$($a$)$ A process control system with a PID controller is as shown in Figure $4,$ where $R(s),Y(s)$ and $D(s)$ are the Laplace transforms of the reference input, the output and the disturbance input, respectively, and $G(s)$ is the process to be controlled. Suppose that $G(s)=\frac{1}{(s+0\mathrm{.}1)(s+5)}.$ Design a PID controller $G_c(s)=\frac{s^2{K}_D+s{K}_P+K_{\text{I}}}{s},$ if it exists, to meet the following performance specifications:

$($i$)$ The steady$-$state error due to a step reference input $R(s)$ is zero.

$($ii$)$ The steady$-$state error due to a step disturbance input $D(s)$ is no more than $5\%.$

$($iii$)$ The closed$-$loop system has a dominant second order response with the dominant poles located at $s_{1,2}=-4\mathrm{.}0+-j3\mathrm{.}0.$

$(15$ Marks$)$