Perform parts $($a$)$ and $($b$)$ only. Do not do part $($c$)$

Set $T_d(s)=0$ to make the analysis easier.

For part $($b$),$ you do not need to discuss the effect of the root separation factor$\mathrm{.}10\mathrm{.}48$ Consider the PID position control system shown in Figure P$10\mathrm{.}42,$ where $I=10$ and $c=2.$ The desired time constant is $=2.$a$.$ Compute the required values of the gains for the following two sets of root locations.

$s=-0\mathrm{.}5,s=-5+-5j$

$s=-0\mathrm{.}5,s=-1,s=-2.$

b$.$ For both cases, use a computer method to plot the command response to a unit step. Discuss the effects of the root separation factor on the response. Compare the results with those of Example $10\mathrm{.}7\mathrm{.}4,$ where the roots are $s=-5$ and $s=-0\mathrm{.}5+-0\mathrm{.}5j.$

c$.$ For both cases, use a computer method to plot the disturbance frequency response. Discuss the effects of the root separation factor on the frequency response. Compare the results with those of Example $10\mathrm{.}7\mathrm{.}4.$