# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 4 Precise Statement and Solution of the Discrete-Time Control Problem
### Static State Feedback for the Extended Discrete-Time Plant  
We start with a simple situation, which turns out to be useful in our later considerations. Combining (79), (82), and (81), we introduce the so-called _extended discrete-time plant_\(\Sigma_{ext}\), defined as follows:  
\[\Sigma_{ext}:\ \left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}x^{d}(i)+B^{d}\tau ^{d}(i)\,,\quad x^{d}(0)=x_{0}\,,\\ r^{d}(i+1)&=&S^{d}r^{d}(i)\,,\quad r^{d}(0)=r_{0}\,,\\ e^{d}(i)&=&C^{d}x^{D}(i)-T^{d}r^{d}(i)+D^{d}\tau^{d}(i)\,.\end{array} \right. \tag{88}\]  
As detailed above, the transformation from \((A,B,C,0)\) to \((A^{d},B^{d},C^{d},D^{d})\) preserves controllability and observability, and the transformation from \((S,T)\) to \((S^{d},T^{d})\) also preserves observability. Therefore, as \((A,B)\) is controllable, so is \((A^{d},B^{d})\); and as \((C,A)\) and  
As \(\sigma(A^{d}-B^{d}F)\subset\mathbb{D}_{1}\), then for any initial states \(x^{d}(0)=x_{0}\) and \(r^{d}(0)=r_{0}\), we get  
\[\lim_{i\to\infty}p^{d}(i)=\lim_{i\to\infty}((A^{d}-B^{d}F)^{i}(x_{0}-\Pi r_{0})=0 \tag{98}\]  
and, hence, \(\lim_{i\to\infty}e^{d}(i)=0\) (i.e., (91) holds). On the other hand, as \(\sigma(S^{d})\subset\mathbb{C}\setminus\mathbb{D}_{1}\), then \(\sigma(S^{d})\cap\sigma(A^{d}-B^{d}F)=\emptyset\) and, for any matrix \(Q\), the Sylvester equation  
\[(A^{d}-B^{d}F)\Pi-\Pi S^{d}+B^{d}Q=0 \tag{99}\]  
has a unique solution \(\Pi\). Now, it follows from (95) that  
\[ \begin{array}{rcl}p^{d}(i+1)&=&(A^{d}-B^{d}F)p^{d}(i)\,,\quad p^{d}(0)=x_{0} -\Pi r_{0}\,,\\ e^{d}(i)&=&(C^{d}-D^{d}F)p^{d}(i)+((C^{d}-D^{d}F)\Pi-T^{d}+D^{d}Q)r^{d}(i)\,. \end{array}  \tag{100}\]  
Moreover, for any \(x^{d}(0)=x_{0}\) and \(r^{d}(0)=r_{0}\), we have \(\lim_{i\to\infty}p^{d}(i)=0\) and, based on (91), we also have \(\lim_{i\to\infty}e^{d}(i)=0\). Thus, for any initial state \(r^{d}(0)=r_{0}\), we obtain the following:  
\[\lim_{i\to\infty}((C^{d}-D^{d}F)\Pi-T^{d}+D^{d}Q)r^{d}(i)=\lim_{i\to\infty}((C ^{d}-D^{d}F)\Pi-T^{d}+D^{d}Q)(S^{d})^{i}r_{0}=0\,. \tag{101}\]  
As \(\sigma(S^{d})\subset\mathbb{C}\setminus\mathbb{D}_{1}\), then (101) implies \((C^{d}-D^{d}F)\Pi-T^{d}+D^{d}Q=0\) for \(\Pi\) from (99). Finally, let us note that the matrix Equation (96) can be written as follows:  
\[\left\{ \begin{array}{l}A^{d}\Pi-\Pi S^{d}+B^{d}(Q-F\Pi)=0\,,\\ C^{d}\Pi-T^{d}+D^{d}(Q-F\Pi)=0\end{array} \right. \tag{102}\]  
which, after the substitution \(\Gamma=Q-F\Pi\), takes the form of the regulator Equation (93). The results we have proved above are a direct extension of the continuous-time general regulation theory (see, e.g., [14]) to the discrete-time scenario. However, an important question now arises regarding _if and when_ the regulator Equation (93) admits a solution in terms of the physical parameters \(I\), \(p\), \(k\), \(b\), and \(\omega_{\tau}\). From Section 3.3, we know that \(D^{d}=C(\mu I-A)^{-1}B\) is invertible, which allows us to express \(\Gamma\) in the following form:  
\[\Gamma=-(D^{d})^{-1}C^{d}\Pi+(D^{d})^{-1}T^{d}\,, \tag{103}\]  
and then to transform the first equation in (93) to the standard Sylvester matrix equation:  
\[(A^{d}-B^{d}(D^{d})^{-1}C^{d})\Pi-\Pi S^{d}+B^{d}(D^{d})^{-1}T^{d}=0\,. \tag{104}\]  
As the existence of a solution \((\Pi,\Gamma)\) to Equation (93) is equivalent to the existence of a solution \(\Pi\) to Equation (104), and it is known that the Sylvester equation has a unique solution if and only if  
\[\sigma(A^{d}-B^{d}(D^{d})^{-1}C^{d})\cap\sigma(S^{d})=\emptyset\,, \tag{105}\]  
then under Condition (105), the regulator equation also has a unique solution \((\Pi,\Gamma)\), where \(\Pi\) is a solution to (104) and \(\Gamma\) is given by (103). We know from Section 3.3, the spectrum \(\sigma(A^{d}-B^{d}(D^{d})^{-1}C^{d})\) coincides with the zeros of the system \(\Sigma^{d}_{G}\). Hence, the regulator equation has a (unique) solution \((\Pi,\Gamma)\) if and only if the spectrum \(\sigma(S^{d})\) is disjoint with the set of zeros of the system \(\Sigma_{\rm G}^{d}\).