A unity negative feedback control system has the plant transfer function given by:

$G_p=\frac{10}{(s+3)(s+5)}$

$($a$)$ Using the root locus method, design a proportional controller so that the system transient response due to a step input $r(t)$ has exactly $4\mathrm{.}3\%$ overshoot.

$($b$)$ Using the root locus method, design a proper controller so that the system transient response due to a step input $r(t)$ has exactly $4\mathrm{.}3\%$ overshoot, time to peak $T_P=1\mathrm{.}57sec,$ and the steady state error $e_s$ due to the step input is zero.

Given information:

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

Time to peak $T_p=\frac{}{{}_d}=\frac{}{{}_n\sqrt[\phantom{2}]{1-{}^2}}$

Settling time $T_n=\frac{4}{{}_n}$

where ${}_n,{}_d$ and $$ is the system natural frequency, system damped natural frequency, and system damping ratio, respectively.

Do NOT use matlab please. Show all steps in a detailed manner pleaaase :)