For the following unity feedback system $G_2(s)$ is a PID controller and $G(s)$ is $G(s)=\frac{1}{s(s+1)(s+4)}.$

a$)$ Draw root locus curve for $K_p$ changes from zero to infinity when $K_1=0$ and $K_d=0.$

b$)$ Obtain the $K_p,K_s$ and $K_d$ values of PID controller by using the $2^{nd}$ Ziegler Nichols Method.

c$)$ Draw the modified root locus curve and step response. Do you have a good step response $?$

d$)$ How could you improve the performance further? Show on root locus and step response curves.

$R(s)$

$Y(s)$