Problem $1$

$(36$ pts$)$ Consider a system that inchudes a conpensator, shown in the figwe below.

Figure $1$: Systen block diagram of problem $1.$

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

$\backslash [$

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

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

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

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

$\backslash ]$

Let usexamine the compensated and unconpensated system open$-$loop trander functions $($OLTF$)$ using Bode and Nyquist plots. The uncompensated system OLTF refers to G$($s$)$ and the compersated OLTF refers to $\backslash ($ G$\_\{$e$\}($s$)$ G$($s$)\backslash ).$

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

$(1)$ Number of asymptotes in magnitude plots

$(2)$ Comar frequencies of asmptotes in magnitode plots

$(3)$ Slopes of asymptotes in magnitude plots

$(4)$ Compare phase angles of sections that are segmented by comer freguencies

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

$(1)$ Number of asymptotes in magnitude plots

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

$(3)$ Slopes of asymptotes in magnitude plots $(4)$ Compare phase angles of sections that are segmented by comer freguencies

$($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$)$ inprowed the system relative stalility?