## 2 The Mathematical Model  
### The Permanent Magnet Synchronous Motor Model  
Taking a surface-mounted PMSM as an example, its angular velocity, mechanical motion, torque, and voltage equations under the \(d\), \(q\) axis are as follows:  
\[\left\{\begin{array}{c}\frac{d\theta}{dt}=\omega\\ \int\frac{d\omega}{dt}=T_{e}-T_{L}-B\omega  
of three components: a tracking differentiator (TD), an extended state observer (ESO), and the nonlinear error feedback control law (NLSEF). The TD helps to smooth the transition of the input command signal and alleviates the trade-off between speed and smoothness in differential operations. The core of ADRC is the ESO, which is responsible for estimating and compensating for internal and external disturbances. The error feedback control law mainly focuses on compensating for and adjusting the error between the TD and ESO outputs. The schematic diagram of the ADRC is as follows:  
As shown in Figure 1, \(v\) is a given reference input, which outputs the tracking signal \(v_{1}\) and the differential \(v_{2}\) of the tracking signal through the TD. At the same time, the output \(y\) passes through the ESO to obtain the extended state \(z_{1}\), \(z_{2}\) of the controlled object, and then the overall disturbance \(z_{3}\) of the system is obtained. Further, it compensates for the target error \(e_{1}\) and the differential error \(e_{2}\) in the system according to feedforward compensation. Finally, the difference between the outputs of the TD and the ESO is passed through the NLSEF and the output \(u_{0}\), together with feedforward compensation. In addition, \(u_{0}\) is the intermediate variable for the output, \(u\) is the final output, \(w\) is the unknown disturbance acting on the controlled object, \(b_{0}\) is the gain, and \(1/b_{0}\) is the compensation coefficient.  
Referring to the control structure in Figure 1, the second-order active disturbance rejection model can be derived as follows (it includes three parts: the TD, the NLSEF, and the ESO):  
\[\text{TD}:\left\{\begin{array}{l}fh=fhan(v_{1}(t)-v(t),v_{2}(t),r,h)\\ v_{1}(t+1)=v_{1}(t)+hv_{2}(t)\\ v_{2}(t+1)=v_{2}(t)+hfh\end{array}\right. \tag{3}\]  
\(h\) in Formula (3) above represents the integral step size. \(v(t)\) is the input signal, \(v_{1}(t)\) corresponds to the tracking signal of \(v(t)\), and \(v_{2}(t)\) denotes the estimated integral of \(v(t)\), \(fh\) represents the fastest control function \(fhan(v_{1}(t)-v(t),v_{2}(t),r,h)\).  
\[\text{The ESO}:\left\{\begin{array}{l}e=z_{1}-y\\ \dot{z_{1}}=z_{2}-\beta_{01}fal(\varepsilon,\alpha_{1},\lambda_{1})\\ \dot{z_{2}}=z_{3}-\beta_{02}fal(\varepsilon,\alpha_{2},\lambda_{2})+bu(t)\\ z_{3}=-\beta_{03}fal(\varepsilon,\alpha_{3},\lambda_{3})\end{array}\right. \tag{4}\]  
Formula (4) includes the error estimate \(e\), where \(y\) represents the output value, \(z_{1}\) is the estimated value of \(y\), \(z_{2}\) is the derivative of \(z_{1}\), and \(z_{3}\) represents the estimated value of the internal and external disturbances in the system. Additionally, \(\beta_{01}\), \(\beta_{02}\), and \(\beta_{03}\) denote the increments for correcting output errors, and \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are filter factors. Lastly, \(b\) is the coefficient for compensating for disturbances.  
\[\text{The NLSEF}:\left\{\begin{array}{l}e_{1}=v_{1}-z_{1}\\ e_{2}=v_{2}-z_{2}\\ u_{0}=\beta_{03}fal(\varepsilon,\alpha_{3},\lambda_{3})\\ u=u_{0}-z_{3}/b\end{array}\right. \tag{5}\]  
Figure 1: Block diagram of the ADRC structure.  
The formula above shows \(e_{1}\) and \(e_{2}\) as error signals, with _fal_ denoting the nonlinear function. The expression is depicted as follows:  
\[fal(e,\alpha,\lambda)=\left\{\begin{array}{l}\frac{e(t)}{\lambda^{a-1}},|e(t)| \leq\lambda\\ |e(t)|^{a}sign(e(t)),|e(t)|>\lambda\end{array}\right. \tag{6}\]  
The interval length \(\lambda\) in the formula represents the linear segment of the function, while \(\alpha\) is the parameter for the nonlinear function where \(0<\alpha<1\). _sign_(\(\cdot\)) denotes a symbolic function.