Problem $1$

A closed$-$loop unity feedback system as in Figure Q$1.$

a$)$ Find the value of the controller gain, $K,$ for the system to operate in the critically damped region.

b$)$ Identify the location of the dominant pole for the system to have the following specifications;

Overshoot, $\%$OS: $20\%$

Settling time, $T_s$:$1\mathrm{.}575s$

c$)$ Validate whether the dominant pole obtained in part $($b$)$ is a closed$-$loop pole of the system. If it is$,$ identify the magnitude of $K.$ If not, proceed to design a controller to achieve the desired response.