Consider the following state$-$space model:

$$x $=$

$10$

$23$

x $+$

$0$

$1$

u

y $=24$ x

$($a$)(20$ points$)$ Check that the system is diagnolizable and transform the system into

diagonal canonical form $($DCF$).$

$($b$)(10$ points$)$ Based on your observation of the transformed system in DCF$,$ could

you intuitively tell whether the system is controllable or not? Why? $($Note: You need to

put an emphasis on the transformed input matrix$)$

$($c$)(30$ points$)$ Use the state matrix in diagonal form from $($a$)$ to compute the matrix

exponential e At $,$ where A $=$

$10$

$23$

$.$ Use the DCF to calculate the system response y$($t$)$

given a unit step u$($t$)$ and the initial condition x$(0)=$

$2$

$1$