$1.$ A magnetic levitation system is shown below:

a$.$ Design a PD compensator $\backslash ($ G$\_\{$c$\}($s$)\backslash )$ to yield a settling time of $0\mathrm{.}1$ seconds or less if the step response is to have more than $\backslash (1\backslash \%\backslash )$ overshoot.

i$.$ Assuming a $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ order approximation is valid, find the design point from the given transient specifications.

ii$.$ Design the PD compensator using this design point, the magnitude and angle conditions.

iii. Write the compensated open$-$loop transfer function in zero$-$pole$-$gain $($ZPK$)$ form.

iv$.$ Use MATLAB to simulate the step response of the compensated system. Include plots of your results as well as your code.

v$.$ Does the compensated system meet the given transient specifications? If not, comment on why this might be the case.

b$.$ Cascade a PI compensator to minimize the steady$-$state error to a step$-$input such that the total settling time does not exceed $0\mathrm{.}5$ seconds. This compensator should not appreciably affect the transient response designed in part $($a$).$

i$.$ Design the PI compensator to satisfy the criteria above.

ii$.$ Write the compensated open$-$loop transfer function in zero$-$pole$-$gain $($ZPK$)$ form.

iii. Use MATLAB to simulate the step response of the compensated system. Include plots of your results as well as your code.