prediction model to forecast the future states of the system; (2) Evaluation: Independently assess each control objective to quantify the effectiveness of different voltage vector control strategies; (3) Sorting: Rank the performance of different voltage vectors based on their control effectiveness, assigning corresponding sorting results accordingly; (4) Optimization solving: Sum up the sorting results obtained by different voltage vectors for each control objective [39]. By obtaining the total sorting result and through minimizing this total sorting result, one determines the optimal voltage vector.  
The MPTC needs to choose the minimization of the cost function of multiple objectives within each sampling period. It can be expressed by two different cost functions:  
\[\begin{split} g_{1}=\left|T^{ref}-\overset{\cdot\cdot\cdot\cdot k+2}{T }\left(v_{\text{s}}^{k+1}\right)\right|\\ g_{2}=\left|\left|\left|v_{\text{s}}^{\text{ref}}\right|-\left| \overset{\cdot\cdot\cdot k+2}{\psi_{\text{s}}^{k+1}}\left(v_{\text{s}}^{k+1} \right)\right|\right|\end{split} \tag{17}\]  
The torque and stator's flux-related errors are represented by \(g_{1}\) and \(g_{2}\), respectively. The MPTC needs to be evaluated for each possible voltage vector.  
During the sorting phase, the sorting criteria are defined as follows: utilizing graph algorithms to reduce the eight basic voltage vectors to four voltage vectors and calculating the cost function values for each voltage vector. In ascending order [40], voltage vectors with smaller errors are assigned lower ranks, starting from 0 and incrementing by 1, with a maximum rank of 4. Based on a unified sorting rule for all control objectives, the voltage vectors have the same range, which is [0, 4], and have been dimensionlessly processed [8]. The torque sorting results for each voltage vector and the stator magnetic flux sorting results are denoted as \(r_{1}\) and \(r_{2}\), respectively, that is,  
\[\begin{split} g_{1}\left(v_{\text{s}}^{k+1}\right)& \longrightarrow r_{1}\left(v_{\text{s}}^{k+1}\right)\\ g_{2}\left(v_{\text{s}}^{k+1}\right)&\longrightarrow r _{2}\left(v_{\text{s}}^{k+1}\right)\end{split} \tag{18}\]  
where \(v_{\text{s}}^{k+1}\) is the voltage vector to be evaluated and \(r_{1}\left(v_{\text{s}}^{k+1}\right)\) and \(r_{2}\left(v_{\text{s}}^{k+1}\right)\) are the ordering values of \(g_{1}\) and \(g_{2}\), respectively; the ordering values are dimensionless variables. Considering the ordering values of the two objective functions comprehensively, the minimum voltage vector of torque and flux errors is selected based on the lowest sorting value. Torque and flux variables are tracked by selecting the voltage vector with the lowest-ranking mean value to select the optimal voltage vector. Then, the proposed optimization method based on ranking is as follows:  
\[\begin{split}\text{v}_{\text{opt}}=\arg\underset{\{\text{v}_{0 },\ldots,\text{v}_{3}\}}{\text{min}}\frac{r_{1}\left(v_{\text{s}}^{k+1}\right) +r_{2}\left(v_{\text{s}}^{k+1}\right)}{2}\end{split} \tag{19}\]  
Based on the genetic algorithm, the concept of average ranking is introduced using average criteria. It can be simplified to the sum of \(r_{1}\) and \(r_{2}\) (Figure 6) because the two optimization objectives are equivalent.