# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 4 Precise Statement and Solution of the Discrete-Time Control Problem
### The Discrete-Time State Space Model of the Exosystem and Its Properties  
We follow the idea of Section 3.2 for the discrete time:  
\[t=ih\,,\quad i\in\mathbb{N}\,, \tag{66}\]  
where \(h>0\) is the discrete-time step. We integrate over the intervals \((ih,(i+1)h)\) on both sides of the state space model (61) and obtain the following:  
\[\begin{array}{rcl}\int_{ih}^{(i+1)h}\dot{r}(t)\mathrm{d}t&=&S\int_{ih}^{(i+1 )h}r(t)\mathrm{d}t\,,\\ \int_{ih}^{(i+1)h}\alpha_{r}(t)\mathrm{d}t&=&T\int_{ih}^{(i+1)h}r(t)\mathrm{d} t\,,\quad i\in\mathbb{N}\,.\end{array} \tag{67}\]  
Approximating the state integral, according to the mid-point rule, and the integral of the output as before; that is,  
\[\int_{ih}^{(i+1)}r(t)\mathrm{d}t\approx\frac{h}{2}(r(ih)+r((i+1)h))\,,\quad \int_{ih}^{(i+1)}\alpha_{r}(t)\mathrm{d}t\approx\sqrt{h}\alpha_{r}^{d}(ih)\,, \tag{68}\]  
we can rewrite (67) as follows:  
\[\begin{array}{rcl}r^{d}((i+1)h)-r^{d}(ih)&=&\frac{h}{2}S(r^{d}(ih)+r^{d}((i +1)h))\,,\\ \sqrt{h}\alpha_{r}^{d}(ih)&=&\frac{h}{2}T(r^{d}(ih)+r^{d}((i+1)h))\,.\end{array} \tag{69}\]  
As the matrix \(S\) satisfies  
\[\sigma(S)\subset j\mathbb{R}\,, \tag{70}\]  
then, for all \(h>0\), the parameter \(\mu\) satisfies \(\mu>0\) and  
\[\mu=\frac{2}{h}\
ot\in\sigma(S)\,. \tag{71}\]  
This implies that there exists an inverse matrix \((\mu I-S)^{-1}\). Through simple manipulations, (69) can be transformed to the following final form of the _discrete-time state space model_\(\Sigma_{S}^{d}\) of the exosystem:  
\[\Sigma_{S}^{d}\colon\left\{\begin{array}{rcl}r^{d}(i+1)&=&S^{d}(\mu)r^{d}(i )\,,\quad r^{d}(0)=r(0)=r_{0}\,,\\ \alpha_{r}^{d}(i)&=&T^{d}(\mu)r^{d}(i)\,,\end{array}\right. \tag{72}\]  
where  
\[\begin{array}{rcl}S^{d}(\mu)&:=&(\mu I-S)^{-1}(\mu I+S)\,,\\ T_{d}(\mu)&:=&\sqrt{2\mu}T(\mu I-S)^{-1}\,,\end{array} \tag{73}\]  
and we have dropped \(h\) in order to simplify the notation for \(r^{d}(ih)\) and \(\alpha_{r}^{d}(ih)\). By repeating the same considerations as in Section 3.3, we see that, for \(\lambda\in\mathbb{C}\setminus\{-1\}\),  
\[\frac{1}{\lambda+1}\det(\lambda I-S^{d})\det(\mu I-S)=\det(\mu\frac{\lambda-1 }{\lambda+1}I-S)\,, \tag{74}\]and as \(\det(\mu I-S)\
eq 0\), we have  
\[\det(\lambda I-S^{d})=0\quad\text{if and only if}\quad\det(\mu\frac{\lambda-1}{ \lambda+1}I-S)=0\,. \tag{75}\]  
If \(-1\
ot\in\sigma(S^{d})\), then the eigenvalues \((\lambda_{l}(S^{d}))_{l=1}^{l=2}\) and \((\mu_{l}(S))_{l=1}^{l=2}\) are related to each other by the following equality:  
\[\mu_{l}(S)=\mu\frac{\lambda_{l}(S^{d})-1}{\lambda_{l}(S^{d})+1}\,,\quad l=1,2\,; \tag{76}\]  
or, equivalently, if \(\mu_{l}(S)\
eq\mu\) (which is always satisfied based on (71)), then  
\[\lambda_{l}(S^{d})=\frac{\mu+\mu_{l}(S)}{\mu-\mu_{l}(S)}\,,\quad l=1,2\,. \tag{77}\]  
Therefore,  
\[|\lambda_{l}(S^{d})|=1\quad\text{if and only if}\quad\operatorname{Re}\mu_{l}(S)=0\,. \tag{78}\]  
The same considerations as in Section 3.3 demonstrate that the observability of the pair \((T,S)\) is equivalent to the observability of \((T^{d},S^{d})\).