# A Deadbeat Current and Flux Vector Control for IPMSM Drive with High Dynamic Performance
## 2 IPMSM Fundamental Model
### Proposed Hybrid Flux Linkage Observer  
In this work, a modified hybrid flux linkage observer (MO-FOB) is developed by simplifying the manipulating process of the Gopinath observing strategy [27]. The complete observer scheme is shown in Figure 4. Unlike other observers, the proposed MO-FOB uses a closed-loop low-pass filter (LPF) estimator [28] for the voltage model and an open-loop linear calculating approach to mix the estimated results from the voltage and current models. The dominance of each model varies linearly according to the operating speed. This hybrid method can replace the conventional model-switching methods (i.e., using PI regulator). The detailed design will be explained later. As reported in a previous study [29], the conventional Gopinath-model-based closed-loop flux linkage observer (CL-FOB) is a hybrid observer capable of combining both voltage and current models with closed-loop feedback for flux linkage estimation to overcome their individual disadvantages. The flux estimator by the conventional Gopinath model can be written as follows:  
\[ \begin{array}{c}\mathbf{\overset{\alpha}{\
abla}}_{\mathbf{\alpha}\beta}^{\text{G}} =\frac{s^{2}}{s^{2}+K_{\text{P}}s+K_{\text{I}}}\mathbf{\overset{\alpha}{\
abla}}_{ \alpha\beta}^{\text{v}}+\frac{K_{\text{P}}s+K_{\text{I}}}{s^{2}+K_{\text{P}} s+K_{\text{I}}}\mathbf{\overset{\alpha}{\
abla}}_{\alpha\beta}^{\text{i}}\end{array}  \tag{16}\]  
where \(\mathbf{\overset{\alpha}{\
abla}}_{\alpha\beta}\) is stator flux linkage estimated by Gopinath method; \(\mathbf{\overset{\alpha}{\
abla}}_{\alpha\beta}=\left[ \begin{array}{cc}\mathbf{ \overset{\alpha}{\
abla}}_{\alpha}^{\text{v}}&\mathbf{\overset{\alpha}{\
abla}}_{ \beta}^{\text{v}}\end{array} \right]^{\text{T}}\) and \(\mathbf{\overset{\alpha}{\
abla}}_{\alpha\beta}^{\text{i}}=\left[ \begin{array}{ cc}\mathbf{\overset{\alpha}{\
abla}}_{\alpha}^{\text{i}}&\mathbf{\overset{ \alpha}{\
abla}}_{\beta}^{\text{i}}\end{array} \right]^{\text{T}}\) are output stator flux linkage vectors and their elements in \(\alpha\)-\(\beta\) frame estimated by the voltage model and current model, respectively; \(K_{\text{P}}=\sqrt{2}\omega_{\text{b}}^{\text{G}}\) and  
Figure 3: Complete block diagram of the proposed DB-CFVC. \(K_{\rm I}=\left(\omega_{\rm g}^{\rm G}\right)^{2}\) are PI gains of the observer, which are determined based on \(\omega_{\rm b}^{\rm G}\) which is the transition speed for switching between the current and voltage models. However, the conventional Gopinath observer is highly sensitive to the estimation errors at medium speed (around the model-switching region) since the voltage model uses a pure integrator [29] and requires careful tuning for the PI coefficients of the observer. Therefore, some studies have modified the voltage model and improved the model-switching method to be more precise in the transition. However, this can make the observers more complicated. In this study, the closed-loop LPF estimator [28] is used to replace the pure integrator in the voltage model. The flux linkage of the voltage model adopting the closed-loop LPF estimator [28] can be expressed by:  
\[\mathbf{\overset{\alpha\,\mathbf{v}}{\mathbf{\overset{\alpha\,\mathbf{\beta} }{\mathbf{\tau}}}}}_{\alpha\beta}=\frac{1}{s+\omega_{\rm c}}\left(\mathbf{v}_ {\alpha\beta}-R_{\rm s}\mathbf{i}_{\alpha\beta}\right)+\frac{\omega_{\rm c}}{ s+\omega_{\rm c}}\mathbf{\overset{\alpha\,\mathbf{\Psi}}{s}}\bigg{[} \begin{array}{c} \cos\hat{\theta}_{\rm s}\\ \sin\hat{\theta}_{\rm s}\end{array} \bigg{]} \tag{17}\]  
with  
\[\mathbf{\overset{\alpha\,\mathbf{v}}{\mathbf{\overset{\alpha\,\mathbf{\Psi}}{ s}}}}=\left\{ \begin{array}{ll}\mathbf{\overset{\alpha\,\mathbf{\Psi}}{s}}, \mathbf{\overset{\alpha\,\mathbf{\Psi}}{s}}>\mathbf{\overset{\alpha\,\mathbf{ \Psi}}{s}}\\ \mathbf{\overset{\alpha\,\mathbf{\Psi}}{s}},\mathbf{\overset{\alpha\,\mathbf{ \Psi}}{s}}<\mathbf{\overset{\alpha\,\mathbf{\Psi}}{s}}\end{array} \right.