Do the mathematical solution as well as the MATLAB script that matches the following exercise:Do the mathematical solution as well as the MATLAB script that matches the following exercise: A Roll Wrapping Machine $($RWM$)$ receives, wraps, and labels large rolls of paper produced in a paper mill. The RWM consists of several main stations: positioning station, waiting station, wrapping station, etc. We will focus on the positioning, station is shown in Figure $($a$),$ the positioning

The station is the first to see a roll of paper. The station is responsible for receiving and weighing the roll, measuring its diameter and width, determining the desired wrapper for the roll, positioning it for further processing and finally ejecting it from the station. station.

Functionally, the RWM can be classified as a complex operation because each functional step $($e$.$g$.,$ measuring the width$)$ involves a large number of actions by the field device and relies on actions of the field device and relies on a number of accompanying sensors$.$The control system for accurate positioning The width measuring arm is shown in figure $($b$).$The pole $p$ of the positioning arm is normally equal to $2,$ but is subject to change due to machine loading and misalignment. $($a$)$ For $p=2.$

Design a compensator so that P$.$O $20\%$ and $Ts1s$ at a unit step input. $($b$)$ Plot $y($t$)$ for an input step $R(s)=\frac{1}{s}.($c$)$ Plot $y(t)$ for a disturbance $Td(s)=1,$ with $R(s)=0.($d$)$ Repeat parts $($b$)$ and $($c$)$ when $p$ changes to $1$ and $Gc($s$)$ remains as designed in part $($a$)$ and compare.