# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 4 Precise Statement and Solution of the Discrete-Time Control Problem
### Derivation of the Discrete-Time Error Feedback Controller Based on a Full-Order Observer  
We rewrite the extended plant model \(\Sigma_{ext}\) (see (109)) in the following form:  
\[\Sigma_{ext}:\left\{ \begin{array}{rcl}\left[\begin{array}{c}x^{d}(i+1)\\ r^{d}(i+1)\end{array}\right]&=&\left[\begin{array}{cc}A^{d}&0\\ 0&S^{d}\end{array}\right]\left[\begin{array}{c}x^{d}(i)\\ r^{d}(i)\end{array}\right]+\left[\begin{array}{c}B^{d}\\ 0\end{array}\right]\tau^{d}(i)\,,\\ &&\left[\begin{array}{c}x^{d}(0)\\ r^{d}(0)\end{array}\right]=\left[\begin{array}{c}x_{0}\\ r_{0}\end{array}\right],\\ &&\epsilon^{d}(i)&=&\left[\begin{array}{cc}C^{d}&-T^{d}\end{array}\right] \left[\begin{array}{c}x^{d}(i)\\ r^{d}(i)\end{array}\right]+D^{d}\tau^{d}(i).\end{array} \right. \tag{125}\]In the research [19], it was shown that, for the pair \(\left(\left[ \begin{array}{cc}C&-T\end{array} \right]\),\(\left[ \begin{array}{cc}A&0\\ 0&S\end{array} \right]\right)\), the observability matrix \(V\) satisfies  
\[\det V=\det\left[ \begin{array}{cc}C&-T\\ CA&-TS\\ \vdots&\vdots\\ CA^{5}&-TS^{5}\end{array} \right]=-\frac{k^{2}\omega_{r}^{2}}{l^{2}}\left( \omega_{r}^{2}\left(\frac{b}{p}+\frac{b}{l}\right)^{2}+\left(\frac{k}{p}+ \frac{k}{l}-\omega_{r}^{2}\right)^{2}\right), \tag{126}\]  
which demonstrates that, for \(b>0\), the pair is observable; meanwhile, if \(b=0\), then it is still observable if and only if \(\frac{k}{p}+\frac{k}{l}\
eq\omega_{r}^{2}\). As we know from Section 3.3, the observability is preserved under our discretization scheme, and it follows that the same condition also holds for the following pair: \(\left(\left[ \begin{array}{cc}C^{d}&-T^{d}\end{array} \right],\left[ \begin{array} []{cc}A^{d}&0\\ 0&S^{d}\end{array} \right]\right)\). More precisely, if \(b>0\) or \(b=0\) and \(\frac{k}{p}+\frac{k}{l}\
eq\omega_{r}^{2}\), then the plant \(\Sigma_{ext}\) is observable. It is a classic result (see, e.g., [22; 24]) that, in this case, the asymptotic full-order state observer for \(\Sigma_{ext}\) exists and has the following form:  
\[ \begin{array}{rcl}\Sigma_{O}:\ \left[\begin{array}{c}\vec{x}^{d}(i+1)\\ \vec{r}^{d}(i+1)\end{array}\right]&=&\left(\left[\begin{array}{cc}A^{d}&0\\ 0&S^{d}\end{array}\right]-\left[\begin{array}{cc}L_{1}\\ L_{2}\end{array}\right]\left[\begin{array}{cc}C^{d}&-T^{d}\end{array}\right] \right)\left[\begin{array}{c}\vec{x}^{d}(i)\\ \vec{r}^{d}(i)\end{array}\right]\\ &&+\left[\begin{array}{c}B^{d}\\ 0\end{array}\right]\tau^{d}(i)+\left[\begin{array}{c}L_{1}\\ L_{2}\end{array}\right]e^{d}(i)\,,\end{array}  \tag{127}\]  
where \(\left[ \begin{array}{c}L_{1}\\ L_{2}\end{array} \right]\in\mathbb{R}^{6\times 1}\) satisfies  
\[\sigma(\left[ \begin{array}{cc}A^{d}&0\\ 0&S^{d}\end{array} \right]-\left[ \begin{array}{c}L_{1}\\ L_{2}\end{array} \right]\left[ \begin{array}{cc}C^{d}&-T^{d}\end{array} \right] )=\sigma(\left[ \begin{array}{cc}A^{d}-L_{1}C^{d}&L_{1}T^{d}\\ -L_{2}C^{d}&S^{d}+L_{2}T^{d}\end{array} \right])\subset\mathbb{D}_{1}\,, \tag{128}\]  
and this spectrum can be freely assigned. We have also proven that, for \(b>0\)  
where  
\[x_{K}(i) = \left[ \begin{array}{cc}\bar{x}^{d}(i)\\ \bar{p}^{d}(i)\end{array} \right],\] \[A_{K} = \left[ \begin{array}{cc}A^{d}&0\\ 0&S^{d}\end{array} \right]-\left[ \begin{array}{cc}L_{1}\\ L_{2}\end{array} \right]\left[ \begin{array}{cc}\mathbf{C}^{d}&-T^{d}\end{array} \right]\] \[+\bigg{(}\left[ \begin{array}{cc}B^{d}\\ 0\end{array} \right]-\left[ \begin{array}{cc}L_{1}\\ L_{2}\end{array} \right]D^{d}\Big{)}\left[ \begin{array}{cc}-F&\Gamma+F\Gamma \mathbf{I}\end{array} \right]\] \[= \left[ \begin{array}{cc}A^{d}-L_{1}\mathbf{C}^{d}-(B^{d}-L_{1}D^ {d})F&L_{1}T^{d}+(B^{d}-L_{1}D^{d})(\Gamma+F\Pi)\\ -L_{2}\mathbf{C}^{d}+L_{2}D^{d}F&S^{d}+L_{2}T^{d}-L_{2}D^{d}(\Gamma+F\Pi)\end{array}  \right],\] \[B_{K} = \left[ \begin{array}{cc}L_{1}\\ L_{2}\end{array} \right],\] \[C_{K} = \left[ \begin{array}{cc}-F&\Gamma+F\Pi\end{array}