Problem $3$

$(34$ pts$)$ Consider the following closed$-$loop control system, where the command is given as a unit step and $\backslash ($ K $\backslash )$ is a scalar positive gain:

Figure $4$: System of problem $3$

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

$($b$)(8$ pts $)$ Find the gain $\backslash ($ K $\backslash )$ that will give $\backslash (7\backslash \%\backslash )$ OS $($maximum percent overshoot$).$ What is the damping ratio? What is the $\backslash (2\backslash \%\backslash )$ settling time?

$($c$)(4$ pts$)$ While maintaining the same level of maximum OS$\backslash \%,$ is it possible to improve the settling time by only changing the gain $\backslash ($ K $\backslash )?$ Why, or why not?

$($d$)(12$ pts$)$ While maintaining the same level of maximum OS$\backslash \%,$ improve the settling time to $0\mathrm{.}2$ seconds by adding a compensator to the loop.