# A Deadbeat Current and Flux Vector Control for IPMSM Drive with High Dynamic Performance
## 2 IPMSM Fundamental Model  
The space vectors of voltage, current, and stator and rotor flux linkages of an IPMSM on different coordinate systems in steady state are shown in Figure 1, where the \(\alpha\)-\(\beta\) frame is the stationary coordinate and the \(d\)-\(q\) and \(f\)-\(t\) frames are the rotor and stator flux linkage rotating coordinates, respectively. The PMSM voltage equation in the stationary coordinate can be given as:  
\[\mathbf{v}_{\alpha\beta}=R_{\mathrm{s}}\mathbf{i}_{\alpha\beta}+\frac{\mathrm{ d}\mathbf{\Psi}_{\alpha\beta}}{\mathrm{d}t} \tag{1}\]  
where \(\mathbf{v}_{\alpha\beta}=\left[ \begin{array}{cc}\mathbf{v}_{\alpha}&\mathbf{ v}_{\beta}\end{array} \right]^{\mathrm{T}}\), \(\mathbf{i}_{\alpha\beta}=\left[ \begin{array}{cc}\mathbf{i}_{\alpha}&\mathbf{ i}_{\beta}\end{array} \right]^{\mathrm{T}}\) and \(\mathbf{\Psi}_{\alpha\beta}=\left[ \begin{array}{cc}\mathbf{\Psi}_{\alpha}& \mathbf{\Psi}_{\beta}\end{array} \right]^{\mathrm{T}}\) are the stator voltage, current, and flux linkage vector in \(\alpha\)-\(\beta\) frame, respectively; \(R_{\mathrm{s}}\) is stator winding resistance. From Equation (1), the stator flux linkage in \(\alpha\)-\(\beta\) frame can be computed as:  
\[\mathbf{\Psi}_{\alpha\beta}=\int\left(\mathbf{v}_{\alpha\beta}-R_{\mathrm{s} }\mathbf{i}_{\alpha\beta}\right)\mathrm{d}t \tag{2}\]  
Figure 1: Phasor diagram used in analysis of IPMSM drive. is directly determined in the stationary frame by the predicted and desired flux linkage vectors without coordinate transformation. Moreover, the desired current delivered from the speed controller is employed as an input command. By adopting MTPA control, the current vector reference in the \(d\)-\(q\) frame is identified and then used for the calculation of stator flux linkage magnitude and load angle reference. For pure deadbeat (pure-DB) control, the voltage command can easily be obtained from Equations (11) and (12) by the measured/estimated and desired values of current and flux linkage. However, the disadvantage of the pure-DB is the inevitable delay caused by factors such as sampling, computation, and voltage modulation in a digital control system. Thus, this causes a delay for the modulated voltage command forwarded onto the VSI and leads to inaccuracies in control, especially in applications with high switching frequency. To avoid the performance drop caused by such a delay, the one-step delay compensation algorithm [14,15] is utilized for the deadbeat control in this work. In this method, the instant of updating the voltage command is delayed by one beat, and simultaneously, the measured/estimated feedback values used in calculating the voltage command are replaced by the predicted ones. By so doing, the calculated voltage command at the present control period will be updated at the beginning instant of the next sampling period. Hence, this resolves the delay problem without much computation effort added. The explanation of the control process and the phasor diagram analysis are illustrated in Figure 2.  
As presented by the graphical timeline shown in Figure 2a, it should be noted that the sampling frequency is synchronous with the switching frequency. Moreover, the superscripts \"\(\wedge\)\", \"\(\star\)\", and \"\(\text{P}\)\" denote the estimation, command/reference, and predicted values, respectively. Assuming at sampling period, \(kT_{\text{s}}\), the voltage command, \(\mathbf{v}_{\boldsymbol{\alpha}\boldsymbol{\beta}}^{*}(k-1)\), is calculated in the preceding sampling, \((k-1)T_{\text{s}}\); then, they are modulated and updated onto VSI at the \(k^{\text{th}}\) period corresponding to output voltage, \(\mathbf{v}_{\boldsymbol{\alpha}\boldsymbol{\beta}}(k)\).