Consider the unity feedback system with $\backslash ($ G$($s$)=\backslash$frac$\{$K$\}\{($s$+2)($s$+4)\}\backslash ).$ The system is operated with $\backslash (4\mathrm{.}32\backslash \%\backslash )$ overshoot. In order to improve the steady$-$state error, $\backslash ($ K$\_\{$P$\}\backslash )$ is to be increased by at least a factor of $5.$ A lag compensator of the form $\backslash ($ G$($s$)=\backslash$frac$\{$s$+0\mathrm{.}5\}\{$s$+0\mathrm{.}1\}\backslash )$ is to be used.

$($a$)$ Find the gain $\backslash ($ K $\backslash )$ required for both the compensated and the uncompensated systems. Use MATLAB to plot root locus.

Hint: see the MALTAB code in the CANVAS to see how to plot root locus. Unlike the step response, the 'rlocus' command in MATLAB requires open$-$loop transfer function GH $($or in this case $\backslash (\backslash$mathrm$\{$H$\}=1\backslash )$ because this is a unity feedback system$)$

$($b$)$ Find the value of $\backslash ($ K$\_\{$P$\}\backslash )$ for both the compensated and the uncompensated systems

$($c$)$ Estimate the percent$-$overshoot and settling time for both the compensated and the uncompensated systems.

$($d$)$ Discuss the validity of the second$-$order approximation used for your results in problem $2-($c$)$

$($e$)$ Use MATLAB or any other computer program to simulate the step response for the uncompensated and compensated systems. What do you notice about the compensated system's response?

Hint: Use a similar code from problem $1.$ Construct two feedback systems: one with compensator and one without. Then plot step responses.

$($f$)$ Design a lead compensator that will correct the objection you notice in problem $2-($e$)$