$5$ Pole Placement

Consider a plant modeled by

$\left[\begin{array}{c}{x}_1^{}\\ x_2^{}\\ y\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ -1& -2& 1\\ 1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right]$

$($a$)$ Suppose we control this plant with the state$-$feedback controller

$u=k_1{x}_1+k_2{x}_2+k_{3}r$

Write down a state$-$space realization for the closed loop system.

$($b$)$ Choose $k_1,k_2,k_3$ so that the closed$-$loop system has poles at $-1,-2$ and a DC gain from $r$ to $y$ of $1.$