An industrial process has the open loop transfer function

$GH(s)=\frac{k}{s(s+2)(s+8)}$

$($a$)$ Using the Nyquist stability criterion, plot the locus of the closed$-$loop system and clearly indicate the gain and phase margins.

$($b$)$ Determine the maximum allowable gain, k $,$ and frequency, $(rad{s}^{-1}),$ for the system to remain stable.

$($c$)$ Sketch the locus of the closed loop system if the open loop transfer function also includes a pure time delay of $10$ ms $.$

$($d$)$ Show that the phase of the time delayed closed loop system is $180$ when

$\frac{}{2}-0\mathrm{.}01=ta{n}^{-1}(\frac{10}{16-{}^2})$

Given

$ta{n}^{-1}x+-ta{n}^{-1}y=ta{n}^{-1}(\frac{x+-y}{1xy})$