# A Deadbeat Current and Flux Vector Control for IPMSM Drive with High Dynamic Performance
## 2 IPMSM Fundamental Model
### Voltage Command Determination with \"Reinforced Flux Phase Angle\" Concept  
As stated in Section 3.1 and depicted schematically in Figure 2a, for the one-step delay compensation implemented on digital devices [14; 15; 16; 17], the voltage commands calculated at the \(k^{\text{th}}\) period will be updated at the beginning of interval \((k+1)T_{\text{s}}\), but they are determined during the \(k^{\text{th}}\) sampling period. These voltage commands will force the flux linkage errors between predicted and desired quantities at the beginning and ending instants of period \((k+1)T_{\text{s}}\), respectively, into zero. Consequently, the voltage commands at  
In this study, a concept to \"reinforce\" the phase angle reference of stator flux linkage is proposed. According to the conclusion in [5], dynamic response is decided by the quality of load angle increment, and the flux phase angle increment can be restricted by the maximum output voltage. It can be noted from Equation (41) that the flux phase angle reference depends on load angle reference. Therefore, to achieve high dynamic performance, the flux phase angle reference should be appropriately reinforced by considering the actual speed error ratio and existing load angle reference. At the same time, this reinforced angle should be bound to ensure the output voltage is confined within an allowable value. Then, the reinforced flux phase angle, during control period \((k+1)T_{\rm s}\), can be proposed as follows:  
\[\theta_{\rm s}^{**}(k+1)=\theta_{\rm s}^{*}(k+1)+F_{\theta}(k+1) \tag{42}\]  
\[F_{\theta}(k+1)=K_{\rm\delta}\frac{\omega_{\rm e}^{*}(k)-\omega_{\rm e}(k)}{ \omega_{\rm e}^{*}(k)}\delta^{*}(k+1) \tag{43}\]  
\[\Delta\theta_{\rm s}^{**}(k+1)=\theta_{\rm s}^{**}(k+1)-\theta_{\rm s}^{P}(k) \tag{44}\]  
where \(\theta_{\rm s}^{**}(k+1)\) is reinforced phase angle reference of stator flux linkage vector at the \((k+1)^{\rm th}\) instant, \(F_{\theta}(k+1)\) is the amount of reinforced angle, \(K_{\rm\delta}\) is the percentage reinforcement of load angle, and \(\Delta\theta_{\rm s}^{**}(k+1)\) is new phase angle increment reference of flux linkage. Note that \(F_{\theta}\) can influence the quality of current and torque generation. Thus, to prevent overshoot of current and high torque ripple, \(K_{\rm\delta}\) in Equation (43) should be chosen properly (here selecting \(K_{\rm\delta}=0.1\)). Further, to ensure that the final flux phase angle increment reference is within a permissible value, the phase angle reference selection, \(\theta_{\rm s}^{*}\), should be concerned with the maximum phase angle increment, i.e., \(\Delta\theta_{\rm s}^{\rm max}\). This maximum angle is constrained by the voltage limitation and magnitudes of the predicted and desired flux linkage vectors. By applying the cosine theorem to the phasor diagram shown in Figure 2b, the maximum phase angle increment calculated at the \((k+1)^{\rm th}\) instant, \(\Delta\theta_{\rm s}^{\rm max}(k+1)\), can be defined by:  
\[\Delta\theta_{\rm s}^{\rm max}(k+1)=\cos^{-1}\left[\frac{\left(\mathbf{\Psi}_ {\rm s}^{*}(k+1)\right)^{2}+\left(\mathbf{\Psi}_{\rm s}^{\rm P}(k)\right)^{2} -\left(V_{\rm s}^{\rm max}T_{\rm s}\right)^{2}}{2\mathbf{\Psi}_{\rm s}^{*}(k+1 )\mathbf{\Psi}_{\rm s}^{\rm P}(k)}\right] \tag{45}\]  
where \(V_{\rm s}^{\rm max}=\frac{V_{\rm s}}{\sqrt{3}}\) is a maximum voltage of VSI. Then, the final desired phase angle, e.g., \(\theta_{\rm s}^{*}(k+1)\), is determined as follows:  
\[\theta_{\rm s}^{*}(k+1)=\left\{ \begin{array}{l}\theta_{\rm s}^{P}(k)+ \Delta\theta_{\rm s}^{\rm max}(k+1),\;\Delta\theta_{\rm s}^{**}(k+1)\geq\Delta \theta_{\rm s}^{\rm max}(k+1)\\ \theta_{\rm s}^{**}(k+1),\Delta\theta_{\rm s}^{**}(k+1)<\Delta\theta_{\rm s}^{ \rm max}(k+1)\end{array} \right. \tag{46}\]  
According to Equations (12)-(14), the \(\alpha\)-\(\beta\)-axis stator flux linkage, \(\mathbf{\Psi}_{\alpha\beta}^{*}(k+1)\), can be derived based on its magnitude and phase angle reference, as given by:  
\[\mathbf{\Psi}_{\alpha\beta}^{*}(k+1)=\left[