Question $1$

Open$-$loop frequency response data for a robotic arm used on a production line is provided in Table $1.$ The production line is in an environment that is liable to vibrations at a frequency of $\backslash (3\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ).$ In order to increase the response time of the arm, the system gain is to be increased. Assume unity feedback.

Table $1$

$($a$)$ Plot the data of Table $1$ on a polar plot and determine the maximum increase in system gain that can be applied while maintaining a gain margin of greater than $4\mathrm{.}44$ dB $.$

$[8$ marks$]$

$($b$)$ With the gain set equal to the value determined in part $($a$),$ calculate new values for the open$-$loop frequency response at $\backslash (\backslash$omega$=1\mathrm{.}6,3\backslash )$ and $\backslash (7\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$ and use them to sketch a new polar plot. $[4$ marks$]$

$($c$)$ For the same gain setting as part $($b$),$ determine the phase margin and estimate the maximum time delay that the system can withstand before it becomes unstable.

$[4$ marks$]$

$($d$)$ To determine the response of the system of part $($b$)$ to vibrations, calculate the closed$-$loop output signal, $\backslash ($ c$($t$)\backslash ),$ for an input signal $\backslash ($ r$($t$)=0\mathrm{.}1\backslash$operatorname$\{$Sin$\}(3$ t$)\backslash ).$

$[4$ marks$]$