Problem $3$

Consider open loop plant

$G(s)=\frac{144K}{s(s+4)(s+12{)}^2}$

and unity feedback.

a$.$ Find $K$ such that overshoot is $20\%.$

b$.$ Design a PD controller $($i$.$e$.$ find zero location$)$ such that $T_p$~~$0\mathrm{.}8sec,$ with the same $20\%$ overshoot.

c$.$ Hand sketch the root locus for the original system and the system with a PD compensator verify with Matlab.

d$.$ Use Matlab to compare the step response for the closed$-$loop compensated and uncompensated systems, transient and steady state response.

e$.$ Find the steady state error for a step for both systems.