$10.$ Given a unity feedback system as shown below with the given $\backslash (\backslash$mathrm$\{$G$\}(\backslash$mathrm$\{$s$\})\backslash )$ :

$\backslash [$

G$($s$)=\backslash$frac$\{$K$\}\{($s$+2)($s$+4)($s$+6)($s$+8)\}$

$\backslash ]$

a$.$ Please sketch the root$-$locus of the system by manual computation. Determine where the root$-$locus on the real axis, and the asymptotes $($if there are any$).$ Please write the way, not just the results. Verify your answer by plotting the root$-$locus by using computer program.

b$.$ For which gain K the system will have damping ratio $0\mathrm{.}5?$ If is a second$-$order system, what will the maximum overshoot for this value of damping ratio?

c$.$ For the value of $\backslash ($ K $\backslash )$ computed in point $\backslash ($ b $\backslash ),$ is it still valid to make second$-$order system approximation for the performance of the system? Please explain your reasons!

d$.$ Design a PID control system that will have the following performance:

i$.$ Has the same maximum overshoot as computed in point $($b$)$

ii$.$ Settling time is $0\mathrm{.}5$ s shorter than the original $($uncontrolled$)$ system

iii. Has zero steady state error for unit step input

e$.$ Make a simulation of controlled and uncontrolled systems to verify your PID control design. Show in the simulation result where you have fulfilled the design requirements of point $($d$)$