Consider a dynamic system whose plant transfer function is given by

\[G(s)=\frac{1}{20 s^{2}}\]

The plant is in a unity feedback arrangement with a controller \(D(s)\) (a cascade controller) as depicted in Figure 4.99.

Consider the following four PID-type controllers

\(P\) controller, \(D(s)=80\)

\(P D\) controller, \(D(s)=80+10 s\)

PI controller, \(D(s)=80+5 / s\)

PID controller, \(D(s)=80+5 / s+10 s\)

(a) For each of the four controllers determine:

(i) the steady-state error for a unit step input 

(ii) the effect of the controller on the stability of the system

(b) Of the P and PD controllers, which one provides damping? Justify your answer.

R(s) C(s) 1 D(s) 20s Figure 4.99 Unity feedback system.