Run the following design of a PID controller for the plant given by:

$sys=\frac{1}{(s+1{)}^3}.$

As a first pass, create a model of the plant and design a simple PI controller for it$.$

C$\_$pi is a pid controller object that represents a PI controller. The fields of info show that the tuning

algorithm chooses an open$-$loop crossover frequency of about $0\mathrm{.}52ra\frac{d}{s}.$

a$)$ Examine the closed$-$loop step response $($reference tracking$)$ of the controlled system. Obtain the

Step response and comment.

The new controller achieves the higher crossover frequency, but at the cost of a reduced phase

margin.

c$)$ Compare the closed$-$loop step response with the two controllers. Comment on the two responses

obtained.

This reduction in performance results because the PI controller does not have enough degrees of

freedom to achieve a good phase margin at a crossover frequency of $1\mathrm{.}0ra\frac{d}{s}.$ Adding a derivative

action improves the response.

Part II

Design a PIDF controller for $Gc$ with the target crossover frequency of $1\mathrm{.}0ra\frac{d}{s}.$

The fields of info show that the derivative action in the controller allows the tuning algorithm to

design a more aggressive controller that achieves the target crossover frequency with a good phase

margin.

d$)$ Compare the closed$-$loop step response and disturbance rejection for the fast PI and PIDF

controllers. Comment on the two responses obtained.

You can compare the input $($load$)$ disturbance rejection of the controlled system with the fast Pl and

PIDF controllers.

e$)$ plot the response of the closed$-$loop transfer function from the plant input to the plant output.