$(36$ pts $)$ Consider a system that includes a compensator, shown in the figure below.

The uncompensated plant transfer function $\backslash ($ G$($s$)\backslash )$ and the compensator transfer function $\backslash ($ G$\_\{$c$\}($s$)\backslash )$ are given as:

$\backslash [$

$\backslash$begin$\{$array$\}\{$l$\}$

G$($s$)=\backslash$frac$\{10\}\{$s$($s$+1)\}\backslash \backslash$

G$\_\{$c$\}($s$)=\backslash$frac$\{$s$+3\}\{$s$+9\}$

$\backslash$end$\{$array$\}$

$\backslash ]$

Let us examine the compensated and uncompensated system open$-$loop transfer functions $($OLTF$)$ using Bode and Nyquist plots. The uncompensated system OLTF refers to $\backslash ($ G$($s$)\backslash )$ and the compensated OLTF refers to $\backslash ($ G$\_\{$c$\}($s$)$ G$($s$)\backslash ).$

$($a$)(10$ pts$)$ Generate Bode plots $($magnitude and phase angle$)$ for the uncompensated OLTF, and make sure the following details are included:

$(1)$ Number of asymptotes in magnitude plots

$(2)$ Corner frequencies of asymptotes in magnitude plots

$(3)$ Slopes of asymptotes in magnitude plots

$(4)$ Compare phase angles of sections that are segmented by corner frequencies

$($b$)(10$ pts $)$ Generate Bode plots $($magnitude and phase angle$)$ for the compensated OLTF, and make sure the following details are included:

$(1)$ Number of asymptotes in magnitude plots

$(2)$ Corner frequencies of asymptotes in magnitude plots

$(3)$ Slopes of asymptotes in magnitude plots

$(4)$ Compare phase angles of sections that are segmented by corner frequencies

$($c$)(6$ pts $)$ Compare $($a$)$ and $($b$)$ in details.

$($d$)(10$ pts$)$ Generate Nyquist plots for both TFs of interest, and compare them by finding and marking the following two items:

$(1)$ gain margin

$(2)$ phase margin

Comment on how the compensator has $($or has not$)$ improved the system relative stability?