# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 3 Modified ADRC Servo Control Design
Firstly, we verify its continuity, taking the point \(x=\delta\) as an example for analysis. Let \(\theta_{1}\sin x+\theta_{2}\text{tan}\,x=\Xi\), \(|x|^{a}\text{sign}(x)=\Gamma\), then:  
\[ \begin{array}{l}\Xi|_{x=\delta}\quad=\theta_{1}\cdot\theta_{2}\text{-}\sin \delta+\theta_{1}\cdot\theta_{3}\cdot\tan\delta\\ \quad=\frac{\delta^{a}-a\delta^{a-1}\sin\delta\text{cos}\,\delta}{\sin^{2} \delta}+\frac{a\delta^{a-1}\sin\delta-\delta^{a}\cos\delta}{\sin^{2}\delta}\\ \quad=\frac{\delta^{a}-a\delta^{a-1}\sin\delta\text{cos}\,\delta}{\sin^{2} \delta}+\frac{a\delta^{a-1}\sin\delta\text{cos}-\delta^{a}\cos^{2}\delta}{\sin^ {2}\delta}\\ \quad=\frac{\delta^{a}-a\delta^{a-1}\delta}{\sin^{2}\delta}\\ \quad=\frac{\delta^{a}\sin^{2}\delta}{\sin^{2}\delta}\\ \quad=\frac{\delta^{a}}{\delta^{a}}\end{array} \]  
\[\Gamma|_{x=\delta}=|\delta|^{a}\text{sign}(\delta)=\delta^{a}\]  
The following can be obtained:  
\[\Xi|_{x=\delta}=\,\Gamma|_{x=\delta}\]  
This implies that the improved fal function is continuous at the point \(x=\delta\). Using the same method, it can be concluded that the improved fal function is also continuous at the point \(x=-\delta\). Secondly, we verify the smoothness of the improved fal function, primarily by checking whether the left and right derivatives of the fal function at points \(x=\delta\) and \(x=-\delta\) are equal. \[ \begin{array}{lcl}\frac{d\Xi}{dx}\Big{|}_{x=\delta}&=\theta_{1}\cdot\theta_ {2}\text{-}\cos\delta+\theta_{1}\cdot\theta_{3}\text{-}\sec^{2}\delta\\ &=\frac{\delta^{a}\cos\delta-a\delta^{a-1}\sin\delta\text{cos}^{2}\delta}{ \sin^{3}\delta}+\frac{a\delta^{a-1}\text{-}\sin\delta-\delta^{a}\cos\delta}{ \sin^{3}\delta}\\ &=\frac{a\delta^{a-1}\text{-}\sin\delta-a\delta^{a-1}\sin\delta\text{cos}^{2} \delta}{\sin^{3}\delta}\\ &=\frac{a\delta^{a-1}\text{-}a\delta^{a}\cos^{2}\delta}{\sin^{2}\delta}\\ &=\frac{a\delta^{a-1}\text{-}\sin^{2}\delta}{\sin^{2}\delta}\\ &=a\delta^{a-1}\\ &=a\delta^{a-1}\\ &\frac{d\Gamma}{dx}\bigg{|}_{x=\delta}=a|x|^{a-1}=a\delta^{a-1}\end{array} \]  
It can be obtained that,  
\[\frac{d\Xi}{dx}\bigg{|}_{x=\delta}=\,\frac{d\Gamma}{dx}\bigg{|}_{x=\delta}\]Given that the left and right derivatives of the improved fal function are equal at \(x=\delta\), it indicates that the improved fal function is smooth at \(x=\delta\). Applying the same method, it is possible to deduce that the improved fal function is also smooth at \(x=-\delta\). Summarizing the above process, the improved fal function is a smooth and continuous function at both \(x=\delta\) and \(x=-\delta\). Next, we verify the symmetry of the improved fal function. When \(x>\delta\), it can be easily derived,  
\[\text{fal}(-x)=-\text{fal}(x)\]  
When \(0<x\leq\delta\),  
\[ \begin{array}{ll}\text{fal}(-x)&=\theta_{1}\cdot\theta_{2}\cdot\sin(-x)+ \theta_{1}\cdot\theta_{3}\cdot\tan(-x)\\ &=-\theta_{1}\cdot\theta_{2}\cdot\sin x-\theta_{1}\cdot\theta_{3}\cdot\tan x \\ &=-(\theta_{1}\cdot\theta_{2}\cdot\sin x+\theta_{1}\cdot\theta_{3}\cdot\tan x )\\ &=-\text{fal}(x)\end{array} \]  
When \(x=0\),  
\[\text{fal}(0)=\theta_{1}\cdot\theta_{2}\cdot\sin 0+\theta_{1}\cdot\theta_{3} \cdot\tan 0=0\]  
In summary, the improved fal function is centrosymmetric about the origin. As shown in Figure 5, it is a comparison of the curves between the conventional fal function and the improved fal function. In the figure, taking \(\delta=0.01\), it can be observed that at two separate points \(x=-0.01\) and \(x=0.01\), the conventional fal function is continuous but not smooth, whereas the improved fal function is both continuous and smooth. Furthermore, both the conventional fal function and the improved fal function are symmetric about the origin, which is consistent with the theoretical deductions previously discussed. Although the improved fal function is a smooth and continuous function, it is much more complex than the conventional fal function when \(|x|\leq\delta\), including nonlinear elements such as trigonometric functions.