Question

A disturbance input d(t) is injected into the unity feedback control loop shown in the figure. Take the referance input r(t) to be a unit step.


If the disturbance is measurable, its effect on the output can be minimized significantly using a feedforward controller $G_{ff}\left(s\right)$. To eliminate the component of the output due to $d\left(t\right)=sint.G_{ff}\left(j\omega \right)|\omega =1$ should be

• A. $\frac{1}{\sqrt{2}}\angle -\frac{3\pi }{4}$
• B. $\frac{1}{\sqrt{2}}\angle \frac{\pi }{4}$
• C. $\sqrt{2}\angle \sqrt{\pi }$
• D. $\sqrt{2}\angle -\frac{\pi }{4}$

Answer 1

The correct option is D $\sqrt{2}\angle -\frac{\pi }{4}$

Output due to D(s)
$=D\left(s\right)+D\left(s\right).G_{ff}\left(s\right).\frac{1}{s(s+1)}$
$=D\left(s\right)[1+\frac{G_{ff}\left(s\right)}{s(s+1)}]$
$0=D\left(s\right)[1+\frac{G_{ff}\left(j\omega \right)}{j\omega (j\omega +1)}]$
$\Rightarrow G_{ff}\left(j\omega \right){|}_{\omega =1}=-j(1+j)=-j+1=1-j$
$=\sqrt{2}\angle -\pi /4$