## 2 Control Structure and State Variable Observers Used in the Research
### Sliding Mode Observer  
Sliding mode observer (SMO)used in article is based on PMSM model in \(\alpha\)-\(\beta\) coordinate system, defined below:  
\[\left[ \begin{array}{c}i_{\alpha}\\ i_{\beta}\end{array} \right]=\frac{1}{L} \begin{bmatrix}-R_{s}&0\\ 0&-R_{s}\end{bmatrix} \left[ \begin{array}{c}i_{\alpha}\\ i_{\beta}\end{array} \right]+\frac{1}{L} \begin{bmatrix}u_{\alpha}-e_{\alpha} \\ u_{\beta}-e_{\beta}\end{bmatrix} , \tag{5}\]  
where  
\(i_{\alpha},i_{\beta}\)--currents in \(\alpha\)-\(\beta\) coordinate;  
\(u_{\alpha},\mu_{\beta}\)--voltages in d-q coordinate. In the case of SMO, the basis for the speed estimation is the estimation of the back-EMF expressed as follows:  
\[e_{\alpha}=-\omega_{e}\psi_{PM}\sin(\theta_{e}); \tag{6}\]  
\[e_{\beta}=\omega_{e}\psi_{PM}\cos(\theta_{e}). \tag{7}\]  
From these equations, it can be concluded that the electrical speed value can be determined if the values of the back-EMF voltage are available. Since these components cannot be directly measured during motor control, an observer is required. For this purpose, the reduced order back-EMF SMO will be used, which is defined as follows:  
\[\left[ \begin{array}{c}\dot{i}_{\alpha}\\ \dot{i}_{\beta}\end{array} \right]=\frac{1}{L_{s}} \begin{bmatrix}-R_{s}&0\\ 0&-R_{s}\end{bmatrix} \left[ \begin{array}{c}\dot{i}_{\alpha}\\ \dot{i}_{\beta}\end{array} \right]+\frac{1}{L_{s}} \begin{bmatrix}u_{\alpha}- \dot{e}_{\alpha}-z_{\alpha}\\ u_{\beta}-\dot{e}_{\beta}-z_{\beta}\end{bmatrix} , \tag{8}\]  
where superscript (') indicates that the value is observed, and \(z_{\alpha}\) and \(z_{\beta}\) are the SMO feedback signals. In the case of this observer, it also attempted to equalize the estimated and measured currents: \(i_{\alpha},i_{\beta}\) and \(\dot{i}_{\alpha},\dot{i}_{\beta}\). The observation errors are defined as follows:  
\[\textbf{s(x)}= \begin{bmatrix}\ddot{i}_{\alpha}\\ \dot{i}_{\beta}\end{bmatrix} =\left[ \begin{array}{c}\dot{i}_{\alpha}-i_{ \alpha}\\ i_{\beta}-i_{\beta}\end{array} \right], \tag{9}\]  
where \(\dot{i}_{\alpha},\dot{i}_{\beta}\) are the errors between the measured and the observed stator currents. The sliding surface defined for the SMO is given by (10). It means that the phase plane is divided into two sections in which the observation errors defined in (9) have different signs. The switching action occurs as follows:  
\[\textbf{s(x)}=0 \tag{10}\]  
Figure 4: Sensorless startup with model reference adaptive system. The current observation errors are used as the input for the switching function. In the early stages of the SMO, the discontinuous sign function was commonly used [25]. However, the sign function introduces a lot of noise and chattering. Modern solutions use continuous functions as the limit, sigmoid or hyperbolic functions [26]. In this article, a hyperbolic function will be used, and it is defined as follows:  
\[\left[ \begin{array}{c}z_{\alpha}\\ z_{\beta}\end{array} \right]=k_{1}\left[ \begin{array}{c}\frac{e^{m\beta_{ \alpha}-e^{-m\beta_{\alpha}}}}{e^{m\beta_{\alpha}-e^{-m\beta_{\alpha}}}}\\ \frac{e^{m\beta_{\beta}-e^{-m\beta_{\beta}}}}{e^{m\beta_{\beta}-e^{-m\beta_{ \beta}}}}\end{array} \right], \tag{11}\]  
where  
\(k_{1}\)--the feedback gain;  
\(m_{\rm{is}}\)--the shaping parameter. The short time interval average values of the feedback signal components in (11) represent the back-EMF components. To obtain these values, a low-pass filter (LPF) can be used, which is defined as follows [26]:  
\[\left[ \begin{array}{c}\dot{e}_{\alpha}\\ \dot{e}_{\beta}\end{array} \right]=\frac{\omega_{c}}{s+\omega_{c}}\left[ \begin{array}{c}z_{\alpha}\\ z_{\beta}\end{array} \right], \tag{12}\]  
where \(\omega_{c}\) is the cutoff frequency of the LPF.