Problem $3$: $(20$ points$)$

Figure $2$ shows a closed$-$loop system using cascade compensation. For a unit$-$step input, the closedloop system must

$-$ have a peak$-$overshoot of less than $\backslash (15\backslash \%\backslash ),$

$-$ and a rise$-$time less than $1$ s $.$

Figure $2$: Feedback control system using cascade compensation.

a$)(5$ points$)$ Can the design objectives be satisfied using proportional compensation $\backslash ($ G$\_\{$c$\}($s$)=$K$\_\{$P$\}\backslash )?$ Justify your answer.

b$)(15$ points$)$ Using the root locus method, design a proportional$-$plus$-$derivative compensator

$\backslash [$

G$\_\{$c$\}($s$)=$K$\_\{$D$\}$ s$+$K$\_\{$P$\}$

$\backslash ]$

that meets the desired transient response requirements. In order to receive full credit for this problem, using the MATLAB publish tool, append the following items to the PDF file containing your solution.

$-$ A root locus sketch a long with an analysis showing how the parameters $\backslash ($ K$\_\{$P$\}\backslash )$ and $\backslash ($ K$\_\{$D$\}\backslash )$ are selected.

$-$ A simulation of the unit$-$step response of the closed$-$loop system that indicates the peak overshoot and rise$-$time.

$-$ The m$-$file uced to onnerato venur renite