Problem $3(14$pt$)$ We'd like to design a PID controller, and the block digram for the compensated system is shown as follows.

$($a$)$ Add a proportional term, a pure integral term, and an ideal derivative term to block $A,B,$ and $C,$ respectively. Express all the three compensators in terms of gain $k_p,k_i,k_d,$ and $s.(7$ pts$)$

$($b$)$ The overall compensator gain can be reformed to the following equation. Express $k_1,k_2$ and $k_3$ in terms of $k_p,k_i$ and $k_d(7$ pts $)$

$G_c(s)=\frac{k_1(s^2+k_{2}s+k_3)}{s}$