Problem $5.$ Considering the control system given in RHS$,$

$($a$)$ Sketch, by hand, its root locus if the controller $G_c(s)=K$ and the plant is

$G_p(s)=\frac{1}{(s-1)(s+4)(s+10)}$

When sketching the root locus, if necessary, make use of the asymptotes finding ${}_a$ and ${}_a$ that are the intersecting point and angles with the real axis, respectively with the following formula,

${}_a=\frac{{\displaystyle \underset{?}{\overset{?}{}}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}-{\displaystyle \underset{?}{\overset{?}{}}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{z}\text{e}\text{r}\text{o}\text{s}}{\text{\#}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}\frac{-}{\text{\#}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{z}\text{e}\text{r}\text{o}\text{s}}$ and ${}_a=\frac{(2k+1)}{\text{\#}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}\frac{-}{\text{\#}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{z}\text{e}\text{r}\text{o}\text{s}},$ where $k=0,+-1,+-2,$dots

$($b$)$ Determine the points where the root locus crosses the imaginary axis, if such intersections exist.

$($c$)$ Write the range of gain $K$ making the system stable.

$($d$)$ Write the gain $K$ that makes the system marginally stable. Then, calculate the frequency of the sustained oscillations in Hertz at this gain.

$($e$)$ Find the steady$-$state values of the error $e_{ss}$ and output $($time response$)c_{ss}$ for this P $-$controlled system for $G_c(s)=80,$ when a unit step input is introduced.

$($f$)$ Propose a PI controller to eliminate the steady$-$state error when the input is a unit step for position control. You may use the gain of the P controller given in $($e$).$ You may locate the zero at $-0\mathrm{.}2.$

$($g$)$ Determine the constant of proportionality $K_p,$ and reset $($integral$)$ time $T_i$ of the PI controller.