Problem $4($a$)$ Use Routh's Stability Analysis to determine the range of the gain K required for the stability of a unity$-$feedback control system whose transfer function is $G(s)=\frac{K}{s^3+4s^2+8s+K}.(5$ Points$)$

$($b$)$ Apply the second method in Ziegler$-$Nichols rules to tune the PID $($i$.$e$.,$ identify PID parameters$)(5$ Points$)$

$($c$)$ Use "Automatic Tuning" function in Simulink to tune the PID and report the parameters as well as data from the scope $(5$ Points$)$