I have been trying for days to find how to convert the linearized Equation of a satellite given by the equation attached in the image to STATE SPACE equation.
Please I have been struggling for days. I am a newbie in control system.
Please do help out.
$$ \ddot\varepsilon+D\,\dot\varepsilon+K\,\varepsilon=B\,u \tag{1} $$
Where $$ \varepsilon= \begin{pmatrix} r \\ \theta-\omega\,t \end{pmatrix} = \begin{pmatrix} \text{radial-deviation-from-R} \\ \text{angular-position} \end{pmatrix} $$ $$ u= \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} \text{radial-thrust} \\ \text{tangential-thrust} \end{pmatrix} $$ $$ D= \begin{pmatrix} 2\omega\over\ R & 0 \\ 0 & 0 \end{pmatrix} $$ $$ K= \begin{pmatrix} -2\omega^2 & 0 \\ 0 & 0 \end{pmatrix} $$ $$ B= \begin{pmatrix} 1\over m & 0 \\ 0 & 1\over m \end{pmatrix} $$
Want to convert above equation $(1)$ to state space equation of the form $$ \dot x = A\,x+B\,u $$
Also if $y=\dot\theta$ is observed at the output what will be the transfer function?
