The linearised equations of motion for a satellite are [ begin{aligned} dot{mathbf{x}} & =mathbf{A x}+mathbf{B u}

Question:

The linearised equations of motion for a satellite are

\[
\begin{aligned}
\dot{\mathbf{x}} & =\mathbf{A x}+\mathbf{B u} \\
\mathbf{y} & =\mathbf{C x}+\mathbf{D u}
\end{aligned}
\]

where

\[
\begin{aligned}
\mathbf{A} & =\left[\begin{array}{cccc}
0 & 1 & 0 & 0 \\
3 \omega^{2} & 0 & 0 & 2 \omega \\
0 & 0 & 0 & 1 \\
0 & -2 \omega & 0 & 0
\end{array}ight], \mathbf{B}=\left[\begin{array}{ll}
0 & 0 \\
1 & 0 \\
0 & 0 \\
0 & 1
\end{array}ight], \mathbf{C}=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0
\end{array}ight] \\
\mathbf{D} & =[0] \\
\mathbf{u} & =\left[\begin{array}{l}
u_{1} \\
u_{2}
\end{array}ight], \mathbf{y}=\left[\begin{array}{l}
y_{1} \\
y_{2}
\end{array}ight] .
\end{aligned}
\]The inputs \(u_{1}\) and \(u_{2}\) are the radial and tangential thrusts, the state variables \(x_{1}\) and \(x_{2}\) are the radial and angular deviations from the reference (circular) orbit and the outputs, \(y_{1}\) and \(y_{2}\), are the radial and angular measurements, respectively.