A CNC machine must position its axes rapidly and precisely so that high$-$quality cutting can be done efficiently. A CNC mill with three axes control can be considered as three independent positioning systems since each axis is orthogonal to the others. Therefore, we'll look at just one axis.

Assuming that the milling head is considered as a single mass and there is viscous friction in the track, the equation of motion for one axis is:

$\backslash [$

m $\backslash$ddot$\{$x$\}+$c $\backslash$dot$\{$x$\}=$F

$\backslash ]$

$($a$)$ Find the open$-$loop transfer function from $\backslash ($ F$($s$)\backslash ),$ the input, to $\backslash ($ X$($s$)\backslash ),$ the position of the head. You can call this the open$-$loop transfer function of the plant, $\backslash ($ G$($s$)\backslash ).$

$($b$)$ Since we want the machine to track specific positions, we need to close the loop with feedback. Assuming a unity$-$gain feedback configuration $($a perfect sensor$)$ and a proportional controller, write the transfer function from $\backslash ($ X$\_\{\backslash$text $\{$desired $\}\}($s$)=$R$($s$)\backslash )$ to $\backslash ($ X$($s$)\backslash ).$ You can use either $\backslash ($ R $\backslash )$ or $\backslash ($ X$\_\{\backslash$text $\{$des $\}\}\backslash ),$ whichever is more intuitive for you. Substitute in the actual transfer functions for $\backslash ($ K$($s$)=$K$\_\{$p$\}\backslash )$ and $\backslash ($ G$($s$)\backslash )$ from part $($a$).$

$($c$)$ The roots of the closed$-$loop system depend on the gain of the proportional controller. Assuming that $\backslash ($ m$=20\backslash$mathrm$\{$~kg$\}\backslash )$ and $\backslash ($ c$=5\backslash$frac$\{$N$\_\{\backslash$pi$\}\}\{\backslash$mathrm$\{$m$\}\}\backslash ),$ use MATLAB to obtain a root locus, plotting all possible roots of the system for $\backslash (0\backslash$leq K$<mi>\backslash </mi>$infty $\backslash ).$

$($d$)$ Using the dominant root approximation, we know that the "slowest" root $($the one nearest the imaginary axis$)$ will dominate the response. To make the system respond rapidly, we want to move this root as far left as possible. However, overshoot would be very bad for a CNC system $($think tools being snapped off$),$ so find the location of the roots that gives the fastest response without any oscillation. Once you have found the roots, find the gain $\backslash ($ K $\backslash )$ that yields these roots and the time constant associated with the roots.

$($c$)$ Find the steady state error response to a step input with this proportional $"$P$"$ controller. Does the system require integral action with an extra pole at the origin $("$PI$"$ control$)$ to cancel out steady state error?