## 2 Control Structure and State Variable Observers Used in the Research
### Model Reference Adaptive System  
The first system used for speed estimation is the model reference adaptive system (MRAS). This type of estimator consists of three basic elements: reference model, adjustable model, and adaptive law. The difference between the value obtained from the reference model and the adjustable model is processed by the adaption law. Subsequently obtained signal can adjust the adjustable model parameters and strive to obtain the same value as the value from the reference model. MRAS used in the article is based on the PMSM model in the \(d\)-\(q\) coordinate system, assuming that the motor is equipped in surface mounted magnets and inductances \(L_{d}=L_{q}\)[22]:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{P_{N}}\) [kW] & \(\mathbf{P_{p}}\) [-] & \(\mathbf{n_{N}}\) [rpm] & \(\mathbf{T_{N}}\) [Nm] & \(\mathbf{I_{N}}\) [A] & \(\mathbf{J}\) [kg\(\cdot\)m\({}^{2}\)] & \(\mathbf{R_{S}}\) [\(\Omega\)] \\ \hline
0.894 & 4 & 6200 & 1.4 & 1.9 & 0.000039 & 4.6615 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of tested motor.  
\[\frac{d}{dt}\begin{bmatrix}i_{d}+\frac{\psi_{f}}{L}\\ i_{q}\end{bmatrix}=\begin{bmatrix}\frac{-R_{s}}{L}&\omega_{e}\\ \omega_{e}&\frac{-R_{s}}{L}\end{bmatrix}\begin{bmatrix}i_{d}+\frac{\psi_{f}}{L} \\ i_{q}\end{bmatrix}+\frac{1}{L}\begin{bmatrix}U_{d}+\frac{R_{s}\psi_{f}}{L}\\ U_{q}\end{bmatrix}, \tag{1}\]  
where  
\(i_{d}\), \(i_{q}\)--currents in \(d\)-\(q\) coordinate;  
\(\psi_{f}\)--magnetic flux;  
\(L\)--stator inductance;  
\(R_{s}\)--stator resistance;  
\(U_{d}\), \(U_{q}\)--voltages in \(d\)-\(q\) coordinate;  
\(\omega_{e}\)--electrical speed.  
In the following, for the sake of simplifying the notation, the following relationship was assumed:  
\[i_{d}^{*}=i_{d}+\frac{\psi_{f}}{L},i_{q}^{*}=i_{q},U_{d}^{*}=U_{d}+\frac{R_{s} \psi_{f}}{L},U_{q}^{*}=U_{q}, \tag{2}\]  
The adjustable model is based on Equations (1) and (2), where the adaptive parameter is electrical speed \(\omega_{e}\). The actual measurement of the \(i_{d}\), \(i_{q}\) currents is selected as the reference model. The excitatory signals for both models are the \(U_{d}\), \(U_{q}\) voltages. The final equation is presented below as follows:  
\[\frac{d}{dt}\begin{bmatrix}\dot{i}_{d}\\ \dot{i}_{q}\end{bmatrix}=\begin{bmatrix}\frac{-R_{s}}{L}&\omega_{e}\\ \tilde{\omega}_{e}&\frac{-R_{s}}{L}\end{bmatrix}\begin{bmatrix}\dot{i}_{d}\\ \dot{i}_{q}\end{bmatrix}+\frac{1}{L}\begin{bmatrix}U_{d}^{*}\\ U_{q}^{*}\end{bmatrix}, \tag{3}\]  
where  
\(\dot{\hat{i}}_{d}\), \(\dot{i}_{q}\)--estimated currents in d-q coordinate;  
\(\tilde{\omega}_{e}\)--estimated electrical speed.  
The last element of the MRAS is adaptive law. In the study, adaptive law based on the following equation was used:  
\[\hat{\omega}=(K_{p}+\frac{K_{i}}{s})(i_{d}\dot{i}_{q}-i_{q}\dot{i}_{d}-\frac{ \psi_{f}}{L}(i_{q}-\dot{i}_{q})), \tag{4}\]  
In the research, adaptive law parameters \(K_{p}=0.6\) and \(K_{i}=200\) were adopted. Figure 2 shows the complete estimation scheme based on MRAS.  
Example transients obtained from the experimental set-up of the estimated and measured speed in the sensor and sensorless mode operation are shown in Figure 3. Speed is estimated with high accuracy. The system works stably in sensorless mode.  
Figure 1: Photos of the experimental set-up.  
Figure 3: Transients of speed, current in \(q\) axis, and rotor position measured and estimated by MRAS in sensor (**a**) and sensorless (**b**) mode.  
Figure 2: Model reference adaptive system used in research structure, where estimated \(\theta_{e}\) is electrical motor position.  
The MRAS system also enables sensorless startup. The course of such a start is shown in Figure 4.