$(6$ pts$)$ Consider the following single input single output $($SISO$)$ control system

$x^{}=Ax+Bu$

$y=Cx+Du$

Note that here SISO only means that the control $u$ and output $y$ are scalars, the state $x$ can have higher dimensions $(xin{R}^n,$ in general$).$

Assume the pair $(A,B)$ is controllable $($i$.$e$.$ the contollability matrix has full rank$).$ We then design the usual feedback controller of the form

$u=-Kx+k_{r}r,$

for some reference input $r(t).$

$($a$)(1$ pt$)$ Write the closed$-$loop state space equations with $r$ as the new input.

$($b$)(3$ pts$)$ Suppose the reference input $r$ is some given constant. Then the output will also reach a static value. Calculate the static output $($i$.$e$.y_{}).$ From this, derive a formula for $k_r$ to achieve a unit static gain, i$.$e$.$ when $y_{}=r.$

$($c$)(2$ pts$)$ Suppose the system matrices are given by Problem $4$ in Homework $8,$ along with

$C=([1,0,0]),D=1.$

Let the feedback matrix $K$ is the same as found in that problem. Use the formula derived in part $($a$)$ to find the gain $k_r$ that yields unit static gain. $($You can use Matlab for computation.$)$