# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 4 Precise Statement and Solution of the Discrete-Time Control Problem
### Characterization of the Discrete-Time Error Feedback Controller
\begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}&0\\ 0&0&S^{d}\\ \hline C^{d}&D^{d}C_{K}&0\end{array} \right]\left[ \begin{array}{c}p^{d}(i)\\ q^{d}(i)\\ r^{d}(i)\end{array} \right]. \tag{119}\]  
Due to (112), this implies that  
\[\lim_{i\rightarrow\infty}e^{d}(i)=\lim_{i\rightarrow\infty}\Bigl{[} \begin{array} []{cc}C^{d}&D^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \Bigr{]}^{i}\left[ \begin{array}{c}p ^{d}(0)\\ q^{d}(0)\end{array} \right]=0\,; \tag{120}\]  
that is, the asymptotic tracking condition (AT) holds. On the other hand, as  
\[\sigma(\left[ \begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \right])\cap\sigma(S^{d})=\mathcal{O }\,, \tag{121}\]then for every \(\left[ \begin{array}{c}0\\ B_{K}T^{d}\end{array} \right]\) the Sylvester Equation (117) has a unique solution \(\left[ \begin{array}{c}\Pi\\ \Sigma\end{array} \right]\). This equation and the asymptotic tracking condition AT, applied to System (116), imply the following:  
\[\lim_{i\to\infty}e^{d}(i) = \lim_{i\to\infty}\left[ \begin{array}{cc}C^{d}&D^{d}C_{K}\\ \end{array} \right]\left[ \begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \right]^{i}\left[ \begin{array}{c} x_{0}-\Pi r_{0}\\ x_{K0}-\Sigma r_{0}\end{array} \right]\] \[+\lim_{i\to\infty}(C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma)(S^{d})^{i}r_{0}\] \[= \lim_{i\to\infty}(C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma)(S^{d})^{i}r_{0}= 0\,.\]  
As \(\sigma(S^{d})\cap\mathbb{D}_{1}=\emptyset\), we get \(C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma=0\). Now let us note that, by substituting (118) into (117), we get an equivalent set of equations:  
\[\left\{ \begin{array}{l}A^{d}\Pi-\Pi S^{d}+B^{d}C_{K}\Sigma=0\,,\\ A_{K}\Sigma-\Sigma S^{d}=0\,,\\ C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma=0\,.\end{array} \right. \tag{123}\]  
Finally, if we introduce \(\Gamma=C_{K}\Sigma\), then (123) can be equivalently written as (113) and (114). Relation (113) is the _regulator equation_ from Theorem 1; hence, we denote it as RE. Furthermore, the second relation (114) is called the _internal model principle_ (see, e.g., [14; 23]), which we denote as IMP. The latter relation reflects the fact that a sensible controller has to include the dynamics of the exosystem. In order to develop a controller \((A_{K},B_{K},C_{K})\), we turn back to the result of Theorem 1 for the static state feedback controller. However, we assume that only the output of the plant can be measured, so, the components of the extended state \(\left[ \begin{array}{c}x^{d}(i)\\ r^{d}(i)\end{array} \right]\) are not available for the state feedback. As we can compute the error signal \(e^{d}(i)\)--that is, the output of the extended plant \(\Sigma_{ext}\)--we can construct a classic full-order asymptotic observer to estimate the extended state and try the approximate state \(\left[ \begin{array}{c}\tilde{x}^{d}(i)\\ r^{d}(i)\end{array} \right]\) in the control law  
\[\tau^{d}(i)=\left[ \begin{array}{cc}-F&\Gamma+F\Pi\end{array} \right]\left[ \begin{array}{c}\tilde{x}^{d}(i)\\ \tilde{r}^{d}(i)\end{array} \right],\quad i\in\mathbb{N}\,. \tag{124}\]  
In the next subsection, we show that if the observer is fed the error \(e^{d}(i)\), we obtain the required discrete-time error feedback controller (110).