# Levitating Control System of Maglev Ruler Based on Active Disturbance Rejection Controller
## 3 A Levitating Control System Based on Active Disturbance Rejection
### Design of Active Disturbance Rejection Controller for Levitating Control System
The smaller the tracking factor, the slower the tracing speed and the slower the tracking curve. However, it is also necessary to set a suitable tracking factor value based on the inertia of the MC to ensure the tracing effect of \(w_{1}(n)\) and \(w_{2}(n)\). \(h_{0}\) is the filtering factor. The effect of TD is determined by the \(h_{0}\). \(h_{0}\) is related to the execution frequency of the controller [2]. \(h\) is the integration step size [2]. The shorter the integration step size, the higher the precision of the integral value, but the stronger the lag. An overly short integration step size can cause system oscillations. However, an overly long integration step size can reduce the precision of the integral value. In this paper, \(h=h_{0}\). 2. Parameter tuning of ESO. By analyzing the inherent laws of ESO parameters and referring to the bandwidth method [22], this section proposes an ESO parameter tuning method for the levitation control system. Suppose:  
\[fal(e,\ a_{j},\ \delta)=\frac{fal(e,\ a_{j},\ \delta)}{e}e=\zeta_{i}(e)e \tag{62}\]where \(i=1,2\). Substituting Equation (62) into Equation (59):  
\[\left\{ \begin{array}{l}e=z_{1}-y_{z}\\ \dot{z}_{1}=z_{1}+h(z_{2}-\beta_{e1}e)\\ \dot{z}_{2}=z_{2}+h[z_{3}-\beta_{e2}\zeta_{1}(e)e+b_{0}u_{z}]\\ \dot{z}_{3}=z_{3}-h\beta_{e3}\zeta_{1}(e)e\end{array} \right. \tag{63}\]  
In order to analyze the characteristics of the function \(\zeta_{i}(e)\) and compare the changes in the output of the function \(\zeta_{i}(e)\) corresponding to different \(a_{j}\) values, this article selects \(a_{j}=0.2\), \(a_{j}=0.4\), \(a_{j}=0.8\), and \(\delta=0.1\), resulting in the output curve of the function \(\zeta_{i}(e)\) shown in Figure 9.  
From Figure 9, it can be seen that as the \(a_{j}\) decreases, the nonlinearity of function \(\zeta_{i}(e)\) increases, and the maximum gain increases. It can be seen that too small \(a_{j}\) may lead to high-frequency oscillations in the observed values, while a larger \(a_{j}\) may make it difficult for ESO to leverage its advantages of fast error attenuation and strong anti-interference capabilities. That is to say, \(a_{j}\) can significantly affect the performance of ESO. In addition, if the error \(e\) is within the range \([-\delta,\delta]\), the function \(\zeta_{i}(e)\) is a constant, and \((\zeta_{i}(e))_{\max}=\delta^{a_{j}-1}\). Additionally, an excessively large linear interval \(\delta\) can cause the nonlinear gain to fail, while a \(\delta\) that is too small can render the observer unstable. Typically, \(\delta\) should fall within the range of \([0.001,\,0.1]\), with \(\delta=h\) being suitable in this context; \(a_{j}\) must satisfy the condition that \(a_{1}>a_{2}\). According to the empirical values, \(a_{1}=0.5\) and \(a_{2}=0.25\). In ESO, there are three additional parameters that require adjustment, namely \(\beta_{e1}\), \(\beta_{e2}\), and \(\beta_{e3}\). \(\beta_{e3}\) is the most important of the three parameters. When tuning the parameters, first select the parameter \(\beta_{e3}\). If \(\beta_{e3}\) is too small, it will cause insufficient observation precision of \(z_{3}(\text{n})\), and \(z_{1}(\text{n})\) and \(z_{2}(\text{n})\) will lag behind \(w_{1}(\text{n})\) and \(w_{2}(\text{n})\). If \(\beta_{e3}\) is too large, it will cause increased system fluctuations and even system oscillations. Therefore, adjust \(\beta_{e3}\) first to ensure the precision of ESO, and then adjust \(\beta_{e1}\) and \(\beta_{e2}\) from small to large to reduce the oscillation of ESO output. Due to the coupling of the levitating system, these two parameter values should be as small as possible while ensuring stable ESO output. This paper employs the bandwidth method and experimental approach to tune the parameters of ESO.