# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 2 Preliminary Formulation of the Control Problem
\begin{table}
\begin{tabular}{l} \hline \hline
**Signal or Parameter [SI Units]** \\ \hline Satellite attitude angle \(\alpha(t)\,\left[\mathrm{rad}\right]\) \\ Panel attitude angle \(\beta(t)\,\left[\mathrm{rad}\right]\) \\ Satellite driving torque \(\tau(t)\,\left[\mathbb{N}\cdot\mathrm{m}\right]\) \\ Satellite rotational inertia \(I\,\left[\mathrm{kg}\cdot\mathrm{m}^{2}\right]\) \\ Panel rotational inertia \(p\,\left[\mathrm{kg}\cdot\mathrm{m}^{2}\right]\) \\ Stiffness coefficient \(k\,\left[\frac{\mathbb{N}\cdot\mathrm{m}}{\mathrm{rad}}\right]\) \\ Damping coefficient \(b\,\left[\mathbb{N}\cdot\mathrm{m}\cdot\mathrm{s}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Signals and parameters for the satellite model. First, we will derive and analyze the system properties of the continuous-time state space model of the plant \(\Sigma_{G}\); that is,  
\[\Sigma_{G}:\quad\left[ \begin{array}{c}\dot{x}(t)\\ \alpha(t)\end{array} \right]=\left[ \begin{array}{cc}A&B\\ C&0\end{array} \right]\left[ \begin{array}{c}x(t)\\ \tau(t)\end{array} \right]. \tag{7}\]  
Then, using the mid-point rule for approximation of the state integral (equivalent to the Cayley-Tustin transformation), we develop a _discrete-time state space model_\(\Sigma_{G}^{d}\) of the plant in the following form:  
\[\Sigma_{G}^{d}:\ \left[ \begin{array}{c}x^{d}((i+1)h)\\ \alpha^{d}(ih)\end{array} \right]=\left[ \begin{array}{cc}A^{d}&B^{d}\\ C^{d}&D^{d}\end{array} \right]\left[ \begin{array}{c}x^{d}(ih)\\ \tau^{d}(ih)\end{array} \right], \tag{8}\]  
where \((x^{d}(ih))_{i\in\mathbb{N}}\subset\mathbb{R}^{n}\) is the state of the plant. We make use of generalized regulation theory to derive a discrete-time state space dynamic error feedback controller in the following form:  
\[\Sigma_{K}:\ \left[ \begin{array}{c}x_{K}((i+1)h)\\ \tau^{d}(ih)\end{array} \right]=\left[ \begin{array}{cc}A_{K}&B_{K}\\ C_{K}&0\end{array} \right]\left[ \begin{array}{c}x_{K}(ih)\\ e^{d}(ih)\end{array} \right], \tag{9}\]  
where \((x_{K}(ih))_{i\in\mathbb{N}}\subset\mathbb{R}^{n_{K}}\), \(n_{K}\) is the order of the controller and the error \((e^{d}(ih))_{i\in\mathbb{N}}\) is the only signal available to the controller. The _control goal_ in the discrete-time domain is to design the error feedback control system to asymptotically track the reference signal \((\alpha_{r}^{d}(ih))_{i\in\mathbb{N}}\) via the plant output \((a^{d}(ih))_{i\in\mathbb{N}}\); that is,  
\[\lim_{i\rightarrow\infty}e^{d}(ih)=0\,. \tag{10}\]  
It follows from (6), (8), and (9) that the discrete-time control system we want to design as the main step is the unit feedback control system shown in Figure 2.  
Our ultimate goal is to implement the discrete-time controller \(\Sigma_{K}\), obtained in the main step, in the _digital control system_ with the continuous-time plant \(\Sigma_{G}\) and the zero-order hold \(ZOH\), as shown in Figure 3.  
Figure 2: The discrete-time unit feedback control system. **Remark 1**. : _Note that in the digital control system in Figure 3, \(\left(\alpha_{r}(ih)\right)_{i\in\mathbb{N}}\) are computed samples of the reference signal (2), \(\left(\alpha(ih)\right)_{i\in\mathbb{N}}\) are measured samples of the real output \(\left(\alpha(t)\right)_{t\geq 0}\), \(\left(e(ih)\right)_{i\in\mathbb{N}}\) is the computed discrete-time signal feeding the discrete-time controller \(\Sigma_{K}\), \(e(ih)\) represents computed samples of the continuous-time error signal \(e(t)=\alpha(t)-\alpha_{r}(t)\), \(\left(\tau(ih)\right)_{i\in\mathbb{N}}\) is the computed discrete-time signal provided by the controller \(\Sigma_{K}\), and \(\left(\tau(t)\right)_{t\geq 0}\) is the continuous-time signal generated by ZOH, which feeds the real continuous-time plant \(\Sigma_{G}\)._