## 3 Weightless Model Predictive Control Based on ESO Observer
### The Design of the Improved ESO (Extended State Observer)  
In order to improve the current error and oscillation problems of the permanent magnet synchronous motor due to parameter mismatch and external perturbations, based on the single vector model predictive torque control, the motor parameter perturbations are observed and compensated for in real time by designing an improved extended observer (ESO), so as to achieve the effect of suppressing the steady-state prediction error and improving the robust performance of the system. Considering the delay effect of the computer instruction cycle, two-step delay compensation is performed, which is designed as follows. The state space equations of the PMSM in the synchronous rotating coordinate system are as follows:  
\[ \begin{array}{l}\frac{\text{d}i_{d}}{\text{d}t}=-\frac{R_{i}}{L_{d}}i_{d}+ \frac{\omega_{c}L_{d}}{L_{d}}i_{q}+\frac{1}{L_{d}}u_{d}\\ \frac{\text{d}i_{q}}{\text{d}t}=-\frac{R_{i}}{L_{q}}i_{q}-\frac{\omega_{c}L_{d }}{L_{q}}i_{d}+\frac{1}{l_{q}}u_{q}-\frac{\omega_{c}\psi_{f}}{l_{q}}\end{array}  \tag{12}\]  
Considering the parameter mismatch, the voltage equation is rewritten as:  
\[ \begin{array}{l}\frac{\text{d}i_{d}}{\text{d}t}=\frac{1}{L_{e}+\Delta L}*(u_ {d}-(R_{s}+\Delta R)i_{d}+\omega_{c}i_{q})+\xi_{d}\\ \frac{\text{d}i_{q}}{\text{d}t}=\frac{1}{L_{s}+\Delta L}*(u_{q}-(R_{s}+\Delta R )i_{q}-\omega_{c}i_{d}+\omega_{e}(\psi_{f}+\Delta\psi_{f})+\xi_{q}\end{array}  \tag{13}\]  
where \(u_{d},u_{q}\) are the voltage quantities of the d-axis and q-axis input to the system, while other state quantities, \(\xi_{d}\) and \(\xi_{q}\) are the values of various parameter perturbations in the d-axis and q-axis and external disturbances. Based on the ESO observation idea, the parameter mismatch perturbation is equated to a new input quantity, i.e., the ESO defines the perturbation as follows:  
\[ \begin{array}{l}f_{\text{d}}=\frac{1}{L_{e}}*(-(R_{s}+\Delta R)i_{\text{d}}- \Delta\frac{\text{d}i_{d}}{\text{d}t}+\omega_{\text{e}}(L_{s}+\Delta L)i_{ \text{q}}-\xi_{d})\\ f_{\text{q}}=\frac{1}{L_{e}}*(-(R_{s}+\Delta R)i_{q}-\Delta\frac{\text{d}i_{q}} {\text{d}t}-\omega_{e}(\psi_{f}+\Delta\psi_{f})-\xi_{q})\end{array}  \tag{14}\]  
Figure 4: Improved MPTC block diagram. where \(f_{\rm d}\), \(f_{\rm q}\) are the total system perturbations in the d- and q-axis, \(R_{s}\), \(L_{s}\), \(\psi_{f}\) are the nominal values of the motor nameplate, \(\Delta R\), \(\Delta\psi_{f}\), \(\Delta\)L are the values of the perturbations of each parameter, and \(\xi\), is the value of the external perturbations other than the resistance, inductance, and magnetic flux. Taking the d-axis as an example, the expression for the second-order ESO of the current \(i_{\rm d}\) is as follows:  
\[\left\{ \begin{array}{l}\mathcal{E}_{i_{\rm d}}=z_{i_{\rm d1}}-i_{\rm d}\\ \hat{z}_{i_{\rm d}1}=z_{i_{\rm d2}}-\hat{p}_{i_{\rm d1}}f{\rm d}(\mathcal{E}_{ i_{\rm d}},\alpha_{1},\lambda_{1})+bu_{q}\\ \hat{z}_{i_{\rm d}2}=-\hat{p}_{i_{\rm d2}}f{\rm d}(\mathcal{E}_{i_{\rm d}}, \alpha_{1},\lambda_{1})\end{array} \right. \tag{15}\]  
where the nonlinear error feedback control rule is as follows:  
\[ \begin{array}{l}f{\rm d}(\mathcal{E}_{i_{\rm d}},\alpha,\delta)=|\mathcal{E }_{i_{\rm d}}\mid^{\alpha}sign(e_{1}),|\mathcal{E}_{i_{\rm d}}\mid>\delta\\ f{\rm d}(\mathcal{E}_{i_{\rm d}},\alpha,\delta)=\frac{e_{\rm d}^{\rm d}}{\delta ^{2-\alpha}},|\mathcal{E}_{i_{\rm d}}\mid\leq\delta\end{array}  \tag{16}\]  
In the formula, \(f{\rm d}(\mathcal{E}_{i_{\rm d}},\alpha_{1},\lambda_{1})\) is a segmented function, which produces the jitter vibration phenomenon at the switching point, while the sliding membrane control system (SMC) can achieve the function of stable tracking of the target by controlling the state of the system on the sliding surface, which can eliminate the jitter vibration phenomenon of the traditional ESO, i.e., the improved _hac_ function is as follows:  
\[