Consider the control system shown in the following diagram.

The controller \(D(s)\) is given by, \[
D(s)=K \frac{s+\alpha}{s^{2}+\omega^{2}} .
\]

(a) Prove that the system is capable of tracking a sinusoidal reference input, \(r(t)=\sin (\omega t)\), with a zero steady-state error.

(b) Find the range of \(K\) and \(\alpha\) such that the closed-loop system remains stable. For the case where \(\alpha=1\), is the closed-loop system asymptotically stable for some set of values of \(K\) ?

Transcribed Image Text:

r(t) 1 y(t) D(s) s(s+1) A Control System