Assignment

You are given the following system

Y(s) = 20 U(s)/ (s² + 12s + 20)

We now are interested in achieving a settling time that is less than 1 sec. You will have to use feedback control to do this.

a. Draw the block diagram of the system with a proportional feedback controller (Kp) and calculate the closed-loop-transfer function (CLTF).

b. Determine the poles of the CLTF with Kp=0, 0.2, 0.4, 0.6, 0.8, 1, 2, 4 , and plot them on a complex plane.

c. Now determine the Kp value that will give you a 1 sec settling time (Make your slow decaying pole settle at 1 sec). Plot the unit step response of the resulting system with this Kp. The settling time should be fast enough, but you should see some steady state error.

d. To help mitigate the steady-state-error problem, look at your closed loop transfer function and calculate the steady state response to a unit step using the final value theorem. What do you need to do to Kp to 'help' with the steady state error problem?

e. Crankyour Kp way up: make it 1000. Now plot the step response again. What happened to the steady state error? Did using a large Kp cause any potential problems?

Consider the following feedback system:

The controller is given by G_c(s) = K (s + 2).

Find:

a. G(s) = Y(s)/(R(s).
b. The poles of the system
c. Damping ratio (ς) and natural frequency (ω_n) as a function of the controller gain K.
d . K that gives a natural frequency of 4 rad/sec.