Problem $5$

A small permanent magnet DC motor drives a large inertial load whose inertia is $\backslash ($ J $\backslash ).$ There is no friction or damping in the system. The motor is driven in current mode, which means the motor torque $\backslash ($ T$=$K$\_\{$T$\}$ I $\backslash )$ is directly contolled by the input. There is no friction or damping in the system.

$($a$)$ Show that the transfer function of this plant between current input and position output is

$\backslash [$

G$\_\{$p$\}($s$)=\backslash$frac$\{\backslash$Theta$($s$)\}\{$I$($s$)\}=\backslash$frac$\{$K$\_\{$T$\}\}\{$J s$^\{2\}\}$

$\backslash ]$

Tip: See "Turning motor into a torque source" section of the lecture notes on DC motors.

$($b$)$ Using whatever design tools you like, including MATLAB, explore PID control forms to design a closed$-$loop system that has a good step response $($you get to define "good"$)$ that is relatively fast without too much overshoot, or too many oscillations, or too much steady$-$state error. For this problem, let $\backslash (\backslash$mathrm$\{$J$\}=0\mathrm{.}01\backslash$mathrm$\{$~kg$\}-\backslash$mathrm$\{$m$\}^\{\backslash$wedge$\}2\backslash )$ and $\backslash (\backslash$mathrm$\{$Ka$\}=0\mathrm{.}1\backslash$mathrm$\{$~N$\}-\backslash$mathrm$\{$m$\}/\backslash )$ Amp. The position feedback signal scaling is $\backslash (1\backslash$mathrm$\{$~V$\}/\backslash$mathrm$\{$rad$\}\backslash ).$ In designing the control system, the maximum input to the motor $($i$.$e$.$ the max input to plant Gp $)$ is $\backslash (\backslash$pm $10\backslash$mathrm$\{$Amps$\}\backslash ),$ so use controller gains that keep the plant input to that limit$.($See note below.$)$

Report your findings so that the reader can understand what you are proposing for your design and why. When reporting your findings, include a step response and a plot of motor current input $($to confirm that it remains under $10$ Amps $).$

Note $1$: Limiting the motor input to $10$ A does not apply for the first few values following the step when using D control as the D part of the controller differentiates the error, which will have a large value when the reference input steps.$)$

Note $2$: You may use P$,$ PI$,$ PD$,$ or PID forms of the controller, whatever you think might work to do the job.