# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 3 Plant State Space Models and Their Properties
### Plant Discrete-Time State Space Model  
For the discrete time  
\[t=ih\,,\quad i\in\mathbb{N}\,, \tag{19}\]  
where \(h>0\) is the discrete-time step, we integrate over the intervals \((ih_{*},(i+1)h)\) on both sides of the state space model (12) and obtain the following:  
\[\begin{array}{rcl}\int_{ih}^{(i+1)h}\dot{x}(t)\mathrm{d}t&=&A\int_{ih}^{(i+1 )h}x(t)\mathrm{d}t+B\int_{ih}^{(i+1)h}\tau(t)\mathrm{d}t\,,\\ \int_{ih}^{(i+1)h}\alpha(t)\mathrm{d}t&=&C\int_{ih}^{(i+1)h}x(t)\mathrm{d}t\,, \quad i\in\mathbb{N}\,.\end{array} \tag{20}\]  
Approximating the state integral according to the mid-point rule and introducing simplified notation for the integrals of input and output,  
\[\int_{ih}^{(i+1)}x(t)\mathrm{d}t\approx\frac{h}{2}(x(ih)+x((i+1)h))\,,\quad \int_{ih}^{(i+1)}\tau(t)\mathrm{d}t=\tau^{int}\,,\quad\int_{ih}^{(i+1)}\alpha( t)\mathrm{d}t=\alpha^{int}\,, \tag{21}\]  
we can rewrite (20) as follows:  
\[\begin{array}{rcl}x^{d}((i+1)h)-x^{d}(ih)&=&\frac{h}{2}A(x^{d}(ih)+x^{d}((i+ 1)h))+B\tau^{int}\,,\\ \alpha^{int}&=&\frac{h}{2}C(x^{d}(ih)+x^{d}((i+1)h))\,.\end{array} \tag{22}\]We use the superscript \((\cdot)^{d}\) to emphasize that \(x^{d}(ih)\) are not exact samples \(x(ih)\) of \(x(i)\) but instead their approximations. As for all values of parameters with physical meaning (\(I\), \(p\), \(k\), and \(b\)), the matrix \(A\) satisfies the following condition:  
\[\sigma(A)\subset\mathbb{C}_{-}\cup j\mathbb{R}\,, \tag{23}\]  
it follows that, for all \(h>0\), the parameter \(\mu\) satisfies \(\mu>0\) and  
\[\mu=\frac{2}{h}\
ot\in\sigma(A)\,. \tag{24}\]  
Consequently, there exists an inverse matrix \((\mu I-A)^{-1}\), where \(I\) is the identity matrix. The meaning of the parameter \(\mu\) is the double sampling frequency. After simple manipulations, (22) takes the equivalent form:  
\[x^{d}((i+1)h) = (\mu I-A)^{-1}(\mu I+A)x^{d}(ih)+\mu(\mu I-A)^{-1}B\tau^{int} \tag{25}\] \[= (-I+2\mu(\mu I-A)^{-1})x^{d}(ih)+\mu(\mu I-A)^{-1}B\tau^{int}\,,\] \[\alpha^{int} = 2C(\mu I-A)^{-1}x^{d}(ih)+C(\mu I-A)^{-1}B\tau^{int}\,.\]  
In order to ensure symmetry between the input and output matrices of the discrete-time state space model, we approximate the last two integrals in (21) as follows:  
\[\tau^{int}\approx\sqrt{h}\tau^{d}(ih)=\sqrt{\frac{2}{\mu}}\tau^{d}(ih)\,,\quad \alpha^{int}\approx\sqrt{h}\alpha^{d}(ih)=\sqrt{\frac{2}{\mu}}\alpha^{d}(ih)\,. \tag{26}\]  
If we now substitute (26) into (25), then we finally obtain the following _discrete-time state space model_\(\Sigma_{G}^{d}\) of the plant:  
\[\Sigma_{G}^{d}:\left\{\begin{array}{rcl}x^{d}(i+1)&=&A^{d}(\mu)x^{d}(i)+B^{d }(\mu)\tau^{d}(i)\,,&x^{d}(0)=x(0)=x_{0}\,,\\ \alpha^{d}(i)&=&C^{d}(\mu)x^{d}(i)+D^{d}(\mu)\tau^{d}(i)\,,\end{array}\right. \tag{27}\]  
where  
\[\begin{array}{rclrclrcl}A^{d}(\mu)&:=&(\mu I-A)^{-1}(\mu I+A)\,,&B^{d}(\mu)& :=&\sqrt{2\mu}(\mu I-A)^{-1}B\,,\\ C_{d}(\mu)&:=&\sqrt{2\mu}C(\mu I-A)^{-1}\,,&D^{d}(\mu)&:=&C(\mu I-A)^{-1}B\,, \end{array} \tag{28}\]  
and we have dropped \(h\) in order to simplify the notation for \(x^{d}(ih)\), \(\tau^{d}(ih)\), and \(\alpha^{d}(ih)\).