A feedback control system shown below is given by:

a$.$ Prove that the closed loop transfer function is given by the following:

$\frac{Y(s)}{R(s)}=\frac{K_P}{s^2+(1+K_2)s+K_P}$

b$.$ The control shown above acts similar to a PD control but is better. How is it better than a PD control?

c$.$ Determine gains $K_p$ and $K_2$ give a closed loop system response with an overshoot of $16\mathrm{.}3\%$ and a natural frequency of $20ra\frac{d}{s}.$

d$.$ What is the steady state error in the closed loop response to a step input? $($Hint: The answer is independent of the gains chosen.$)$