Problem #$5(20$ marks$-($a$)-10$marks, $($b$)-10$marks$)$

A unity feedback system has a plant transfer function given by:

$G_p(s)=\frac{1}{s(s+1)}$

$($a$)$ Using root$-$locus method $($based on characteristic equation$),$ design a cascade proportional derivative $($PD$)$ compensator $G_c(s)=K(s+a)$ so that the following design specifications are met:

Percent overshoot P$.$O$.=4\mathrm{.}32\%$

Natural frequency ${}_n=7\mathrm{.}07ra\frac{d}{sec}$

$($b$)$ Using root$-$locus method $($based on characteristic equation$),$ design a cascade "lead compensator" $G_{lc}(s)=\frac{K(s+z_0)}{s+p_0}$ so that the above specifications are met.

Given information:

Percent overshoot P$.$O$.=100e^{-\frac{}{\sqrt[\phantom{2}]{1-{}^2}}}$

where $$ is the system damping ratio.

Please answer problem $5$ as if it where an exam.

Thank you