# 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  
If we interconnect the _extended discrete-time plant_\(\Sigma_{ext}\), defined in the following form:  
\[\Sigma_{ext}:\ \left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}x^{d}(i)+B^{d} \tau^{d}(i)\,,\quad x^{d}(0)=x_{0}\,,\\ r^{d}(i+1)&=&S^{d}\tau^{d}(i)\,,\quad r^{d}(0)=\tau_{0}\,,\\ e^{d}(i)&=&C^{d}x^{d}(i)-T^{d}r^{d}(i)+D^{d}\tau^{d}(i)\,,\end{array} \right. \tag{109}\]  
and the discrete-time controller:  
\[\Sigma_{K}:\left\{ \begin{array}{rcl}x_{K}(i+1)&=&A_{K}x_{K}(i)+B_{K}e^{d}(i )\,,\quad x_{K}(0)=x_{K0}\,,\\ \tau^{d}(i)&=&C_{K}x_{K}(i)\,,\end{array} \right. \tag{110}\]  
then we obtain the _discrete-time forced closed loop system_\(\Sigma_{cl}\):  
\[\Sigma_{cl}:\left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}x^{d}(i)+B^{d}C_{K} x_{K}(i)\,,\quad x^{d}(0)=x_{0}\,,\\ x_{K}(i+1)&=&(A_{K}+B_{K}D^{d}C_{K})x_{K}(i)+B_{K}C^{d}x^{d}(i)-B_{K}T^{d}r^{d }(i)\,,\quad x_{K}(0)=x_{K0}\,,\\ r^{d}(i+1)&=&S^{d}r^{d}(i)\,,\quad r^{d}(0)=r_{0}\,,\\ e^{d}(i)&=&C^{d}x^{d}(i)+D^{d}C_{K}x_{K}(i)-T^{d}r^{d}(i)\,.\end{array} \right. \tag{111}\]  
As we already know, IS holds if and only if  
\[\sigma(\left[ \begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \right])\subset\mathbb{D}_{1}\,. \tag{112}\]  
As \((A^{d},B^{d})\) is controllable and \((C^{d},A^{d})\) is observable, there always exists a controller \((A_{K},B_{K},C_{K},0)\) satisfying (112). In the continuous-time case, for a controller satisfying IS, the general regulation theory (see, e.g., [14]) provides necessary and sufficient conditions to be satisfied for the asymptotic tracking condition AT to hold. We extend these results to the discrete-time case and, in order to keep our exposition complete, we provide the corresponding results with proofs. **Theorem 2**. : _If, for a given controller \((A_{K},B_{K},C_{K},0)\), the condition IS holds, then the closed loop system \(\Sigma_{cl}\) satisfies the asymptotic tracking condition AT if and only if there exist \(\Pi\in\mathbb{R}^{4\times 2}\), \(\Gamma\in\mathbb{R}^{1\times 2}\), and \(\Sigma\in\mathbb{R}^{n_{K}\times 2}\) such that_  
\[\text{RE}:\ \left\{ \begin{array}{l}A^{d}\Pi-\Pi\SS^{d}+B^{d}\Gamma=0\,,\\ C^{d}\Pi-T^{d}+D^{d}\Gamma=0\,,\end{array} \right. \tag{113}\]  
_and_  
\[\text{IMP}:\ \left\{ \begin{array}{l}\Gamma=C_{K}\Sigma\,,\\ \Sigma\SS^{d}=A_{K}\Sigma\,.\end{array} \right. \tag{114}\]  
Proof. : Let (112) hold. For the closed-loop system (111), we introduce new state variables:  
\[\left[ \begin{array}{c}p^{d}(i)\\ q^{d}(i)\\ r^{d}(i)\end{array} \right]=\left[ \begin{array}{ccc}I&0&-\Pi\\ 0&I&-\Sigma\\ 0&0&I\end{array} \right]\left[ \begin{array}{c}x^{d}(i)\\ x_{K}(i)\\ r^{d}(i)\end{array} \right], \tag{115}\]  
where \(\Pi\in\mathbb{R}^{4\times 2}\) and \(\Sigma\in\mathbb{R}^{n_{K}\times 2}\) can be arbitrary, and we obtain the following:  
\[\Sigma_{cl}:\left\{ \begin{array}{lcl}p^{d}(i+1)&=&A^{d}p^{d}(i)+B^{d}C_{K}q ^{d}(i)+(A^{d}\Pi-\Pi\SS^{d}+B^{d}C_{K}\Sigma)r^{d}(i)\,,\\ &&p^{d}(0)=x_{0}-\Pi r_{0}\,,\\ q^{d}(i+1)&=&B_{K}\mathbb{C}^{d}p^{d}(i)+(A_{K}+B_{K}D^{d}C_{K})q^{d}(i)\\ &&+[(A_{K}+B_{K}D^{d}C_{K})\Sigma-\Sigma\SS^{d}+B_{K}(C^{d}\Pi-T^{d})]r^{d}(i) \,,\\ &&q^{d}(0)=x_{K0}-\Sigma r_{0}\,,\\ r^{d}(i+1)&=&S^{d}r^{d}(i)\,,\quad r^{d}(0)=r_{0}\,,\\ e^{d}(i)&=&C^{d}p^{d}(i)+D^{d}C_{K}q^{d}(i)+(C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma)r^ {d}(i)\,.\end{array} \right. \tag{116}\]  
If there exist matrices \((\Pi,\Sigma)\) such that  
\[\left[ \begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \right]\left[ \begin{array}{c}\Pi \\ \Sigma\end{array} \right]-\left[ \begin{array}{c}\Pi\\ \Sigma\end{array} \right]\SS^{d}=\left[ \begin{array}{c}0\\ B_{K}T^{d}\end{array} \right], \tag{117}\]  
and  
\[C^{d}\Pi-T^{d}+D^{d}C_{K}\Sigma=0\,, \tag{118}\]  
then (116) simplifies to the following form:  
\[\Sigma_{cl}:\left[ \begin{array}{c}p^{d}(i+1)\\ q^{d}(i+1)\\ r^{d}(i+1)\end{array} \right]=\left[