Problem $4(7$ points$)$

Consider the following state$-$space model:

$x^{}=\left[\begin{array}{ccc}-2& 0& 3\end{array}\right]$

$1x+\left[\begin{array}{c}0\end{array}\right]$

$1u,x(0)=x_0=\left[\begin{array}{c}2\end{array}\right]$

$-1$

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

$($a$)(2$ points$)$ Transform the state$-$space model into diagonal canonical form $($DCF$).$

$($b$)(2$ points$)$ Use the state$-$space model in DCF to find the system transfer function $G(s).$

$($c$)(3$ points$)$ Use the state matrix in diagonal form from $($a$)$ to compute the matrix exponential $e^{At},$

where $A=\left[\begin{array}{cc}-2& 0\\ 1& 3\end{array}\right].$ Calculate the system response $y(t)$ for a unit step $u(t).$