# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 3 Modified ADRC Servo Control Design
\begin{array}{l}\frac{x}{\delta^{1-\delta}}, \left|x\right|\leq\delta\\ \text{sign}(x)\left|x\right|^{\alpha},\left|x\right|>\delta\end{array} \right. \tag{22}\]  
Nonlinear state error feedback (NLSEF):  
In PID control, the proportional, integral and differential elements are linearly weighted and summed. Studies show that this is not an efficient method for designing control systems. ADRC adopts a nonlinear feedback scheme and practical applications have shown that, compared to linear feedback, nonlinear feedback can enhance the control precision of the system and the feedback coefficients exhibit stronger adaptability. In PID control, the integral term is used to eliminate steady-state error and suppress the adverse effects of disturbances on the system. When the system is subject to constant or persistent disturbances, the integral element can accumulate and gradually eliminate the resulting errors, ensuring that the system's output can gradually return to the set point. This means that the integral element helps the control system better cope with persistent interference, thereby improving the system's stability and robustness. However, in ADRC, because of the presence of the extended state observer, the total disturbance of the system can be estimated and the system can compensate for the adverse effects of the total disturbance. Therefore, in ADRC, the integration of the error signal is usually not used and only the error signal and its differential are needed. By using the input signal and its differential extracted by the tracking differentiator and the estimated values of the system's state variables and their differential obtained through the extended state observer, the error signal and its differential can be obtained by taking the difference. Nonlinear state error feedback generally has two forms of expression, one of which is:  
\[u_{0}=\beta_{1}\text{fal}(\varepsilon_{1},a_{1},\delta)+\beta_{2}\text{fal}( \varepsilon_{2},a_{2},\delta) \tag{23}\]  
Another form of expression is:  
\[u_{0}=\text{fhan}(\varepsilon_{1},\varepsilon_{2},r,h_{1}) \tag{24}\]  
In Equations (23) and (24), \(e_{1}\) is the error signal, \(e_{2}\) is the differential of the error signal and \(\beta_{1}\), \(\beta_{2}\), \(a_{1}\), \(a_{2}\), \(h_{1}\), \(c\) and \(\delta\) are adjustable parameters of the controller. In ADRC, the performance of the extended state observer significantly impacts the control effect of the control system. The control system needs to compensate for the total disturbance, which is observed and estimated by the extended state observer. The fal function, derived from engineering practice through fitting, does not have a unique form. In ADRC, the fal function is a continuous function, but it has non-differentiable points, namely at \(x=\delta\) and \(x=-\delta\) in Equation (22), where the left and right derivatives of the fal function are not equal. These points are cusp points, indicating that it is a continuous but non-smooth function. Studies have shown that the continuous smoothness of the fal function significantly affects the control effect of ADRC [36]. Thus, the fal function may adversely affect the control system, there is a need to develop a new fal function [37; 38; 39]. Therefore, this paper introduces a new fal function as presented in Equation (25):  
\[\text{fal}(x,a,\delta)=\left\{ \begin{array}{l}\theta_{1}\cdot\theta_{2}\text {-}\sin x+\theta_{1}\cdot\theta_{3}\text{-}\tan x,\ |x|\leq\delta\\ \text{sign}(x)|x|^{a},\ \ \ |x|>\delta\end{array} \right. \tag{25}\]  
where  
\[\theta_{1}=\text{csc}^{3}\,\delta\]  
\[\theta_{2}=\delta^{a}-a\delta^{a-1}\text{sin}\,\delta\text{cos}\,\delta\]  
\[\theta_{3}=a\delta^{a-1}\text{sin}\,\delta\text{cos}^{2}\,\delta-\delta^{a} \text{cos}^{3}\,\delta\]  
In this paper, the fal function described by Equation (25) is referred to as the improved fal function, whereas the ADRC based on this improved fal function is referred to as the modified ADRC. Next, we verify whether the improved fal function is a smooth continuous function.