PLEASE SOLVE ALL STEP BY STEP

Problem $\backslash (1(600\backslash$mathrm$\{$pts$\})\backslash )$ : Consider a unity feedback control system with the following open$-$loop plant transfer function $\backslash (\backslash$mathrm$\{$G$\}\_\{\backslash$mathrm$\{$p$\}\}(\backslash$mathrm$\{$s$\})\backslash )$ given as$,$

$\backslash [$

G$\_\{$p$\}($s$)=\backslash$frac$\{1\}\{$s$($s$+2)($s$+5)\}$

$\backslash ]$

The goal is to design a controller to achieve the following design specifications: $1)$ M$.$O$.(\backslash \%)$ less than equal $\backslash (40\mathrm{.}0,2\backslash ))$ a settling time $(\backslash (2\backslash \%\backslash ))$ of less than and equal $2\mathrm{.}5$ sec $,$ and $3)$ zero steady$-$state error.

a$)$ For a simple proportional $\backslash ((\backslash$mathrm$\{$P$\})\backslash )$ controller with a transfer function of $\backslash (\backslash$mathrm$\{$G$\}\_\{\backslash$mathrm$\{$c$\}\}(\backslash$mathrm$\{$s$\})=$K$=50\backslash ),$ use MATLAB to plot the unit step response and determine the time$-$response characteristics $($e$.$g$.,$ overshoot, settling time, steady$-$state errors$)$ of the closed$-$loop system.

b$)$ Use a proper Ziegler$-$Nichol's method to determine control gains of a PID controller for the closedloop system in part $($a$)$; plot the unit step response of the PID closed$-$loop system and determine the new time response characteristics.

c$)$ Use the MATLAB Root$-$locus editing tools of Control System design toolbox to further tune the PID gains $($found in part $($b$))$ to achieve the design specifications.