$4.$ Find the closed$-$loop transfer function in terms of $\backslash ($ C$($S$)\backslash )$ and $\backslash ($ G$($S$)\backslash ).($In other words reduce Fig $2$ into a single block$)$

$5.$ The controller chosen is a proportional controller $\backslash (($C$($S$)=10)\backslash ).$ Find the closed$-$loop transfer function with controller being $10.$

$6.$ Use MATLAB to simulate and plot the step response of the system. Provide your MATLAB code and the Plot.

Fig. $2.$ Feedback control loop. C$($S$)$ the Controller and $\backslash ($ G$($S$)\backslash )$ the open loop transfer function of the DC motor and driver.

Extra credit Problems:

$7.$ Find the system response to the sinesoidal input, $\backslash ($ R$($S$)=5\backslash$sin $(0\mathrm{.}2$ t$)\backslash )$

$8.$ Use MATLAB to generate a Bode plot for the system.