## 2 Theoretical Analysis of the Integral Backstepping Controller and the ELESO
### Design of ELESO  
In the static coordinate system, the voltage equation of salient pole PMSM is as follows:  
\[ \begin{bmatrix}u_{\alpha}\\ u_{\beta}\end{bmatrix} = \begin{bmatrix}R_{s}+PL_{d}&\omega_{e}(L_{d}-L_{q})\\ -\omega_{e}(L_{d}-L_{q})&R_{s}+PL_{d}\end{bmatrix}\begin{bmatrix}i_{\alpha}\\ i_{\beta}\end{bmatrix} + \begin{bmatrix}E_{\alpha}\\ E_{\beta}\end{bmatrix}  \tag{2}\]  
where \(u_{\alpha}\) and \(u_{\beta}\) are the stator voltages in the static coordinate system. \(i_{\alpha}\) and \(i_{\beta}\) are \(\alpha\) and \(\beta\)-axis stator currents. \(\omega_{e}=n_{p}\omega\) is the rotor's electrical angular speed. \(L_{\alpha}\), \(L_{\beta}\) are the stator inductance. \(p\) is the differential operator. \(E_{\alpha}\) and \(E_{\beta}\) are the extended back EMF, which can be expressed as follows:  
\[ \begin{bmatrix}E_{\alpha}\\ E_{\beta}\end{bmatrix} =\Big{(}\omega_{e}\Big{[}(L_{d}-L_{q})i_{d}+\psi_{f} \Big{]}-\big{(}L_{d}-L_{q}\big{)}\,pi_{q}\Big{)} \begin{bmatrix}-\sin\theta_{e }\\ \cos\theta_{e}\end{bmatrix}  \tag{3}\]  
It can be seen from Equation (3) that the extended back EMF contains rotor position information. Therefore, the estimation of back EMF in static coordinates will directly affect the estimation accuracy of the rotor position. The ESO can estimate the lumped disturbance of the system accurately and efficiently. The basic form of the second-order LESO is expressed as:  
\[\left\{ \begin{array}{l}\varepsilon=z_{1}-x_{1}\\ z_{1}=z_{2}+f+b_{0}u-\beta_{1}\varepsilon\\ \dot{z}_{2}=-\beta_{2}\varepsilon\end{array} \right. \tag{4}\]  
where \(\varepsilon\) is the estimated error. \(x_{1}\) is the actual state variable. \(z_{1}\) is the estimation of \(x_{1}\). \(z_{2}\) is the estimated disturbance. \(u\) is the system input. \(f\) is the known disturbance, which contains the known model information of the system. \(b_{0}\) is the system parameter. \(\beta_{1}\) and \(\beta_{2}\) are the observer gains. If the \(\alpha\) and \(\beta\)-axis back EMF components with non-ideal conditions are considered as external interference, the state equation of stator current is as follows:  
\[pi_{x}=f_{\varepsilon_{x}}+f_{x}+u_{x}/L_{d},x=\alpha,\beta \tag{5}\]  
where \(f_{\varepsilon_{x}}\) represents the lumped interference of the system, which is the sum of the external interference and the internal interference. \(f_{x}\) represents known model information, which can be expressed as follows:  
\[\left\{ \begin{array}{l}f_{\alpha}=\hat{\omega}_{e}(L_{q}-L_{d})i_{\beta}/L_{ d}-R_{s}i_{\alpha}/L_{d}\\ f_{\beta}=\hat{\omega}_{e}(L_{d}-L_{q})i_{\alpha}/L_{d}-R_{s}i_{\beta}/L_{d} \end{array} \right. \tag{6}\]  
where \(\hat{\omega}_{e}\) is the estimated value of rotor electrical angular speed. Then, according to the definition of the state equation, the external disturbance state observer LESO1 is designed as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{x}=\hat{t}_{x}-i_{x}\\ p_{x}^{\dagger}=\hat{f}_{\varepsilon_{x}}+f_{x}+u_{x}/L_{d}-\beta_{x1} \varepsilon_{x}\\ p_{x}^{\dagger}=-\beta_{x2}\varepsilon_{x}\end{array} \right. \tag{7}\]  
where \(\hat{t}_{x}\) is the estimated value of the current. \(\beta_{x1}\) and \(\beta_{x2}\) are the gains of the LESO1. According to Equations (2) and (5), the estimated back EMF is as follows:  
\[\left[ \begin{array}{l}\hat{\varepsilon}_{\alpha}\\ \hat{\varepsilon}_{\beta}\end{array} \right]=-L_{d}\left[ \begin{array}{l} \hat{f}_{\varepsilon_{\alpha}}\\ \hat{\varepsilon}_{\beta}\end{array} \right] \tag{8}\]  
Based on the estimated back EMF, the position and the speed of the rotor under sensorless conditions can be obtained through the phase-locked loop (PLL). However, in practice, the inverter nonlinearity will generate different 5 th and 7 th harmonics in the phase current. As a result, \((6\mathrm{k}+1)\) subharmonics will appear in the back EMF estimation, which results in six harmonic fluctuations in the estimated rotor position and speed [24]. The sixth harmonic will reduce the performance of the control system.