$($a$)(10$ pts$)$ Determine the bounds on gain K to ensure stable closed$-$loop performance.

$($b$)(8$ pts$)$ Find the gain K that will give $7\%$ OS $($maximum percent overshoot$).$ What is the

damping ratio? What is the $2\%$ settling time?

$($c$)(4$ pts$)$ While maintaining the same level of maximum OS$\%,$ is it possible to improve the

settling time by only changing the gain K$?$ Why, or why not?

$($d$)(12$ pts$)$ While maintaining the same level of maximum OS$\%,$ improve the settling time to

$0\mathrm{.}2$ seconds by adding a compensator to the loop.