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Which of the following options is the correct reduced block diagram (complete reduction) of the given block diagram?

Block diagram: input R, two summing points, blocks G1 and G4 in series, then a parallel pair G3 and G2 summed to give output C, with feedback H1 around G1, G4 and the take-off, and feedback H2 from C to the first summing point

  • A.Two-loop diagram: G1G4 then G2+G3, inner feedback H1 through 1/(G2+G3), outer feedback H2
  • B.Two-loop diagram: G1G4 then G2+G3, inner feedback H1 around G1G4, outer feedback H2
  • C.Single positive feedback loop: G1G4 with feedback H1
  • D.Single block G1G4(G2+G3)/(1-G1G4H1) with feedback H2

Answer: D

Reduce the diagram step by step. The blocks \(G_2\) and \(G_3\) are in parallel, so they combine to \(G_2 + G_3\), and the blocks \(G_1\) and \(G_4\) in series combine to \(G_1G_4\).


The feedback path \(H_1\) is taken from a point before the block \(G_2 + G_3\). Moving that take-off point after the block means that \(H_1\) must be divided by \(G_2 + G_3\), giving the inner loop with forward gain \(G_1G_4(G_2 + G_3)\) and feedback \(H_1/(G_2 + G_3)\), with positive feedback.


\[ \begin{aligned} \frac{G_1G_4(G_2 + G_3)}{1 - G_1G_4(G_2 + G_3)\cdot\dfrac{H_1}{G_2 + G_3}} \\ &= \frac{G_1G_4(G_2 + G_3)}{1 - G_1G_4H_1} \end{aligned} \]


This is a single block with the outer feedback \(H_2\) around it, which is figure (D). Figures (A) and (B) are intermediate stages, not the complete reduction, and (C) leaves out \(G_2 + G_3\) and \(H_2\). The answer is option D.

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