# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 3 Modified ADRC Servo Control Design
In the optimal synthesis function \(\text{fhan}(x_{1},x_{2},r,h)\) described by Equation (18), sign is the sign function and the specific expression of the fgg function is shown in Equation (19):  
\[\text{fgg}(y,d)=\frac{\text{sign}(x+d)-\text{sign}(x-d)}{2} \tag{19}\]  
Extended state observer (ESO):  
In PID control, its ability to suppress system disturbances is relatively limited. When disturbance signals do not have specific characteristics and are unknown, PID struggles to effectively suppress all disturbances in the control system. In ADRC, the concepts of the nominal model and total disturbance are introduced. The nominal model is an expected model that deviates slightly from the system's actual state, typically modeled as a series integral type. The difference between the actual model of the object and the desired nominal model is considered as the system's internal disturbance, generally including inaccuracies in the model establishment and the unmodeled parts of the actual system contained within the established model. The starting point of ADRC in suppressing disturbances is to model the object as the desired nominal model, no longer distinguishing between internal and external disturbances, but rather treating them uniformly as the system's total disturbance. In the actual implementation process, an extended state observer was designed, which can observe the total disturbance of the system and expand it as a state to be output by the extended state observer. Leveraging the convergence of the extended state observer, it can accurately observe and extract the total disturbance of the system, making the observed total disturbance close to the true value. This design scheme does not differentiate between internal and external disturbances, reducing the complexity of disturbance analysis from a design perspective. The extended state observer is a focal point in the design of ADRC, addressing how to acquire system disturbances during the control process. It not only observes the state and the differential of state variables to differentiate from the output of the tracking differentiator, but also can expand the observed total internal and external disturbances of the system into a state. The extended state observer observes total disturbances without targeting disturbances with specific characteristics and does not require measurement through sensors to obtain a more accurate disturbance, simplifying the control system's process of dealing with disturbances. Moreover, because the extended state observer can observe the total system disturbance in real-time accurately, appropriate compensation can be applied to enhance the system's disturbance suppression performance significantly. The continuous nonlinear extended state observer model is:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=z_{1}-y\\ \dot{z}_{1}=z_{2}-\beta_{01}\varepsilon_{1}\\ \dot{z}_{2}=z_{3}-\beta_{02}\text{fal}(\varepsilon_{1},0.5,\delta)+b_{0}u\\ \dot{z}_{3}=-\beta_{03}\text{fal}(\varepsilon_{1},0.25,\delta)\end{array} \right. \tag{20}\]  
The corresponding discrete nonlinear extended state observer model is:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=z_{1}(k)-y(k)\\ z_{1}(k+1)=z_{1}(k)+h\cdot(z_{2}(k)-\beta_{01}\varepsilon_{1})\\ z_{2}(k+1)=z_{2}(k)+h\cdot(z_{3}(k)-\beta_{02}\text{fal}(\varepsilon_{1},0.5, \delta)+b_{0}u)\\ z_{3}(k+1)=z_{3}(k)-h\cdot\beta_{03}\text{fal}(\varepsilon_{1},0.25,\delta) \end{array} \right. \tag{21}\]  
Wherein \(z_{1}\) is the estimated value of the system state observed by the extended state observer, \(z_{2}\) is the estimated value of the differential of the system state observed by the extended state observer, \(z_{3}\) is the total disturbance of the system observed by the extended state observer, \(h\) is the sampling period of the discrete system. In Equations (20) and (21), \(\beta_{01}\), \(\beta_{02}\), \(\beta_{03}\) and \(\delta\) are adjustable parameters of the controller. The specific mathematical description of the conventional fal function is:  
\[\text{fal}(x,a,\delta)=\left\{