$1.$ Lead compensator: Consider a second$-$order plant model given by $\backslash ($ G$($s$)=\backslash$frac$\{40\}\{$s$($s$+2)\}\backslash ).$

Figure $1$: Cascade compensation.

For a closed$-$loop system as in Figure $1,$ we desire overshoot to be approximately $\backslash (26\backslash \%\backslash )$ for a step input and the settling time $($with a $\backslash (2\backslash \%\backslash )$ criterion$)$ of the system to be about $800$ ms $.$ Design a phase$-$lead compensator $\backslash ($ G$\_\{$c$\}($s$)=\backslash$frac$\{$K$($s$+$z$)\}\{($s$+$p$)\}\backslash )$ using the root locus method.

$($a$)$ Determine the desired dominant conjugate pair of poles.

$($b$)$ Use Matlab to view the root locus of the uncompensated system; does the root locus come close to the desired pole locations?

$($c$)$ Select a zero location $\backslash ($ s$=-$z $\backslash )$ informed by the desired pole location. Use the angles of loop gain poles and zero to the desired pole location to determine compensator pole location $\backslash ($ s$=-$p $\backslash ).($A sketch of the loop gain poles and zeros will help you with the trigonometry of the angles; note that in Matlab atand and tand operate in units of degrees.$)$

$($d$)$ Determine the gain $\backslash ($ K $\backslash )($again$,$ a sketch of the loop gain poles and zeros will help you apply Pythagorus's formula$).$

$($e$)$ Plot the root locus for your design, e$.$g$.,$ rlocus $($GcG$)$

$($f$)$ Report the closed$-$loop pole locations.

$($g$)$ Use Matlab to determine the step response characteristics, e$.$g$.,$ stepinfo$($feedback$($GcG$,1))$

$($h$)$ How do your P$.$O$.$ and $\backslash ($ T$\_\{$s$\}\backslash )$ compare to the desired values? If they differ, comment briefly why.

$($i$)$ For your design, determine the error constant, $\backslash ($ K$\_\{$v$\}\backslash ),$ for a ramp input.