## 2 Theoretical Analysis of the Integral Backstepping Controller and the ELESO
### Design of Backstepping Control
\tag{25}\]  
The current tracking error is selected as the virtual control variable. The current error is defined as follows:  
\[ \begin{cases}e_{q}=i_{qref}-i_{q}+i_{qe}\\ e_{d}=i_{dref}-i_{d}+i_{de}\end{cases}  \tag{26}\]  
where \(i_{de}\) and \(i_{qe}\) are the integral terms, which are expressed as follows:  
\[i_{xe}=K_{x}\int\limits_{0}^{t}{(i_{xref}-i_{x})dt},x=d,q \tag{27}\]  
At this time, a new system is being constructed. Consider the following candidate Lyapunov function:  
\[V_{q}=V_{\omega}+\frac{1}{2}e_{q}^{2}+\frac{1}{2}i_{qe}^{2} \tag{28}\]The derivative of Equation (28) can be obtained by combining Equations (1), (24) and (26):  
\[ \begin{array}{l}\dot{V}_{q}=-K_{1}e_{\omega}^{2}+e_{q}\Big{[}\frac{di_{qref}}{ dt}+\frac{R_{S}}{L_{q}}i_{q}+\frac{L_{d}}{L_{q}}n_{p}\omega i_{d}+\frac{\dot{V}_{q}m_{ p}}{L_{q}}\omega-\frac{1}{L_{q}}u_{q}+K_{q}(i_{qref}-i_{q})\Big{]}\\ +i_{qe}K_{q}(i_{qref}-i_{q})\end{array}  \tag{29}\]  
According to Equation (29), the control quantity \(u_{q}\) is selected as follows:  
\[u_{q}=L_{q}\bigg{[}K_{2}e_{q}+\frac{di_{qref}}{dt}+\frac{R_{S}}{L_{q}}i_{q}+ \frac{L_{d}}{L_{q}}n_{p}\omega i_{d}+\frac{\dot{V}_{q}n_{p}}{L_{q}}\omega\bigg{]} \tag{30}\]  
Substituting Equation (30) into Equation (29) yields:  
\[ \begin{array}{l}\dot{V}_{q}=-K_{1}e_{\omega}^{2}-K_{2}e_{q}^{2}+(e_{q}+i_{qe })K_{q}(i_{qref}-i_{q})\\ =-K_{1}e_{\omega}^{2}-K_{2}e_{q}^{2}+(e_{q}+i_{qe})K_{q}(e_{q}-i_{qe})\\ =-K_{1}e_{\omega}^{2}-(K_{2}-K_{q})e_{q}^{2}-K_{q}^{2}i_{qe}^{2}\end{array}  \tag{31}\]  
The result of Equation (31) is as follows:  
\[\dot{V}_{q}\leq-K_{Q}V_{q},K_{Q}=\min\big{\{}K_{1};(K_{2}-K_{q});K_{q}\big{\}} \tag{32}\]  
For the d-axis current \(i_{d}\), the target \(i_{dref}=0\). The control value is \(u_{d}\). Similar to the previous subsystem construction process, the candidate Lyapunov function is defined as:  
\[V_{d}=\frac{1}{2}e_{d}^{2}+\frac{1}{2}i_{de}^{2} \tag{33}\]  
The derivative of Equation (33) yields:  
\[ \begin{array}{l}\dot{V}_{d}=e_{d}\Big{(}\frac{di_{qref}}{dt}-\frac{di_{d}}{ dt}-K_{d}i_{d}\Big{)}+i_{de}K_{d}(i_{dref}-i_{d})\\ =e_{d}\Big{(}\frac{R_{S}}{L_{d}}i_{d}-\frac{L_{q}}{L_{d}}n_{p}\omega i_{q}- \frac{1}{L_{d}}u_{d}\Big{)}+K_{d}(e_{d}+i_{de})(e_{d}-i_{de})\end{array}  \tag{34}\]  
According to Equation (34), the control quantity \(u_{d}\) is selected as follows:  
\[u_{d}=L_{d}\bigg{(}K_{3}e_{d}+\frac{R_{S}}{L_{d}}i_{d}-\frac{L_{q}}{L_{d}}n_{ p}\omega i_{q}-\frac{1}{L_{d}}u_{d}\bigg{)} \tag{35}\]  
Substituting Equation (35) into Equation (34) yields:  
\[\dot{V}_{d}=-(K_{3}-K_{d})e_{d}^{2}-K_{d}i_{de}^{2} \tag{36}\]  
Similar to Equation (32), \(\dot{V}_{d}\) is represented as follows:  
\[\dot{V}_{d}\leq-K_{D}V_{d},K_{D}=\min\{(K_{3}-K_{d});K_{d}\} \tag{37}\]  
Through the design of the backstepping controllers, the speed controller and current controllers are cascaded to ensure that all virtual control variables will eventually converge to their reference values. In practice, some parameters of the PMSM are not fixed. Its value changes over time. For example, the stator resistance will change as the temperature changes [34]. However, in sensorless control of using equation model, motor parameters such as resistance and inductance are used in calculation. The system operation is based on the assumption that these parameters are predetermined and conform to actual motor parameters. Therefore, the difference of motor parameters may affect the stability of the sensorless control system. To solve this problem, in [35], the author introduced the on-line parameter identification as a compensation method for motor parameter difference. Specifically, the Recursive LeastSquares method (RLS) is used as the core, and the appropriate forgetting matrix is selected to identify. The feasibility of the method is verified by simulation.