## 3 Weightless Model Predictive Control Based on ESO Observer
### Unweighted Cost Function Design  
To address the difficulty of tuning the weight coefficients in the traditional value function of single-vector MPTC (model predictive torque control), a value function that eliminates the weight coefficients has been proposed. This new value function contains only the torque angle, thereby eliminating the need for weight coefficient design and reducing the complexity of the value function computation. The calculation is as follows:  
\[g=\left[\delta^{ref}-\delta(k+2)\right]^{2}+I_{m} \tag{28}\]  
where \(I_{m}\) is the current limit value,  
\[\text{and}:\ I_{m}=\left\{\begin{array}{ll}0,&|i_{s}|\leq|I_{\max}|\\ \infty,&|i_{s}|>|I_{\max}|\end{array}\right.,\ |i_{s}|=\sqrt{i_{d}^{2}(k+2)+i_{q}^{2} (k+2)}.\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Sector**} \\ \cline{2-7}  & \(\mathbf{S}_{1,12}\) & \(\mathbf{S}_{2,3}\) & \(\mathbf{S}_{4,5}\) & \(\mathbf{S}_{6,7}\) & \(\mathbf{S}_{8,9}\) & \(\mathbf{S}_{10,11}\) \\ \hline \(d\psi_{s}>0\), \(dT_{e}>0\) & \(\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{0}\) & \(\mathrm{U}_{2},\mathrm{U}_{3},\mathrm{U}_{0}\) & \(\mathrm{U}_{3},\mathrm{U}_{4},\mathrm{U}_{0}\) & \(\mathrm{U}_{4},\mathrm{U}_{5},\mathrm{U}_{0}\) & \(\mathrm{U}_{5},\mathrm{U}_{6},\mathrm{U}_{0}\) & \(\mathrm{U}_{6},\mathrm{U}_{1},\mathrm{U}_{0}\) \\ \(d\psi_{s}>0\), \(dT_{e}<0\) & \(\mathrm{U}_{5},\mathrm{U}_{6},\mathrm{U}_{0}\) & \(\mathrm{U}_{6},\mathrm{U}_{1},\mathrm{U}_{0}\) & \(\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{0}\) & \(\mathrm{U}_{2},\mathrm{U}_{3},\mathrm{U}_{0}\) & \(\mathrm{U}_{3},\mathrm{U}_{4},\mathrm{U}_{0}\) & \(\mathrm{U}_{4},\mathrm{U}_{5},\mathrm{U}_{0}\) \\ \(d\psi_{s}<0\), \(dT_{e}>0\) & \(\mathrm{U}_{2},\mathrm{U}_{3},\mathrm{U}_{0}\) & \(\mathrm{U}_{3},\mathrm{U}_{4},\mathrm{U}_{0}\) & \(\mathrm{U}_{4},\mathrm{U}_{5},\mathrm{U}_{0}\) & \(\mathrm{U}_{5},\mathrm{U}_{6},\mathrm{U}_{0}\) & \(\mathrm{U}_{6},\mathrm{U}_{1},\mathrm{U}_{0}\) & \(\mathrm{U}_{1},\mathrm{U}_{0}\) & \(\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{0}\) \\ \(d\psi_{s}<0\), \(dT_{e}<0\) & \(\mathrm{U}_{4},\mathrm{U}_{5},\mathrm{U}_{0}\) & \(\mathrm{U}_{5},\mathrm{U}_{6},\mathrm{U}_{0}\) & \(\mathrm{U}_{6},\mathrm{U}_{1},\mathrm{U}_{0}\) & \(\mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{0}\) & \(\mathrm{U}_{2},\mathrm{U}_{3},\mathrm{U}_{0}\) & \(\mathrm{U}_{3},\mathrm{U}_{4},\mathrm{U}_{0}\) \\ \hline \end{tabular} Where \(\mathrm{U}_{1..6}\) is the non-zero voltage vector.  
\end{table}
Table 1: New candidate voltage vector 2-D selection table.  
Figure 5: Space voltage vector division diagram.  
where \(I_{\text{max}}\) is the maximum peak current, and \(i_{s}\) is the stator current.  
\[\delta(k+2)=\arcsin\Bigg{[}\frac{2L}{3p\psi_{f}}\bigg{(}\frac{T_{e}(k+2)}{|\psi_{ s}(k+2)|}\bigg{)}\Bigg{]} \tag{29}\]  
\[\delta^{ref}=arcsin\Bigg{[}\frac{L}{\psi_{f}}\bigg{(}\frac{\psi_{a}^{ref}i_{ \beta}^{ref}-\psi_{\beta}^{ref}i_{a}^{ref}}{\left|\psi_{s}^{ref}\right|}\bigg{)} \Bigg{]} \tag{30}\]  
where \(\psi_{a}^{ref}\), \(\psi_{\beta}^{ref}\) are the reference magnetic chains of the \(\alpha\), \(\beta\) axes, and \(i_{a}^{ref}\), \(i_{\beta}^{ref}\) are the reference currents of the \(\alpha\), \(\beta\) axes, which can be calculated from the reference torque.