## 3 Control System Architecture
### Modified Observer with Improved QPLL  
Position estimation accuracy using direct back-EMF and the arc-tangent method can be affected by noise and harmonics, particularly at zero-crossing points. Therefore, an improved back-EMF is built, as shown in Figure 4. The QPLL structure is designed to accurately estimate rotor position and speed by leveraging the relationship between the back-EMF and rotor position. Its effectiveness lies in synchronizing its internally generated phase with the phase of the back-EMF to extract the rotor's angular position with high precision. The quadrature components of the back-EMF in the stationary \(\alpha\beta\) frame are computed using three sensed back-EMF signals, \(e_{a}\), \(e_{b}\) and \(e_{c}\) for phases 'a', 'b' and c', respectively, through the general Clarke transform, according to Equation (17):  
\[ \begin{bmatrix}e_{\alpha}\\ e_{\beta}\end{bmatrix} =\frac{2}{3}\cdot \begin{bmatrix}1&-\frac{1}{2}&-\frac{ 1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{bmatrix} \cdot \begin{bmatrix}e_{a} \\ e_{b}\\ e_{c}\end{bmatrix}  \tag{17}\]  
These orthogonal components can also be expressed as per Equation (18). The magnitude of the back-EMF is designated by \(E\), and the actual rotor angular position is identified by \(\theta\). \[ \begin{bmatrix}e_{\alpha}\\ e_{\beta}\end{bmatrix} =E\cdot \begin{bmatrix}cos\theta\\ sin\theta\end{bmatrix}  \tag{18}\]  
Figure 4: Modified back-EMF observer with improved QPLL. The QPLL generates an estimate of the back-EMF \(\alpha\beta\) components from the estimated rotor position \(\hat{\theta}\). These components can be expressed based on the same space construction, as per Equation (19):  
\[ \begin{bmatrix}\dot{\varrho}_{\alpha}\\ \dot{\theta}_{\beta}\end{bmatrix} =E\cdot \begin{bmatrix}cos\hat{\theta}\\ sin\hat{\theta}\end{bmatrix}  \tag{19}\]  
The objective of the QPLL is to minimize the phase difference between the actual rotor position (\(\theta\)) and the estimated rotor position (\(\hat{\theta}\)), denoted by \(\Delta\theta\), which is achieved by aligning the estimated and actual back-EMF components in the stationary \(\alpha\beta\) frame. The phase difference is captured by an error function \(\epsilon(t)\). To effectively extract \(\epsilon(t)\), the QPLL employs a signal transformation leveraging the trigonometric relationship expressed by Equation (20):  
\[sin\big{(}\hat{\theta}-\theta\big{)}=sin(\Delta\theta)=sin\hat{\theta}\cdot cos \hat{\theta}-cos\hat{\theta}\cdot sin\theta \tag{20}\]  
The QPLL uses this information to calculate an error signal that drives the feedback loop. For small phase errors, \(sin(\Delta\theta)\ \approx\Delta\theta\), and additionally using the space transformation described by Equation (18) and scaling the error quadratically to avoid dependence on \(E\), the final error function is achieved as follows:  
\[\Delta\theta\ \approx\epsilon(t)=\Big{(}e_{\alpha}{}^{2}-e_{\beta}{}^{2} \Big{)}\cdot sin\hat{\theta}-2\cdot e_{\alpha}\cdot e_{\beta}\cdot cos\hat{\theta} \tag{21}\]  
The error function \(\epsilon(t)\) measures the misalignment between the estimated and actual back-EMF components in the \(\alpha\beta\) frame. The first term emphasizes imbalances between the orthogonal components relative to the estimated phase, while the second term corrects for coupling effects and ensures synchronization. Assuming small error signals (\(\epsilon(t)\approx 0\)), Equation (22) provides a tangent estimation. The estimated position does not include speed information remaining accurate for any speed without delays. \[tan\hat{\theta}\ \approx\ \frac{2\cdot e_{\alpha}\cdot e_{\beta}}{e_{\alpha}{}^{2 }-e_{\beta}{}^{2}} \tag{22}\]  
The QPLL uses a PI controller to drive \(\epsilon(t)\) to zero, ensuring synchronization between the estimated and actual rotor positions. The PI controller processes \(\epsilon(t)\) to adjust the estimated rotor speed \(\hat{\omega}\) and the rotor angular position \(\hat{\theta}\) according to Equations (23) and (24), respectively.