G$5.$ A coordinate measuring machine $($CMM$)$ measures coordinates on three$-$dimensional objects. The

accuracy of CMMs is affected by temperature changes as well as by mechanical resonances due to joint

elasticity. These resonances are more pronounced when the machine has to go over abrupt changes of

dimension, such as sharp corners at high speed. Each of the machine links can be controlled in a closed$-$

loop configuration, such as the one shown in Figure below for a specific machine with prismatic $($sliding$)$

links. In the figure, $x_{\text{r}\text{e}\text{f}}(s)$ is the commanded position and $x(s)$ is the actual position. The minor loop

uses a tachometer generator to obtain the joint speed, while the main loop controls the joint's position

a$.$ Find the value of $K$ that will result in a minor loop with $=0\mathrm{.}5.$

b$.$ Use a compensator, $G_c(s),$ for the external loop so that it results in a closed$-$loop damping factor of

$=0\mathrm{.}7$ with $T_s$~~$4$ seconds.

c$.$ Use MATLAB to simulate the compensated system's closed$-$loop step response.