$1.$ Part I. Mention the differences between a PI and PID controller by writing the expressions of the controller$$s actuating signal. Also, draw the closed loop responses of a second$-$order under$-$damped and over$-$damped systems by showing the effect of varying the damping coefficient on the output response.

Part II$.$ The characteristic equation of a system controlled using a P controller is given as follows:

$3+92+26+12(2+$c$)=0$

Determine the condition on Kc for which the system will exhibit a stable response using the Routh$-$Hurwitz criteria.