Problem $2$

Consider open loop plant

$G(s)=\frac{2000K}{(s+10{)}^2(s+20)}$

and unity feedback.

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

b$.$ Design a PI controller with the same $20\%$ overshoot such that steady state error is $0,$ with $T_{s}1$ sec$.$

c$.$ Hand sketch the root locus for the original system and the system with a PI compensator, and 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.