$[8$ marks$]$ Consider a second$-$order system whose response to a unit step input has $20\%$

overshoot, an approximate $2\%$ settling time of $5$ seconds, and final $($steady$-$state$)$ value of $4.$

Write a MATLAB script to do the following:

Compute $\backslash$xi and $\backslash$omega $\_($n$)$ precisely. Print them to the Command Window to $6$ decimal places.

Define a vector of time points from $0$ to $20$ seconds, spaced $1$ millisecond apart.

Evaluate the step response y$($t$)$ at each of these $20001$ time points using the expression

from row $24$ of the Laplace Transform Pairs table and your precise calculations of $\backslash$xi

and $\backslash$omega $\_($n$).$

Using the y$($t$)$ vector you computed above, find the maximum value of the step

response and calculate the percent overshoot.

Calculate the precise settling time by finding the last point in y$($t$)$ that falls outside

$+-2\%$ of the final value. There are several possible approaches for doing this. An

algorithmic strategy is to examine each point in the step response from beginning to

end and update a "bookkeeping" variable if y$($t$\_($k$))>1\mathrm{.}02$y$\_($final $)$ or y$($t$\_($k$))<mn>0</mn><mn>.</mn><mn>9</mn><mn>8</mn>$y$\_($final $)$ at

time point t$\_($k$).$ When the algorithm terminates, this bookkeeping variable will hold the

settling time.

Print percent overshoot and settling time to the Command Window to $6$ decimal places.

Compare your calculations of percent overshoot and settling time to the expected values

of $20\%$ and $5$ seconds, respectively. Briefly explain any discrepancies. Submit a printout

of your code and a screenshot of the output in the Command Window.