## 3 Sensorless FOC with Luenberger-Type Back-EMF Observer
### Luenberger-Type Back-EMF Observer  
The primary purpose of this sensorless approach is to estimate the back-EMF voltage, which contains information about the rotor position. It is a method based on the mathematical model of the PMSM machine. Therefore, the mathematical model introduced in Section 1 is mandatory to describe this sensorless approach. The voltage equations of the PMSM in rotational reference frame \(dq\) depicted by (5) can be rearranged to the following form:  
\[ \begin{bmatrix}v_{d}\\ v_{q}\end{bmatrix} = \begin{bmatrix}R_{s}+L_{d}s&-\omega_{e}L_{q}\\ \omega_{e}L_{d}&R_{s}+L_{q}s\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix} +\omega_{e}\psi_{pm} \begin{bmatrix}0\\ 1\end{bmatrix}  \tag{24}\]  
where \(s\) is the Laplace operator. However, (24) cannot be applied for a sensorless control, because the motor position is unavailable, and thus, the back-EMF voltage cannot be observed. Therefore, it is necessary to employ a new estimated rotational reference frame \(\gamma\delta\) lagging from the origin \(dq\) rotational reference frame by an estimation position error \(\theta_{err}\), evident from Figure 10.  
Figure 10: Vector diagram of PMSM defining stationary and rotating reference frames. However, (24) has to be rewritten to the form with the symmetrical impedance matrix, as follows:  
\[ \begin{bmatrix}v_{d}\\ v_{q}\end{bmatrix} = \begin{bmatrix}R_{s}+L_{d}s&-\omega_{e}L_{q}\\ \omega_{e}L_{q}&R_{s}+L_{d}s\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{d}\end{bmatrix} +E_{ex} \begin{bmatrix}0\\ 1\end{bmatrix}  \tag{25}\]  
where \(E_{ex}\) is the extended EMF (EEMF) term equal to:  
\[E_{ex}=\omega_{e}\big{[}(L_{d}-L_{q})i_{d}+\psi_{pm}\big{]}-(L_{d}-L_{q})si_{q} \tag{26}\]  
Afterward, (25) can be transformed to the estimated reference frame \(\gamma\delta\), yielding voltage equations:  
\[ \begin{bmatrix}v_{\gamma}\\ v_{\delta}\end{bmatrix} = \begin{bmatrix}R_{s}+L_{d}s&-\omega_{e}L_{q}\\ \omega_{e}L_{q}&R_{s}+L_{d}s\end{bmatrix}\begin{bmatrix}i_{\gamma}\\ i_{\delta}\end{bmatrix} + \begin{bmatrix}e_{\gamma}\\ e_{\delta}\end{bmatrix}  \tag{27}\]  
where \(e_{\gamma}\), \(e_{\delta}\) are the \(\gamma\delta\) components of EEMF:  
\[ \begin{bmatrix}e_{\gamma}\\ e_{\delta}\end{bmatrix} =E_{ex} \begin{bmatrix}-\sin(\theta_{err})\\ \cos(\theta_{err})\end{bmatrix} +(\hat{\omega}_{e}-\omega_{e})L_{d} \begin{bmatrix} -i_{\delta}\\ i_{\gamma}\end{bmatrix}  \tag{28}\]  
where \(\hat{\omega}_{e}\) is the estimated rotor speed. Assuming that error between the estimated speed \(\hat{\omega}_{e}\) and actual rotor speed \(\omega_{e}\) under steady-state condition is zero, the second part of (28) can be neglected. Afterward, the estimated \(\gamma\delta\) components of EEMF become:  
\[ \begin{bmatrix}\hat{\omega}_{\gamma}\\ \hat{\varepsilon}_{\delta}\end{bmatrix} =E_{ex} \begin{bmatrix}-\sin(\theta_{ err})\\ \cos(\theta_{err})\end{bmatrix}  \tag{29}\]  
containing estimated position error \(\theta_{err}\). This variable can be extracted from (29) by employing arctan function as follows [12]:  
\[\theta_{err}=\tan^{-1}\bigg{(}\frac{-\hat{\varepsilon}_{\gamma}}{\hat{ \varepsilon}_{\delta}}\bigg{)} \tag{30}\]  
This also reduces the effect of the back-EMF observer parameter variation. For example, if the \(d\)-axis inductance \(L_{d}\) is not accurate, it will affect the dynamic performance of the observer. On the other hand, stator resistance \(R_{s}\) will not influence the control dynamics. Instead, it will directly affect the resulting amplitude of the estimated back-EMF. However, the effects of the unprecise \(L_{d}\), \(R_{s}\) are the same in both \(\gamma\) and \(\delta\) axis, because the controller and observer models are identical in these axes. Therefore, by performing the division of EEMF \(\gamma\delta\) components as in (30), the effects of the inaccurate \(L_{d}\), \(R_{s}\) are compensated. Nevertheless, the \(q\)-axis inductance \(L_{q}\) is not compensated with such division, because the change in one axis is not proportional to the other one.