$24.$ A servo system is described by following equation dot x$($t$)=$ Ax$($t$)+$ Bu$($t$)+$ B $*$ d$($t$)$ nu$($t$)=$ Cx$($t$)$

Where A $=[[-3,2],[4,-5]]-$C$-10$

d $=$ input to the system The control law is u$($t$)=-$ K $*$ x$($t$)+$ K$\_\{$x$\}($t$)$ where r$($t$)=$constant reference input

Find $($a$)$ Feedback state matrix gain & so that the eigenvalues of $($A $-$ BK$)$ are$-4$ and $-7.$

$($b$)$ Find the value of overline K$\_\{$g$\}$ have zero steady state error to the reference input.

$($c$)$ Prove that the steady state error to disturbance input d is non$-$zero assuming K$\_\{1\}=7$

$($d$)$ If an integrator dot x$\_\{1\}($t$)=$ y$($t$)-$ r$($t$)$ added, find matrix K$,$ so that the control law u$=-$ K$\_\{1\}*$ x$\_\{1\}($t$)-$ K$\_\{1\}*$x$\_\{2\}($t$)-$K $1$ x $1($t$)|$ places the closed loop poles at $-1.-2$ and $-7.$

$($e$)$ Find the transfer function $($Y$($s$))/($D$($s$))|$