# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 3 Modified ADRC Servo Control Design
Likewise, when \(|x|>\delta\), the improved fal function involves nonlinear elements such as exponential calculations. The equation for the improved fal function is known and from a mathematical standpoint, its implementation is relatively straightforward and does not require further discussion. However, from the perspective of implementation in SOTM systems, due to the presence of the aforementioned  
Figure 5: Conventional fal function and the improved fal function. complex calculations, microprocessors need to perform repeated floating-point multiplications, divisions and other operations when processing these calculations, which typically require more clock cycles to complete, consuming more computing resources and time. Especially for low-cost, low-power microprocessors, this requires more computational resources and time. Although the performance of microprocessors has improved, enabling high-end microprocessors to feature specialized hardware or instruction sets optimized for these complex calculations to enhance performance and computational efficiency. For most microprocessors, exponential and trigonometric calculations require substantial computing resources and time, presenting a significant challenge for microprocessors serving as controllers. Based on this, this paper proposes a piecewise linearization method for the improved fal function by approximating segmented straight lines to simplify calculations, thereby reducing the processing difficulty for microprocessors, enhancing their computational efficiency and decreasing the consumption of computational resources and time on the microprocessor. A schematic diagram of the piecewise linearization method for the improved fal function is shown in Figure 6. It should be noted that the method of selecting points described in this paper is not the only one. It can be adjusted according to different parameters of the improved fal function and verified by experiments or simulations. Figure 6 provides only one example. In Figure 6, several points, namely \(A\), \(B\), \(\cdots\), \(K\), are selected on the curve of the fal function, with line segments \(a\), \(b\), \(\cdots\), \(j\) connecting two adjacent points. Since the coordinates of points \(A\), \(B\), \(\cdots\), \(K\) are known, the equations for line segments \(a\), \(b\), \(\cdots\), \(j\) can be easily obtained:  
\[ \begin{cases}f(a)=\rho_{a}x+\sigma_{a}\\ f(b)=\rho_{b}x+\sigma_{b}\\ \vdots\\ f(j)=\rho_{j}x+\sigma_{j}\end{cases}  \tag{26}\]  
From Figure 6, it can be seen that the line formed by connecting segments \(a\), \(b\), \(\cdots\), \(j\) closely approximates the curve of the improved fal function. For the aforementioned piecewise linearization method, it is sufficient to store the parameters of each line and the coordinates of each point locally in the microprocessor to easily compute the improved fal function. Since the expression for each segment is a linear function, it can be calculated quickly and efficiently to obtain the value of the improved fal function by microprocessors  
Figure 6: A schematic diagram of the piecewise linearization method for the improved fal function. with hardware floating-point capabilities. This method consumes less computing resources and time for microprocessors. Additionally, as mentioned earlier, since the improved fal function is symmetrical about the origin, it is only necessary to select points and connect them with line segments either after or before the \(x=0\) point of the improved fal function. The other half can be easily obtained by utilizing the central symmetry property of the improved fal function, which can reduce the consumption of the microprocessor's local storage resources to some extent.