## 2 Theoretical Analysis of the Integral Backstepping Controller and the ELESO
### Design of ELESO
The back EMF formula including the sixth harmonic component is as follows:  
\[\left\{ \begin{array}{l}\hat{\varepsilon}_{\alpha}=-E_{ex}\sin(\omega_{e}t+ \theta_{e})-E_{6\mathrm{k}\pm 1}\sin((6\mathrm{k}+1)\omega_{e}t+\theta_{6 \mathrm{k}\pm 1})\\ \hat{\varepsilon}_{\beta}=-E_{ex}\sin(\omega_{e}t+\theta_{e})+E_{6\mathrm{k} \pm 1}\cos((6\mathrm{k}+1)\omega_{e}t+\theta_{6\mathrm{k}\pm 1})\end{array} \right. \tag{9}\]  
The rotor angle estimation error becomes:  
\[\varepsilon_{\theta}=\theta_{e}-\hat{\theta}_{e}+E_{6\mathrm{k}}\sin(6k\omega_ {e}t+\theta_{6\mathrm{k}}) \tag{10}\]  
where \(E_{ex}=\omega_{e}\cdot\psi_{f}\). In general, there is only the harmonic effect when \(k=1\). Therefore, after the whole FOC control finishes, the output rotor angle error will contain the sixth harmonic component. The ADRC does not require an accurate motor model. According to this characteristic, this paper proposes an internal disturbance observer LESO2 to estimate the internal disturbance of the system. The LESO2 is cascaded with the LESO1. With this scheme, the influence of higher harmonics and motor parameter changes of the system performance can be eliminated as far as possible. At this time, the stator current model is shown as follows:  
\[pi_{x}=f_{\varepsilon_{x}}+f_{x}+u_{x}/L_{d}+f_{\varepsilon_{bx}},x=\alpha,\beta \tag{11}\]  
where \(f_{\varepsilon_{bx}}\) represents the internal disturbance in the static coordinate system, which is the high-order harmonic back EMF. According to the current formula, the LESO2 is designed as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{hx}=\hat{i}_{hx}-i_{x}\\ p_{hx}^{\dagger}=\hat{f}_{x}+f_{x}+\hat{f}_{\varepsilon_{hx}}+u_{x}/L_{d}- \beta_{x3}\varepsilon_{hx}\\ p\hat{f}_{\varepsilon_{hx}}=-\beta_{x4}\varepsilon_{hx}\end{array} \right. \tag{12}\]  
where \(\hat{i}_{hx}\) is the estimated current component under the higher harmonic. \(\varepsilon_{hx}\) is the estimation error. \(\hat{f}_{\varepsilon_{hx}}\) is the estimated internal disturbance. \(\beta_{x3}\) and \(\beta_{x4}\) are the gains of the LESO2. According to Equations (3) and (8), the estimated rotor position can be calculated as:  
\[\hat{\theta}_{e}=\arctan\left(\frac{-\hat{\varepsilon}_{x}}{\hat{\varepsilon} _{\beta}}\right) \tag{13}\]  
In Equation (13), the estimated rotor position can be obtained directly. However, the arctangent function is very sensitive to noise. The position extraction method based on the arctangent function directly brings this noise into the operation, which leads to the error being amplified, thus causing greater angle estimation error. This uncertainty will reduce the accuracy and stability of the control system, especially in the middle and low speed range. Therefore, it is necessary to use the PLL to extract speed and position information from the estimated back EMF [25]. \(\beta_{x1}\), \(\beta_{x2}\), \(\beta_{x3}\) and \(\beta_{x4}\) are the gains of the LESO1 and the LESO2. If the gains are not properly selected, it will lead to the loss of position in the process of controller evaluation, which will lead to system instability [26; 27]. Therefore, the selection of gains is an important issue. Taking the LESO1 as an example, the tracking error is defined as follows:  
\[e=\left[ \begin{array}{c}\hat{i}_{x}-i_{x}\\ \hat{f}_{\varepsilon_{x}}-f_{\varepsilon_{x}}\end{array} \right] \tag{14}\]  
According to Equations (5) and (7), the error state equation can be derived as:  
\[\dot{e}=C_{m}e,C_{m}=\left[ \begin{array}{cc}-\beta_{x1}&1\\ -\beta_{x2}&0\end{array} \right] \tag{15}\]  
Equation (15) shows that the eigenvalues of \(C_{m}\) determine the behavior of the LESO1. If and only if \(\beta_{x2}>0\), Equation (15) is asymptotically stable. Therefore, the gains of the LESO1 can be selected according to the bandwidth of the LESO1 [28]. Specifically, the gains of the LESO1 are designed such that \(C_{m}\) has a double eigenvalue \(\lambda\) that is equal to the bandwidth of the LESO1 [29]. The characteristic equation is as follows:  
\[|\lambda E-C_{m}|=\left|