## 3 Proposed MPTC with Multi-Objective Ranking  
Figure 2 shows the block diagram of the improved MPTC frame. This algorithm is primarily composed of four components: motor output observation, delay compensation, prediction process, and cost function calculation. (1) Motor output observation: The three-phase voltage quantities and current flow information of the motor are measured, which are then fed into the Lomborg observer. This calculates the stator flux \(\psi_{\mathrm{s}}(k)\) and the torque \(T_{e}(k)\) at the current time. Simultaneously, the reference speed \(\omega_{r}\)ref and actual speed \(\omega_{r}\) are input into the PI controller, yielding the current electric torque reference value \(T_{e}^{ref}\). In the case of utilizing a conventional PI controller, the expression for the speed loop controller is as follows:  
\[i_{q}^{*}=\left(K_{\mathrm{P}}+\frac{K_{i}}{s}\right)\left(\omega_{\mathrm{m} }^{*}-\omega_{\mathrm{m}}\right)-B_{\mathrm{n}}\omega_{\mathrm{m}} \tag{9}\]  
\[ \begin{array}{c}K_{\mathrm{P}}=\frac{\beta I}{1.5\omega_{\mathrm{m}}\psi_{i} }\\ K_{i}=\beta K_{\mathrm{P}}\end{array}  \tag{10}\]  
Figure 1: Two-level three-phase inverter. where \(\omega_{\rm m}\) represents the mechanical angular velocity of the motor; \(J\) denotes the moment of inertia; \(B\) is the damping coefficient; \(\psi_{f}\) is the stator flux; \(B_{a}\) is active damping; \(p_{n}\) is the polar logarithm; \(K_{\rm P}\) represents the proportional gain of the PI controller; and \(K_{i}\) is the integral gain of the PI controller. Compared to the typical method of tuning PI controller parameters using classical two-degree-of-freedom systems, this approach to parameter tuning is straightforward, and the relationship between parameter adjustments and the system's dynamic quality is clear;  
(2) Delay compensation: In the case of model predictive torque control, the selection of the optimal voltage vector for the next moment based on the sampled value at the current moment will \"expire\" due to the delay due to the existence of the computer command cycle. As a result, the optimal voltage vector acting at time \(k\) will be delayed to act at time \((k+1)\), which is inaccurate for the motor control system. Therefore, delay compensation is usually used to calculate the optimal voltage vector at the \((k+1)\) moment in advance to eliminate the inaccuracy of the voltage vector due to the delay in calculation. To further enhance the precision of model predictions and achieve complete compensation, this study employs a two-step computational delay compensation approach: firstly, estimating the stator current \(i_{\rm s}^{k}\) and the stator flux \(\psi_{\rm s}^{k}\) at the \(k\) instant, and then predicting \(i_{\rm s}^{k+1}\) and \(\psi_{\rm s}^{k+1}\) based on Equations (4) and (5). Subsequently, the variables at time \((k+1)\) serve as initial conditions for predicting \((k+2)\);  
(3) Forecasting process: The traditional MPTC control algorithm requires rapidly substituting 8 fundamental voltage vectors into the prediction model to calculate the stator flux and torque values generated under the influence of each voltage vector. However, this computational task is evidently complex. In this study, a graph algorithm is employed to reduce the 8 fundamental voltage vectors to 4. These reduced vectors are then applied to the prediction model to calculate the stator flux \(\psi_{\rm s}^{k+2}\) and torque values \(T_{e}^{k+2}\) at time \((k+2)\) for subsequent cost function screening;  
(4) The calculation of cost function. In the context of multi-objective optimization problems for permanent-magnet synchronous motors (PMSM), the weighting coefficients in traditional model-predictive torque control (MPTC) serve as a crucial method to coordinate various control objectives. These coefficients enable standardization of control variables while allowing for the assignment of varying importance levels to different controlled variables based on practical requirements.