# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 3 Modified ADRC Servo Control Design
By real-time estimation and elimination of this equivalent total disturbance, the control of complex controlled objects is simplified to that of a basic series integral type controlled object. Figure 4 shows the block diagram of the ADRC structure. ADRC mainly consists of three major parts: the tracking differentiator (TD), the extended state observer (ESO) and the nonlinear state error feedback (NLSEF). The following sections will introduce these three parts of ADRC individually. Tracking differentiator (TD):  
For PID controllers, since the differential element is non-regular, an ideal differentiator is essentially physically unrealizable and can only be approximated by a differentiator with a filter or by a differencer. In the case of SOTM antenna systems, which use digital control technology, the differentiator is typically approximated by a differencer. Furthermore, due to the presence of significant random noise signals in SOTM antenna systems, random noise signals may be amplified, making the approximate differentiator unusable due to excessive deviation. Therefore, in the engineering application of SOTM antenna systems, PI controllers are commonly used. Due to the absence of the differential element, the dynamic performance of the system may be reduced. The main function of the tracking differentiator is to extract discontinuous or signal with random noise from the actual system as well as the differential of the signal. Additionally, it mitigates the contradiction between the rapidity and overshoot of the control system by arranging a transient process. ADRC establishes an optimal synthesis function, based on which the Tracking Differentiator is designed to achieve rapid and synchronous extraction of the input signal and its differential, thereby reducing the amplification of random noise in the signal. Moreover, through the tracking differentiator, a smooth differential signal can be extracted. Simultaneously, by coordinating with the optimal synthesis function to arrange the transient process, it is possible to achieve rapid tracking of the input signal with minimal or even no overshoot. The continuous tracking differentiator model is:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\text{fhan}(x_{1}-v,x_{2},r_{0},h_{0})\end{array} \right. \tag{16}\]  
The corresponding discrete tracking differentiator model is:  
\[\left\{ \begin{array}{l}x_{1}(k+1)=x_{1}(k)+h\cdot x_{2}(k)\\ x_{2}(k+1)=x_{2}(k)+h\cdot fh\\ fh=\text{fhan}(x_{1}(k)-v(k),x_{2}(k),r_{0},h_{0})\end{array} \right. \tag{17}\]  
In Equations (16) and (17), \(v\) is the input signal, \(x_{1}\) is the input signal extracted by the tracking differentiator, which is the tracking of the input signal, \(x_{2}\) is the extracted  
Figure 4: Structure of the ADRC. differential signal and \(h\) is the sampling period. fhan function refers to the optimal synthesis function mentioned earlier and its expression is as follows:  
\[\text{fhan}(x_{1},x_{2},r,h)=\left\{ \begin{array}{l}d=r\hbar^{2}\\ a_{0}=hx_{2}\\ y=x_{1}+a_{0}\\ a_{1}=\sqrt{d(d+8|y|)}\\ a_{2}=a_{0}+((a_{1}-d)/2)\text{sign}(y)\\ a=(a_{0}+y)\text{fgg}(y,d)+a_{2}(1-\text{fgg}(y,d))\end{array} \right. \tag{18}\]  
In the \(\text{fhan}(x_{1},x_{2},r,h)\) function, \(r\) is the response to the tracking speed of the input signal, known as the speed factor. while \(h\) filters the noise carried by the signal, known as the filtering factor. Both the speed factor \(r\) and the filtering factor \(h\) are adjustable parameters. It is important to note that the filtering factor \(h\) in the fhan function and the sampling period \(h\) in Equation (17) are not the same variable. The sampling period \(h\) is determined at the beginning of system design and does not change, while the filtering factor \(h\) is an adjustable parameter that can be modified according to changes in the control objective.