Problem $2.$ Consider the system from Problem $1.$ a$)$ Determine the transfer function of the plant with $$ as input and $$ as output. Sketch the corresponding Bode diagrams using straight line approximation. b$)$ Design a PID controller $($i$.$e$.,$ determine $$P$,$I $,$ and $$D $)$ so that the phase margin becomes $$m $=75.($Hint: choose the $$P and $$I gains so that you eliminate the two rightmost poles of the plant. Formulate the transfer functions $()$ and $()$ of the controller and the open loop. Calculate the phase crossover frequency and the gains.$)$ c$)$ Use Matlab to simulate the system using: P control with: $$P $=10,$I $=0,$D $=0,$ PD control with: $$P $=10,$I $=0,$D $=0\mathrm{.}5,$ PI control with: $$P $=10,$I $=50,$D $=0,$ PID control with: $$P $=10,$I $=50,$D $=0\mathrm{.}5,$ PID control with the gains designed above, for zero initial conditions. Plot $()$ for the five cases in the same graph. Show the range $$ in $[0,/4],$ in $[0,2],$ and plot $$r $.$ Submit the figure and your codes.