Suppose that we are asked to design a unity$-$feedback closed loop position controller for the plant $($a spring$-$mass$-$damper system$)$ below:

$\backslash [$

G$($s$)=\backslash$frac$\{1\}\{0\mathrm{.}5$ s$^\{2\}+4$ s$+5\}$

$\backslash ]$

where $\backslash (\backslash$mathrm$\{$G$\}(\backslash$mathrm$\{$s$\})\backslash )$ is the transfer function between the position output and the actuator force input.

a$)$ Design a proportional controller that satisfies a steady$-$state error of $0\mathrm{.}1$ for a unity step reference input. Prove that your design works by plotting the resultant closed loop system's response $($in time$)$ to the mentioned reference input on MATLAB. Include your code and plot to your submission pdf file.

Later, we noticed that the actuator of the system $($the component that generates the input when commanded by the controller$)$ has a delay! The actuator itself has a first$-$order dynamics, making the plant as shown below:

$\backslash [$

G$\_\{-\}$ n e w$($s$)=\backslash$frac$\{0\mathrm{.}5\}\{($s$+0\mathrm{.}5)\}\backslash$frac$\{1\}\{\backslash$left$(10$ s$^\{2\}+6$ s$+3\backslash$right$)\}$

$\backslash ]$

b$)$ Check whether the previously designed proportional controller would work on this new system by sketching the root$-$locus of the new system by hand! Prove whether the controller you designed in part a works or not!

c$)$ Design a controller for the system with delay $($G$\_$new$)$ such that the closed loop system would be stable no matter what the scalar gain of the compensator is$.$