Consider the 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 4.75.

(a) Use a \(P\) controller, \(D(s)=80\). Does this controller provide any damping?

(b) Use a PD controller, \(D(s)=80+10\) s. Determine the tracking property for a unit step input.

(c) Use a PI controller, \(D(s)=80+5 /\) s. Discuss the effect of this controller on the stability of the system.

(d) Use a PID controller, \(D(s)=80+5 / s+10 s\). Discuss the effect of this controller on the stability and steady-state error (for a unit step input) of the system.

R(s) C(s) 1 D(s) 20s2 Figure 4.75 Unity feedback system.