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GATE 2023 EC – Question 42

Control Systems · Transfer Function, Block Diagram and Signal Flow Graph · 2 marks · Multiple choice

In the following block diagram, $R(s)$ and $D(s)$ are two inputs. The output $Y(s)$ is expressed as $Y(s)=G_1(s)R(s)+G_2(s)D(s)$. $G_1(s)$ and $G_2(s)$ are given by

Diagram for GATE 2023 EC question 42
  1. $G_1(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$ and $G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$
  2. $G_1(s)=\dfrac{G(s)}{1+G(s)+H(s)}$ and $G_2(s)=\dfrac{G(s)}{1+G(s)+H(s)}$
  3. $G_1(s)=\dfrac{G(s)}{1+G(s)+H(s)}$ and $G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$
  4. $G_1(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$ and $G_2(s)=\dfrac{G(s)}{1+G(s)+H(s)}$

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Correct answer: (A) $G_1(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$ and $G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$

Explanation

The input to $G$ is $(R-Y)+D-HY$: the outer unity feedback subtracts $Y$ at the first summer, the inner $H$ feedback subtracts $HY$ at the second, and $D$ is added there. So $Y=G[R-Y+D-HY]$, giving $Y(1+G+GH)=GR+GD$. Hence $G_1=G_2=\dfrac{G}{1+G+GH}$.