Consider a third order system described by:

$\frac{d}{dt}\left[\begin{array}{c}{x}_1(t)\end{array}\right]$

$x_2(t)$

$x_3(t)=\left[\begin{array}{ccccccc}-5& 1& 0& 0& 1& 0& 0\end{array}\right]$

$-5$

$-9\left[\begin{array}{c}{x}_1(t)\end{array}\right]$

$x_2(t)$

$x_3(t)+\left[\begin{array}{c}2\end{array}\right]$

$4$

$1u(t)$

$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\end{array}\right]$

$x_2(t)$

$x_3(t)$

$($a$)$ Obtain the transfer function.

$(4$ marks$)$

$($b$)$ Find the state space realization of the system in controllable canonical form. $(4$ marks$)$

$($c$)$ Given that one eigenvalue is $-1,$ find the other eigenvalues and convert matrix

$A=\left[\begin{array}{ccc}-5& 1& 0\\ -9& 0& 1\\ -5& 0& 0\end{array}\right]$ to the modified canonical form $J.$

$(10$ marks$)$

$($d$)$ Find the transformation matrix $T$ such that $A=TJ{T}^{-1}.$

$(10$ marks$)$

$($e$)$ Given $u(t)=0,$ Find all initial conditions $($of the original controllable canonical form$)$ so that the free response $y(t)$ is exponentially decaying.

$(12$ marks$)$

$($f$)$ Analyze the stability of $\frac{d}{dt}\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]=\left[\begin{array}{ccc}-5& 1& 0\\ -9& 0& 1\\ -5& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right].$

$(3$ marks$)$