Problem $4$

Consider open loop plant $G(s)$ with unity feedback:

$G(s)=\frac{K(15)(13)}{(s+15)(s^2+6s+13)}$

a$.$ Find the gain $K$ such that $\frac{G}{1+G}$ has overshoot less than $20\%($Matlab ok$).$

b$.$ Design a PID controller $G_c(s)$ such that overshoot is less than $20\%,$ with with zero steady state error for a step. Specify open and closed$-$loop poles, zeros and gains.

c$.$ Hand sketch the root locus for both $G(s)$ and $G_c(s)G(s),$ andverifywithMatlab$).$

d$.$ Show before and after compensation step response on same plot using Matlab.