Problem $3$

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

Figore $4$: Systen of problem $3$

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

$($b$)(8$ pts$)$ Find the gain $\backslash ($ K $\backslash )$ that will give $?\backslash \%$ OS $($macimum 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 navimun OS$\backslash \%,$ is it possible to improwe the settling time by only changing the gain $\backslash ($ K $\backslash )?$ Why, or why not?

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