## 3 Sensorless FOC with Luenberger-Type Back-EMF Observer
### Sensorless FOC
#### 3.1.1 Current Loop  
FOC is based on independent control of the machine torque and flux. Generally, the torque-producing component is _q_-axis current, and the component responsible for machine flux production is _d_-axis current. These components are part of the inner current loop, with PI controllers placed for each path, as shown in Figure 3. By compensating the cross-coupling and back-EMF terms in (5), simplified voltage equations can be defined:  
\[ \begin{bmatrix}v_{d}\\ v_{q}\end{bmatrix} =R_{s} \begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix} + \begin{bmatrix}L_{d}&0\\ 0&L_{q}\end{bmatrix} \frac{d}{dt} \begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}  \tag{8}\]  
It is clear from (8) that the equations are identical in structure. Therefore, the same principle can be employed to calculate the gains of the controllers in both axes. The only \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Control Structure** & **PI** & **I** & **Park** & **Clarke** & **Atan** & **Filters** \\ \hline Sensorless FOC & & & & & & & \\ with the Luenberger-type & 6 & 1 & 4 & 1 & 1 & 2 \\ back-EMF observer & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sensorless FOC with the Luenberger-type back-EMF observer—complexity. Figure 4: Transformations in FOC scheme. difference lies in \(L_{d}\) and \(L_{q}\), which leads to different proportional and integral gains \(K_{P}\), \(K_{I}\). To accurately describe the current control loop, the transfer functions of A/D converters and the voltage source inverter with a PWM modulator have to be evaluated. Moreover, the sampling time of the current loop and discrete form of the controllers have to be considered. However, the effects of A/D converter and inverter transportation delay are neglected to simplify the controller design. The closed-current loop can then be designed with the following transfer functions representing RL circuit and current controller:  
\[F_{RL}(s)=\frac{1}{Ls+R_{s}} \tag{9}\]  
\[F_{Pl_{i}}(s)=\frac{K_{P_{i}}s+K_{I_{i}}}{s} \tag{10}\]  
where \(K_{P_{i}}\), \(K_{I_{i}}\) are the proportional and integral gains of the current controller. The transfer function of a simplified closed-current loop can be derived based on (9) and (10):  
\[F_{i}(s)=\frac{F_{Pl_{i}}(s)F_{RL}(s)}{1+F_{Pl_{i}}(s)F_{RL}(s)}=\frac{\frac{K_ {P_{i}}}{L}s+\frac{K_{I_{i}}}{L}}{s^{2}+\bigg{(}\frac{K_{P_{i}}+R_{s}}{L} \bigg{)}s+\frac{K_{I_{i}}}{L}} \tag{11}\]  
The block diagram of the simplified current control loop with compensated cross-coupling and back-EMF terms is illustrated in Figure 5.  
The characteristic Equation of (11) reveals a second-order system. However, the nominator of the transfer function introduces zero due to the PI controller. It increases the system overshoot, lowering the potential closed-loop bandwidth. The zero can be compensated by a zero-cancellation block in the feedforward. The zero-cancellation block can be considered as a first-order filter with the following transfer function:  
\[F_{ZC_{i}}(s)=\frac{1}{\frac{K_{P_{i}}}{K_{I_{i}}^{2}}s+1} \tag{12}\]  
Introducing (12) to (11), the closed-current loop transfer function will be rearranged as:  
\[F_{i}(s)=\frac{F_{Pl_{i}}(s)F_{RL}(s)}{1+F_{Pl_{i}}(s)F_{RL}(s)}=\frac{\frac{K _{I_{i}}}{L}}{s^{2}+\bigg{(}\frac{K_{P_{i}}+R_{s}}{L}\bigg{)}s+\frac{K_{I_{i} }}{L}} \tag{13}\]  
The block diagram of the simplified closed-current loop with zero-cancellation is presented in Figure 6.  
Figure 5: Block diagram of the simplified current loop scheme. Figure 4:0 Block diagram of the simplified current loop scheme with zero-cancellation. Finally, the proportional and integral gains of the current controller can be designed by comparing the closed-loop characteristic polynomial with the ideal second-order system:  
Figure 4:1  
where \(\omega_{0_{i}}\) is the natural frequency of the current loop, and \(\xi_{i}\) is the damping factor of the current loop. The proportional and integral gains of the current controller in continuous domain can be then derived from (14) as follows:  
\[