# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 5 Numerical Simulations  
In order to verify the applicability of the results, we performed numerical simulations using MATLAB R2024b and its Control System Toolbox [25]. We applied the controller \(\Sigma_{K}\) given by (130), designed for the discrete-time unit feedback control system of Figure 2, to the digital control system shown in Figure 3. The data assumed for the computations correspond to the class of microsatellites with weight of 50-100 kg, as detailed in Table 2.  
For the reference signal \(\alpha_{r}(t)=a\sin(\omega_{r}t)\), we assume the following:  
\[a=1\left[ \begin{array}{cc}\mathrm{rad}\end{array} \right],\quad\omega_{r}=1 \left[ \begin{array}{c}\mathrm{deg}\\ \mathrm{s}\end{array} \right]=\frac{\pi}{180}\left[ \begin{array}{c}\mathrm{ rad}\\ \mathrm{s}\end{array} \right]. \tag{132}\]  
For the continuous-time plant \(\Sigma_{G}\), we obtain the following state space matrix:  
\[\left[ \begin{array}{cc|cc}A&B\\ \hline\mathbf{C}&0\end{array} \right]=\left[ \begin{array}{cccc|cc}0&0&1&0&0\\ 0&0&0&1&0\\ -441.17&441.17&-0.0059&0.0059&0.5882\\ 7500&-7500&0.1&-0.1&0\\ \hline 1&0&0&0&0\end{array} \right]. \tag{133}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(k\left[\begin{array}{c}\mathrm{N}\cdot\mathrm{m}\\ \mathrm{rad}\end{array}\right]\) & 750 \\ \(b\left[\begin{array}{c}\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}\end{array}\right]\) & 0.01 \\ \(I\left[\begin{array}{c}\mathrm{kg}\cdot\mathrm{m}^{2}\end{array}\right]\) & 1.7 \\ \(p\left[\begin{array}{c}\mathrm{kg}\cdot\mathrm{m}^{2}\end{array}\right]\) & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters for the satellite with a solar panel used in the computations. The continuous-time exosystem \(\Sigma_{S}\) has the following state space matrix:  
\[\left[\frac{S}{T}\right]=\left[ \begin{array}{cc}0&1\\ -0.0003&0\\ \hline 1&0\end{array} \right], \tag{134}\]  
with  
\[\sigma(S)=\left[ \begin{array}{cc}j0.0175&-j0.0175\end{array} \right]^{T}, \tag{135}\]  
which generates the continuous-time reference \(\alpha_{r}(t)\) in the following form:  
\[\alpha_{r}(t)=\sin 0.0175t\,. \tag{136}\]  
The discrete-time step \(h>0\) is determined according to the sampling frequency of the actual satellite's attitude angle \(\alpha(t)\), which depends on its on-board sensors. It is known from the literature that, for small \(h\) values, the obtained discrete-time models are usually very accurate. In our computations, we assumed the following two values of \(h\) and the resulting double sampling frequency \(\mu\):  
\[h = 0.1\,\mathrm{s}\,,\quad\mu = \frac{2}{h}=20\,\mathrm{Hz}\,, \tag{137}\] \[h = 0.05\,\mathrm{s}\,,\quad\mu = \frac{2}{h}=40\,\mathrm{Hz}\,.\]  
For these data, we obtain two discrete-time state space matrices:  
\[\left[ \begin{array}{c|c}A^{d}(20)&B^{d}(20)\\ \hline C^{d}(20)&D^{d}(20)\end{array} \right]=\left[ \begin{array}{cccc}0.8942 &0.1058&0.0947&0.0053&0.0088\\ 1.797&-0.7979&0.0899&0.0101&0.0084\\ -2.1151&2.1151&0.8942&0.1058&0.1762\\ 35.9570&-35.9570&1.7983&-0.7983&0.1673\\ \hline 0.2995&0.0167&0.0150&0.0008&0.0014\end{array} \right] \tag{138}\]  
and  
\[\left[ \begin{array}{c|c}A^{d}(40)&B^{d}(40)\\ \hline C^{d}(40)&D^{d}(40)\end{array} \right]=\left[ \begin{array}{c|cccc}0.90 76&0.0924&0.0477&0.0023&0.0031\\ 1.5714&-0.5714&0.0393&0.0107&0.0026\\ -3.6975&3.6975&0.9075&0.0925&0.1255\\ 62.8574&-62.8574&1.5723&-0.5723&0.1034\\ \hline 0.2133&0.0103&0.0053&0.0003&0.0003.5\end{array} \right], \tag{139}\]  
and two corresponding discrete-time exosystem matrices:  
\[\left[ \begin{array}{c}S^{d}(20)\\ \hline T^{d}(20)\end{array} \right]=\left[ \begin{array}{cc}1&0.1\\ 0&1\\ \hline 0.3162&0.0158\end{array} \right] \tag{140}\]  
and  
\[\left[ \begin{array}{c}S^{d}(40)\\ \hline T^{d}(40)\end{array} \right]=\left[ \begin{array}{cc}1&0.05\\ 0&1\\ \hline 0.2236&0.0056\end{array} \right]. \tag{141}\]  
In order to solve the regulator Equation (93), we used the _lyap_ procedure from the MATLAB R2024b Control System Toolbox [25] to compute \(\Pi\) from (104), after which \(\Gamma\)follows from (103).