# 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 Error Feedback Control System \(\Sigma_{efcs}\)  
We are ready to carry out the design of a discrete-time error feedback system. Let us recall the main equations describing the systems in discrete-time (see (27), (9), (6) and (72)); that is, the discrete-time plant:  
\[\Sigma_{G}^{d}:\left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}(\mu)x^{d}(i)+B^{d }(\mu)\tau^{d}(i)\,,\quad x^{d}(0)=x(0)=x_{0}\,,\\ \alpha^{d}(i)&=&C^{d}(\mu)x^{d}(i)+D^{d}(\mu)\tau^{d}(i)\,,\end{array} \right. \tag{79}\]  
the discrete-time controller:  
\[\Sigma_{K}:\left\{ \begin{array}{rcl}x_{K}(i+1)&=&A_{K}x_{K}(i)+B_{K}e^{d}(i )\,,\quad x_{K}(0)=x_{K0}\,,\\ \tau^{d}(i)&=&C_{K}x_{K}(i)\,,\end{array} \right. \tag{80}\]  
the discrete-time error:  
\[e^{d}(i)=\alpha^{d}(i)-\alpha_{r}^{d}(i)\,,\quad i\in\mathbb{N} \tag{81}\]  
and the discrete-time exosystem:  
\[\Sigma_{S}^{d}:\left\{ \begin{array}{rcl}r^{d}(i+1)&=&S^{d}(\mu)r^{d}(i)\,, \quad r^{d}(0)=r(0)=r_{0}\,,\\ a_{r}^{d}(i)&=&T^{d}(\mu)r^{d}(i)\,.\end{array} \right. \tag{82}\]  
In order to simplify the notation, we drop \(\mu\) in all system matrices. Combining (79)-(81), we obtain the basic model of the _discrete-time error feedback control system_ (see Figure 2), denoted by \(\Sigma_{efcs}\), with \((a_{r}^{d}(i)_{i\in\mathbb{N}}\) as an external reference signal. Its model takes the following form:  
\[\Sigma_{efcs}:\left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}x^{d}(i)+B^{d}C_{ K}x_{K}(i)\,,\quad x^{d}(0)=x_{0}\,,\\ x_{K}(i+1)&=&B_{K}C^{d}x^{d}(i)+(A_{K}+B_{K}D^{d}C_{K})x_{K}(i)-B_{K}a_{r}^{d}( i)\,,\quad x_{K}(0)=x_{K0}\,,\\ e^{d}(i)&=&C^{d}x^{d}(i)+D^{d}C_{K}x_{K}(i)-a_{r}^{d}(i)\,.\end{array} \right. \tag{83}\]  
The error feedback control system \(\Sigma_{efcs}\) with zero input (i.e., \(a_{r}^{d}\equiv 0\)) and without the output equation is referred to as the _discrete-time unforced closed loop system_\(\Sigma_{uf}\), and its description simplifies to the following autonomous form:\[\Sigma_{uf}:\left[ \begin{array}{c}x^{d}(i+1)\\ x_{K}(i+1)\end{array} \right]=\left[ \begin{array}{cc}A^{d}&B^{d}C_{K}\\ B_{K}C^{d}&A_{K}+B_{K}D^{d}C_{K}\end{array} \right]\left[ \begin{array}{c}x^{d }(i)\\ x_{K}(i)\end{array} \right],\quad\left[ \begin{array}{c}x^{d}(0)\\ x_{K}(0)\end{array} \right]=\left[ \begin{array}{c}x_{0}\\ x_{K0}\end{array} \right]. \tag{84}\]  
If we take into account the fact that the reference \(\alpha_{r}^{d}(i)\) is generated by the discrete-time exosystem \(\Sigma_{S}^{d}\) and combine equations (83) and (82), then we obtain the following _discrete-time forced closed loop system_\(\Sigma_{cl}\):  
\[\Sigma_{cl}:\left\{ \begin{array}{rcl}x^{d}(i+1)&=&A^{d}x^{d}(i)+B^{d}C_{K}x _{K}(i)\,,\quad x^{d}(0)=x_{0}\,,\\ x_{K}(i+1)&=&(A_{K}+B_{K}D^{d}C_{K})x_{K}(i)+B_{K}C^{d}x^{d}(i)-B_{K}T^{d}r^{d} (i)\,,\quad x_{K}(0)=x_{K0}\,,\\ r^{d}(i+1)&=&S^{d}r^{d}(i)\,,\quad r^{d}(0)=r_{0}\,,\\ e^{d}(i)&=&C^{d}x^{d}(i)+D^{d}C_{K}x_{K}(i)-T^{d}r^{d}(i)\,.\end{array} \right. \tag{85}\]  
Now, we are ready to precisely define--in mathematical terms--the error feedback controller that we are looking for. We require the controller (80) to guarantee that the following two conditions hold:  
* _Internal stability_. The error feedback control system \(\Sigma_{efcs}\) is said to be _internally stable_ if the unforced closed system \(\Sigma_{uf}\) is asymptotically stable; that is, for all \(x^{d}(0)=x_{0}\), \(x_{K}(0)=x_{K0}\), we have the following: \[\lim_{i\to\infty}\left[ \begin{array}{c}x^{d}(i)\\ x_{K}(i)\end{array} \right]=0\,.\] (86)
* _Asymptotic tracking_ (or _regulation_). The error feedback control system \(\Sigma_{efcs}\) is said to satisfy the _asymptotic tracking_ condition if, for all \(r^{d}(0)=r_{0}\), \(x^{d}(0)=x_{0}\), and \(x_{K}(0)=x_{K0}\), the forced closed loop system \(\Sigma_{cl}\) satisfies the following: \[\lim_{i\to\infty}e^{d}(i)=0\,.\] It can be seen from (84) that 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}