## 3 Control System Architecture
### Modified Observer with Improved QPLL
\[\hat{\omega}=\bigg{(}K_{P}+\frac{K_{I}}{s}\bigg{)}\cdot\epsilon(t) \tag{23}\]  
\[\hat{\theta}=\frac{1}{s}\cdot\hat{\omega} \tag{24}\]  
The dynamics of the QPLL can be modeled as a second-order system. Combining the PI control law with the error signal dynamics gives the transfer function shown in Equation (25). \[\frac{\hat{\theta}}{\hat{\theta}}=\frac{K_{P}\cdot s+K_{I}}{s^{2}+K_{P}\cdot s +K_{I}} \tag{25}\]  
From Equation (25), it is deduced that increasing the proportional gain \(K_{P}\) enhances the system's ability to quickly respond to changes in the back-EMF phase but may increase sensitivity to noise and high-frequency disturbances. On the other hand, increasing the integral gain \(K_{I}\) helps eliminate residual phase error over time, ensuring precise rotor position estimation in the steady state. The ratio \(K_{P}/K_{I}\) affects the damping of the QPLL system. A well-balanced ratio ensures minimal oscillations and fast convergence to the actual rotor position. There exists a trade-off between the dynamic performance of the QPLL and its disturbance rejection capability, which depends on the values of \(K_{P}\) and \(K_{I}\). It is worth noting that the speed-dependent variable loop gain exhibits different dynamic responses at varying speeds. To achieve a consistent dynamic response, in the proposed method, the magnitude of the fundamental components is kept constant across different speeds using a magnitude normalization method. For this, the \(\alpha\beta\) components of the back-EMF are divided by the peak flux magnitude to generate normalized back-EMF components, which are then fed to the QPLL. This approach mitigates higher-order harmonics, reduces noise propagation, and enables smooth and continuous speed estimation. Using Equations (14)-(16), virtual Hall signals are estimated once the position information is available. The proposed QPLL enhances sensorless control by making it adaptive, eliminating delays in estimated positions. The Bode plot of the closed-loop QPLL, shown in Figure 5, demonstrates that the system is stable in the closed loop. Furthermore, Figure 6 illustrates that the rotor position is accurately tracked relative to the actual position signals. It should be noted that the proposed estimator is based on the \(\alpha\beta\) components of the back-EMF, which stem from direct processing of the back-EMF signals through the QPLL. Thereby, the proposed method avoids reliance on speed-related information for position estimation, which allows for ensuring accurate and delay-free rotor position detection even at low speeds. This is the actual origin of the superiority of the proposed method compared  
Figure 5: Bode plot of the improved QPLL. Figure 6: Observer output virtual Hall pulses with rotor position. to other back-EMF-based sensorless control methods. Due to this, the performance of the proposed method closely matches that of sensor-based control. However, in the case of fast dynamic conditions or large system disturbances, the constraints associated with the use of the QPLL would lead the proposed method to underperform sensor-based control.