A mass spring damper system represented by the following transfer function exhibits oscillatory

behaviour with minimal damping

$G(s)=\frac{1}{0\mathrm{.}02s^2+s+1}$

$($a$)$ Design a PID controller $($Zeigler$-$Nichols method$)$ and simulate the closed$-$loop step response

using MATLAB script.

$($b$)$ Design a model predictive controller $($MPC$)$ with sampling time: $0\mathrm{.}08$ seconds, prediction hori$-$

zon: $10$ steps, control horizon: $2$ steps. the control effort is constrained between $-10$ and $10$ units

$($MATLAB script$).$

$($c$)$ Evaluate and compare the performance of the both the controllers in terms of rise time, settling

time and peak overshoot and state your inference.