Consider the following transfer function for a DC motor that relates the rotor$$s position

m in radians to the armature voltage va in volts.

$$m$($s$)/$Va$($s$)=$ Km$/($s$($JmLas$2+($JmRa $+$ mLa$)$s $+($mRa $+$ KmKb$)))$ rad$/$V

Use the values provided below for parameters that were taken from a Pittman $24$V

DC$030$C$-1$ brushed DC motor$$s data sheet.

$$ Jm $=2106$ Nms$2$

$$ m $=3106$ Nm$/($rad$/$s$)$

$$ Ra $=9\mathrm{.}5$

$$ La $=0\mathrm{.}0073$ H

$$ Km $=0\mathrm{.}0316$ Nm$/$A

$$ Kb $=0\mathrm{.}0316$ V$/($rad$/$s$)$

$1.$ Given the following constraints for a step response, determine $($and sketch$)$ the regions

where the dominant two closed$-$loop poles $($those with real parts closest to the origin$)$

should be placed.

$($a$)$ Ts $0\mathrm{.}1$ s

$($b$)$ TP $0\mathrm{.}025$ s

$($c$)$ P O $15\%$

$($d$)|$ess$|0\mathrm{.}002$ rad

$2.$ Consider P$,$ PD$,$ PI and PID control to determine one viable controller and its gains

for the DC motor such that the criteria above are satisfied when va$($t$)=$ u$($t$)=1$ rad.

Provide the following to support your chosen controller.

$($a$)$ Type $($P$,$ PD$,$ PI or PID$)$ of controller selected.

$($b$)$ Root locus for your chosen type of controller applied to the DC motor. Identify the

controller gain that was varied and provide values for any zeros chosen.

$($c$)$ Identify the locations of the closed loop poles that were selected along with the

corresponding value of the controller gain.

$($d$)$ Calculation of other gains for the controller. Specify values for all controller gains.

$($e$)$ Plot of the step response of the closed loop system that shows all the specifications

are met.

$($f$)$ Provide values of the specifications taken from the step response