Question $4$

$[20$ Marks$]$

An antenna is to track the movement of a satellite by appropriate control of its elevation. The dynamic behaviour of the antenna system can be described by the following model.

$x^{}=Ax+Bu$;$,y=Cx+Du$

$A=\left[\begin{array}{ccc}1\mathrm{.}9& -1& 0\end{array}\right]$

$0\mathrm{.}9$;$B=\left[\begin{array}{c}1\end{array}\right]$

$-0\mathrm{.}9$;$C=\left[\begin{array}{cc}0& 1\end{array}\right]$

$($a$).$ Is the system stable, controllable and$/$or observable?

$[4$ Marks$]$

$($b$).$ Design a state feedback controller such that the closed$-$loop system has eigenvalues of $0\mathrm{.}4$ and $0\mathrm{.}5$

$[5$ Marks$]$

$($c$).$ Design a full$-$state observer with eigenvalues of $0\mathrm{.}1,0\mathrm{.}2$ and show its implementation via a block diagram.

$[6$ Marks$]$

$($d$).$ Write a short note limitations of state variable approach

$[5$ Marks$]$