Question $3$

Measured open$-$loop frequency response data for a unity feedback system using PID control is given in Table $2.$

Table $2$

$($a$)$ Plot the data of Table $2$ on a Nichols chart and determine resulting values for the peak amplitude response, $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$p$\}\}\backslash ),$ and the resonant frequency, $\backslash (\backslash$omega$\_\{\backslash$mathrm$\{$r$\}\}\backslash ),$ under closed$-$loop conditions.

$[6$ marks$]$

$($b$)$ Given that $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$o$\}\}=0\backslash$mathrm$\{$~dB$\}\backslash ),$ estimate the percentage step$-$response overshoot and the $\backslash (\backslash$pm $5\backslash \%\backslash )$ settling time, $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$mathrm$\{$s$\}\backslash$pm $5\backslash \%\}\backslash ).$

$[6$ marks$]$

$($c$)$ Using your plot of part $($a$),$ determine the maximum time delay $($in seconds$)$ that the system can withstand while maintaining a phase margin of greater than $\backslash (15^\{\backslash$circ$\}\backslash ).$

$[4$ marks$]$

$($d$)$ Calculate how the delay of part $($c$)$ changes the Nichols chart plot at frequencies of $\backslash (\backslash$omega$=2\backslash )$ and $3\backslash (\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$ and determine the resulting gain margin.

$[4$ marks$]$