# A Deadbeat Current and Flux Vector Control for IPMSM Drive with High Dynamic Performance
## 2 IPMSM Fundamental Model
### Stator Flux Linkage and Current Predictions  
As presented above, the predicted values at the end moment of the \(k^{\rm th}\) period are the response values obtained after the \(\alpha\)-\(\beta\)-axis voltages, \(\mathbf{v}_{\alpha\beta}(k)\), are applied during period \(kT_{\rm s}\). Thus, the stator flux linkage vector and current in \(\alpha\)-\(\beta\) axes predicted at the ending instant of the control period \(kT_{\rm s}\) can be calculated by the measured current, i.e., \(\mathbf{i}_{\alpha\beta}(k)\), the updated voltage, \(\mathbf{v}_{\alpha\beta}(k)\), and estimated stator flux linkage, i.e., \(\overset{\overset{\alpha}}{\mathbf{\
abla}}_{\alpha\beta}(k)\), based on the voltage Equation (11) that is transformed into:  
\[\mathbf{\Psi}_{\alpha\beta}^{\mathbf{P}}(k)=\overset{\overset{\alpha}}{ \mathbf{\
abla}}_{\alpha\beta}(k)+(\mathbf{v}_{\alpha\beta}(k)-R_{\rm s} \mathbf{i}_{\alpha\beta}(k))T_{\rm s} \tag{22}\]  
\[\theta_{\rm s}^{\rm p}(k)=\tan^{-1}\bigg{(}\frac{\mathbf{\Psi}_{\beta}^{\rm p }(k)}{\mathbf{\Psi}_{\rm a}^{\rm p}(k)}\bigg{)} \tag{23}\]  
where \(\mathbf{\Psi}_{\alpha\beta}^{\mathbf{P}}(k)=\big{[}\ \ \mathbf{\Psi}_{\alpha}^{\rm p }(k)\ \ \ \mathbf{\Psi}_{\beta}^{\rm p}(k)\ \big{]}^{\rm T}\) is the predicted stator flux linkage vector and its elements in \(\alpha\)-\(\beta\) frame at the ending instant of period \(kT_{\rm s}\), and \(\theta_{\rm s}^{\rm p}(k)\) is phase angle of predicted stator flux linkage vector. Note that \(\overset{\alpha}{\
abla}_{\alpha\Theta}^{\alpha}(k)\) is estimated at the beginning of the \(k^{\text{th}}\) period from the voltage \(\mathbf{v}_{\alpha\Theta}(k-1)\) and current \(\mathbf{i}_{\alpha\Theta}(k)\) using the proposed MO-FOB.  
Since the control period \(T_{\text{s}}\) is sufficiently short compared to mechanical time constant of the motor, the rotor speed can be considered constant during one control period, resulting in the rotor angle being predicted by periodically adding an incremental angle \(\omega_{\text{e}}T_{\text{s}}\) over each control period. Thus, the load angle prediction can be deduced from Equation (8) by:  
\[\delta^{P}(k)=\theta_{\text{s}}^{P}(k)-(\theta_{\text{e}}(k)+\omega_{\text{e} }(k)T_{\text{s}}) \tag{24}\]  
where \(\delta^{P}(k)\) is the predicted load angle of flux linkage vector.  
It is known that the voltage drops of resistance in Equation (11), i.e., \(R_{\text{s}}\mathbf{i}_{\alpha\Theta}(k)\), are insignificant to the output voltage, i.e., \(\mathbf{v}_{\alpha\Theta}(k)\), in operation at medium and high speed but should not be neglected at low speed. Nevertheless, the change of current magnitude, \(i_{\text{s}}\), is relatively small in each control period. Thus, the magnitude of the current vector can be considered constant in one control period. Its phase angle after each sampling period is approximately equal to the incremental angle produced by the rotor movement during one control period, i.e., \(\omega_{\text{e}}T_{\text{s}}\). As a result, the \(\alpha\)-\(\beta\)-axis currents can be approximately predicted by a simple method, given by:  
\[\mathbf{i}_{\alpha\Theta}^{\mathbf{p}}(k)=\mathbf{T}(\omega_{\text{e}}(k)T_{ \text{s}})\mathbf{i}_{\alpha\Theta}(k) \tag{25}\]  
As observed in (22) and (25), the prediction of \(\alpha\)-\(\beta\)-axis stator flux linkage and current in this work is simplified and is nearly independent of motor parameters (excluding \(R_{\text{s}}\)). This is different from other methods [14; 15; 17].