# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 3 Plant State Space Models and Their Properties
### Plant Continuous-Time State Space Model and Its Properties  
The plant is described by the system of second-order differential equations in (1). We introduce the following standard state variables:  
\[x(t)=\left[\begin{array}{c}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\\ x_{4}(t)\end{array}\right]=\left[\begin{array}{c}\alpha(t)\\ \beta(t)\\ \dot{\alpha}(t)\\ \dot{\beta}(t)\end{array}\right],\quad t\geq 0\,, \tag{11}\]  
and we obtain the _plant state space model_\(\Sigma_{G}\) in the following form:  
\[\Sigma_{G}:\quad\left[\begin{array}{c}\dot{x}(t)\\ \alpha(t)\end{array}\right]=\left[\begin{array}{cc}A&B\\ C&0\end{array}\right]\left[\begin{array}{c}x(t)\\ \tau(t)\end{array}\right], \tag{12}\]  
where the state space matrix is given as follows:  
\[\left[\begin{array}{c}A\\ \hline C\end{array}\right]=\left[\begin{array}{cccc|c}0&0&1&0&0\\ 0&0&0&1&0\\ \dfrac{k}{I}&\dfrac{k}{I}&-\dfrac{b}{I}&\dfrac{b}{I}&\dfrac{1}{I}\\ \dfrac{k}{p}&-\dfrac{k}{p}&\dfrac{b}{p}&-\dfrac{b}{p}&0\\ \hline 1&0&0&0&0\end{array}\right]. \tag{13}\]  
The characteristic polynomial of \(A\) is given as follows:  
\[\det(\lambda I-A)=\lambda^{2}(\lambda^{2}+(\dfrac{b}{p}+\dfrac{b}{I})\lambda+ \dfrac{k}{p}+\dfrac{k}{I})\,, \tag{14}\]  
Figure 3: The digital control system with the continuous-time plant \(\Sigma_{G}\) and \(ZOH\).  
and the spectrum \(\sigma(A)\)--namely, the set \((\lambda_{l}(A))_{l=1}^{4}\) of eigenvalues of \(A\)--is given as follows:  
\[\sigma(A)=\left[\begin{array}{c}\lambda_{1}(A)\\ \lambda_{2}(A)\\ \lambda_{3}(A)\\ \lambda_{4}(A)\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2}\Big{(} \frac{b}{p}+\frac{b}{I}\Big{)}+\sqrt{\Big{(}\frac{1}{2}\Big{)}^{2}\Big{(} \frac{b}{p}+\frac{b}{I}\Big{)}^{2}-\Big{(}\frac{k}{p}+\frac{k}{I}\Big{)}}\\ -\frac{1}{2}\Big{(}\frac{b}{p}+\frac{b}{I}\Big{)}-\sqrt{\Big{(}\frac{1}{2} \Big{)}^{2}\Big{(}\frac{b}{p}+\frac{b}{I}\Big{)}^{2}-\Big{(}\frac{k}{p}+\frac{ k}{I}\Big{)}}\\ 0\\ 0\end{array}\right]. \tag{15}\]  
For \(b>0\), we have \(\sigma(A)\subset\mathbb{C}_{-}\cup j\mathbb{R}\), and for \(b=0\), we only have \(\sigma(A)\subset j\mathbb{R}\), where \(\mathbb{C}_{-}\) denotes the open left half-plane and \(j\mathbb{R}\) is the imaginary axis. Moreover, for the plant \(\Sigma_{G}\), the controllability of \((A,B)\) can be checked using the controllability matrix \(W\), and the observability of \((C,A)\) can be checked using the observability matrix \(V\). Namely, we have  
\[\det W=-\frac{k^{2}}{I^{4}p^{2}}\
eq 0\,,\quad\det V=-\frac{k^{2}}{I^{2}}\
eq 0\,, \tag{16}\]  
for all values of parameters \(I\), \(p\), \(k\), and \(b\) with physical meaning. The controllability and observability imply that the zeros of \(\Sigma_{G}\) are the roots \((s_{l}(\Sigma_{G}))_{l=1}^{l=2}\) of the following polynomial:  
\[\text{Cadj}\,(sI-A)B=\frac{1}{I}(s^{2}+\frac{b}{p}s+\frac{k}{p})\,, \tag{17}\]  
which are given as follows:  
\[s_{1}(\Sigma_{G})=-\frac{b}{2p}-\sqrt{\Big{(}\frac{b}{2p}\Big{)}^{2}-\frac{k} {p}}\,,\quad s_{2}(\Sigma_{G})=-\frac{b}{2p}+\sqrt{\Big{(}\frac{b}{2p}\Big{)} ^{2}-\frac{k}{p}}\,. \tag{18}\]  
For \(b>0\), we have \((s_{l}(\Sigma_{G}))_{l=1}^{l=2}\subset\mathbb{C}_{-}\cup j\mathbb{R}\) and for \(b=0\) we have \((s_{l}(\Sigma_{G}))_{l=1}^{l=2}\subset j\mathbb{R}\).