## 3 Control System Architecture
### Sensorless Control with Conventional Back-EMF Observer  
The trapezoidal back-EMF waveform of BLDCM serves as a reliable indicator for determining commutation instants. In this context, the line back-EMF is estimated using a nonlinear observer, as illustrated in Figure 3.  
Considering phases A and B, along with their respective phase resistance \(R\) and inductance \(L\), the BLDCM can be modeled following Equation (9), as a function of the phase currents (\(i_{a}\) and \(i_{b}\)), voltages (\(v_{a}\) and \(v_{b}\)), and back-EMFs (\(e_{a}\) and \(e_{b}\)). In Equation (9), only the back-EMF, \(e_{ab}=e_{a}-e_{b}\), is unknown, while all other variables are known.  
\[\frac{d}{dt}(i_{a}-i_{b})=\frac{R}{L}(i_{a}-i_{b})+\frac{1}{L}(v_{a}-v_{b})- \frac{1}{L}(e_{a}-e_{b}) \tag{9}\]  
Using state-space observer modeling, the back-EMF \(\hat{e}_{ab}\) is estimated according to Equations (10) and (11), where \(l\) and \(\hat{l}\) are the corresponding actual and estimated currents, respectively, and \(K_{1}\) and \(K_{2}\) are constants. Other line back-EMFs are estimated similarly using these equations.  
\[\dot{\hat{l}}=-\frac{R}{L}\cdot\hat{l}+\frac{1}{L}\cdot v_{a}-\frac{1}{L}\hat{ e}_{ab}+K_{1}\cdot(I-\hat{l}) \tag{10}\]  
\[\dot{\hat{e}_{ab}}=K_{2}\cdot(I-\hat{l}) \tag{11}\]  
The gains of the back-EMF observer, \(K_{1}\) and \(K_{2}\), are tuned using the pole-placement method [25]. It is important to note that higher values of observer gains tend to stabilize the observer; however, excessively high gains can amplify noise and cause chattering. Therefore, careful tuning is required to achieve an optimal balance in observer gain selection [17].  
Commutation instants are determined by directly measuring the zero-crossing points of the line back-EMF of the BLDCM, which is the most conventional method for sensorless commutation. However, this approach suffers from significant lag at low speeds, leading  
Figure 3: Back-EMF observer block diagram.  
to erroneous commutation. To address this issue, the estimated rotor position is further corrected as follows:  
\[\hat{\theta}=\int_{k}^{k+1}\hat{\omega}_{e}dt+\theta_{0} \tag{12}\]  
where the estimated rotor position \(\hat{\theta}\) is updated at every (k + 1)th instant, through the estimated electrical speed \(\hat{\omega}_{e}\), while the initial rotor position is \(\theta_{0}\). The rotor position is determined using Equation (12) rather than relying solely on basic back-EMF zero-crossing points. The estimated electrical speed \(\hat{\omega}_{e}\) of the motor is defined as in Equation (13). The motor speed is primarily calculated using the estimated back-EMF, \(\hat{\omega}_{ab}\), and the machine constant \(K\).  
\[\hat{\omega}_{e}=\frac{\hat{\omega}_{ab}}{K} \tag{13}\]  
Finally, the virtual Hall position signals (\(H_{a}^{*}\), \(H_{b}^{*}\), \(H_{c}^{*}\)) are generated based on the rotor position estimations in Equation (12), in accordance with the following logic:  
\[\hat{\theta}\geq-60^{\circ}\otimes\ \hat{\theta}\leq 120^{\circ}=H_{a}^{*} \tag{14}\]  
\[\hat{\theta}\geq 60^{\circ}\otimes\ \hat{\theta}\leq-120^{\circ}=H_{b}^{*} \tag{15}\]  
\[\hat{\theta}\geq-180^{\circ}\otimes\ \hat{\theta}\leq 0^{\circ}=H_{c}^{*} \tag{16}\]