# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 2 Preliminary Formulation of the Control Problem  
We start with an explanation of the control problem in the continuous-time setting, then we use the mid-point rule for integration to develop discrete-time models for the satellite and the controller. For more details regarding the continuous-time case control problem, we refer the reader to [19; 20]. We consider the simplified model of an earth observation satellite with a solar panel shown in Figure 1.  
The rotation of a satellite with an attached solar panel around the \(y\)-axis can be modeled as an under-actuated mechanical system with two degrees of freedom. Hence, the system is governed by the following differential equations:  
\[\Sigma_{G}:\ \left\{ \begin{array}{rcl}I\dot{\alpha}(t)&=&k(\beta(t)-\alpha(t))+b( \dot{\beta}(t)-\dot{\alpha}(t))+\tau(t)\,,\\ p\ddot{\beta}(t)&=&-k(\beta(t)-\alpha(t))-b(\dot{\beta}(t)-\dot{\alpha}(t))\,, \end{array} \right. \tag{1}\]  
where \(t\geq 0\) denotes the time. Descriptions of all signals and parameters are given in Table 1.  
Figure 1: Visualization of the observation satellite Sentinel-2 with a solar panel (image: ©ESA/Astrium (**left**)) and its simplified sketch (**right**). For the purpose of control, we refer to the satellite with a solar panel as the _plant_, to the driving torque \(\tau(t)\) as the _plant input_ (or _control_), and to the satellite attitude angle \(\alpha(t)\) as the _plant output_. We assume that the attitude \(\alpha(t)\) is the only signal to be measured. According to the physical meaning of the model parameters, we assume that \(I>0\), \(p>0\), \(k>0\), and \(b\geq 0\). We could assume \(b>0\), but we deliberately allow \(b=0\) to see how a lack of damping influences the properties of the plant. A short derivation of the model (1) can be found in Appendix A. In order to state the control problem in continuous time, we assume that we are given the following sinusoidal _reference signal_:  
\[\alpha_{r}(t)=a\sin\omega_{r}t\,,\quad a>0\,,\quad\omega_{r}>0\,, \tag{2}\]  
and we define the _control error_ as follows:  
\[e(t)=\alpha(t)-\alpha_{r}(t)\,. \tag{3}\]  
The _control goal_ in continuous time is to design an error feedback control system to asymptotically track the reference signal \(\alpha_{r}(t)\) via the plant output \(\alpha(t)\); that is,  
\[\lim_{t\to\infty}e(t)=0\,. \tag{4}\]  
As we want our results to be ready for digital implementation, we carry out the design work in the discrete-time domain:  
\[t=ih\,,\quad i\in\mathbb{N}=0,1,2,3,\ldots\,, \tag{5}\]  
where \(h>0\) is the discrete time step. In order to keep everything precise, we will always use the superscript \((\cdot)^{d}\) to distinguish between an approximated sample \(f^{d}(ih)\) and the exact sample \(f(ih)\) of the continuous-time signal \(f(t)\); that is, \(f(t)|_{t=ih}=f(ih)\approx f^{d}(ih)\). As the main step in our considerations, we will formulate and study the control problem in the discrete-time domain. For this reason, we introduce the discrete-time input torque \((\tau^{d}(ih))_{i\in\mathbb{N}}\), the discrete-time output angle \((\alpha^{d}(ih))_{i\in\mathbb{N}}\), the discrete-time reference \((\alpha^{d}_{r}(ih))_{i\in\mathbb{N}}\), and the discrete-time control error  
\[e^{d}(ih)=\alpha^{d}(ih)-\alpha^{d}_{r}(ih)\,,\quad i\in\mathbb{N}\,, \tag{6}\]  
as the discrete-time approximations of the discrete-time samples \((\tau(ih))_{i\in\mathbb{N}}\), \((\alpha_{r}(ih))_{i\in\mathbb{N}}\), \((\alpha(ih))_{i\in\mathbb{N}}\) and \((e(ih))_{i\in\mathbb{N}}\) of the continuous-time counterparts \((\tau(t))_{t\geq 0}\), \((\alpha_{r}(t))_{t\geq 0}\), \((\alpha(t))_{t\geq 0}\), and \((e(t))_{t\geq 0}\), respectively.