Consider the simple first$-$order model described by $x^{}=ax+bu,y=cx$ with $a=-2,b=2,c=1.$ A closed$-$loop system has been designed through feedback and integral control. The state feedback has the form:

$u=-K\left[\begin{array}{c}x\\ x_i\end{array}\right]+k_{r}r$

Where $K=[k_1{k}_2]$ and $x_i^{}=y-r.$ The closed$-$loop system has a pole in $s=-6$ and another pole in $s=-8.($Hint $x_i^{}=x-r$ and write $\left[\begin{array}{c}{x}^{}\\ x_i^{}\end{array}\right]=A[]+$dots $)$

a$)$ Represent the block diagram of the closed$-$loop system $($starting from the block diagram of the state$-$space representation$)$;

b$)$ Compute $K$;

c$)$ Represent the time response of the closed$-$loop system to a step input $($in Simulink$).$