Consider a unity$-$feedback system shown in Figure $1$ with the open$-$loop transfer function.

$G(s)=\frac{1}{s(s+0\mathrm{.}5)}$

We want to design a lead compensator in the form of

$C(s)=K\frac{s+z}{s+p}$

to place closed$-$loop poles to desired pole location represented by $\%OS16\mathrm{.}3\%$ and ${}_0=5.$ Answer the following questions.

$($a$)$ Compute the desired poles from the design specifications.

$($b$)$ Compute the angle deficiency $$ in degree.

$($c$)$ Find the compensator pole and zero using the bisection method.

$($d$)$ Compute the gain $K.$

$($e$)$ Sketch the root locus of the compensated system. Make sure you include the calculations of

Asymptotes

Break$-$away point

Find b$,$ c$,$d$,$e