Q$5.$ Consider the system given by

$x^{}=\left[\begin{array}{ccccccc}2& 1& 1& 3& 6& -1& -4\end{array}\right]$

$-5$

$-5$

$5x+\left[\begin{array}{c}1\end{array}\right]$

$0$

$0u$

$y=\left[\begin{array}{ccc}1& 1& 2\end{array}\right]x$

$($a$)$ Determine which modes are controllable and$/$or observable using the controllable Kalman

decomposition. Use any computational tool $($e$.$g$.,$ MATLAB$)$ to aid in the calculations.

$($b$)$ Find the transfer function of the system and observe which modes appear as system poles.