## 3 Control Algorithm
### Current Observer  
In the proposed sensorless scheme, an electric angular velocity tracking error associated with the \(d\)-\(q\)-axes current tracking error is an important argument. Hence, in order to obtain the needed current tracking error, an estimated value rather than the measured value of the \(d\)-\(q\)-axes current should first be obtained.  
In this case, an extended current model of the \(d\)-\(q\)-axes is adopted in the proposed scheme. According to (2), the differential equation of the \(d\)-\(q\)-axes current can be rewritten as  
\[\frac{d}{dt}\left[\begin{array}{c}i_{d}+\psi_{f}/L_{d}\\ i_{q}\end{array}\right]=\left[\begin{array}{cc}-R/L_{d}&\omega_{e}L_{q}/L_{d }\\ -\omega_{e}L_{d}/L_{q}&-R/L_{q}\end{array}\right]\left[\begin{array}{c}i_{d}+ \psi_{f}/L_{d}\\ i_{q}\end{array}\right]+\left[\begin{array}{cc}1/L_{d}&0\\ 0&1/L_{q}\end{array}\right]\left[\begin{array}{c}u_{d}+R\psi_{f}/L_{d}\\ u_{q}\end{array}\right] \tag{8}\]  
with the definition as  
\[\left[\begin{array}{c}i_{d}t\\ i_{q}t\\ u_{d}t\\ u_{q}\end{array}\right]=\left[\begin{array}{c}i_{d}+\psi_{f}/L_{d}\\ i_{q}\\ u_{d}+R\psi_{f}/L_{d}\\ u_{q}\end{array}\right] \tag{9}\]  
Afterward, combining (8) and (9), the adjustable current model can be built as  
\[\frac{d}{dt}\left[\begin{array}{c}i_{d}t^{\prime}\\ i_{q}t^{\prime}\end{array}\right]=\left[\begin{array}{cc}-R/L_{d}&\omega_{e} L_{q}/L_{d}\\ -\omega_{e}L_{d}/L_{q}&-R/L_{q}\end{array}\right]\left[\begin{array}{c}i_{d}t^{ \prime}\\ i_{q}t^{\prime}\end{array}\right]+\left[\begin{array}{cc}1/L_{d}&0\\ 0&1/L_{q}\end{array}\right]\left[\begin{array}{c}u_{d}t^{\prime}\\ u_{q}t^{\prime}\end{array}\right] \tag{10}\]  
Then, the current observer could be designed to estimate the \(d\)-\(q\)-axes currents, as  
\[\frac{d}{dt}\left[\begin{array}{c}\hat{i}_{d}\\ \hat{i}_{q}\end{array}\right]=\left[\begin{array}{cc}-R/L_{d}&\hat{\omega}_ {e}L_{q}/L_{d}\\ -\hat{\omega}_{e}L_{d}/L_{q}&-R/L_{q}\end{array}\right]\left[\begin{array}{c }\hat{i}_{d}\\ \hat{i}_{d}\end{array}\right]+\left[\begin{array}{cc}1/L_{d}&0\\ 0&1/L_{q}\end{array}\right]\left[\begin{array}{c}\hat{u}_{d}\\ \hat{u}_{q}\end{array}\right] \tag{11}\]  
where hat \"\({}^{\cdots}\)\" represents the estimated value; \(\hat{u}_{d}=u_{d}\)', \(\hat{u}_{q}\)= \(u_{q}\)', and it has been proved to be asymptotically stable in [19].  
It is obvious that estimated values of the \(d\)-\(q\)-axes currents can be obtained when \(\hat{\omega}_{e}\), \(\hat{u}_{d}\), and \(\hat{u}_{q}\) are all clear. As such, a current observer can be constructed according to the adjustable model of the \(d\)-\(q\)-axes current, as shown in Figure 5, where \(\hat{\omega}_{e}\), \(\hat{u}_{d}\), \(\hat{u}_{q}\) are defined as the input value.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Positive** & **\(U_{0}\)** & **Negative** & **\(U_{0}\)** & **Medium** & **\(U_{0}\)** & **Large** & **\(U_{0}\)** & **Zero** & **\(U_{0}\)** \\
**Small Vector** & **Small Vector** & **Small Vector** & **Vector** & **Vector** & **Vector** & **Vector** & **Vector** & **\(U_{0}\)** \\ \hline
100 & + & 0\(-\)1\(-\)1 & \(-\) & 10\(-\)1 & \(-\) & 1\(-\)1\(-\)1 & \(=\) & 111 & \(=\) \\
110 & + & 00\(-\)1 & \(-\) & 01\(-\)1 & \(-\) & 11\(-\)1