$G(s)=\frac{(s)}{T_D(s)}=\frac{12}{s^2+3s+5\mathrm{.}5}$

a$)$ Calculate the natural frequency $($w$),$ damping ratio $($zeta$),$ settling time $($Ts$),$ peak time $($Tp$)$ and percent overshoot $($PO$)$ values of the system.

b$)$ Plot the unit step response of the system in Matlab$/$Simulink environment. Set the simulation time $($stop time$)$ to $25$ seconds and the sample time to $0\mathrm{.}1.$ There will be no delay in the step signal.

c$)$ Design a proportional control that will allow the system to overshoot by $20\%$ and a relevant controller that will reduce the settling time by $\frac{1}{3}$ compared to the proportional controller.

d$)$ Plot the unit step response of the system for the controller design $($obtained in option c$)$ in Matlab $/$ Simulink environment.

e$)$ Design a controller that will reduce the steady$-$state error by $10$ times compared to proportional control.

f$)$ For the final state of the controller design, plot the unit step response of the system in Matlab $/$ Simulink environment.