(i) Plot the step responses of the following second order systems and state the nature of each system. For each case, find the poles and plot the location of the poles in the complex plane by hand. Draw a general conclusion about the position of the system poles and the nature of system response. Calculate the damping ratio for each system.

An unstable system is highly undesirable in control engineering. What we want is that the system could behave smoothly and have its output follow (track) our desired (reference) input. For this purpose, we use "feedback control". Let us feed the output of the system G(s) back to the input position and compare it with our desired (reference) input to construct the following closed-loop system.

The closed-loop transfer function of the above system is calculated as  follows:

The nature of this system will now depend on the value of constant gain controller K.

(ii) Let K be 0.5, 1.0, 1.5 and 15 respectively and plot the corresponding step responses. From your plots determine the steady error for the two cases of K=1.5 and K=15. Compare them to see what conclusion you can draw.