Part 1 · Chapter 4

Block Diagram Algebra and Reduction

A real control system is rarely one block — it is a web of forward paths, feedback loops, and cross-couplings. Because the Laplace transform turned convolution into multiplication, that whole web can be collapsed, block by block, into a single transfer function using a small set of algebraic rules. This chapter is the toolkit for doing exactly that, cleanly and without error.

Control Systems Prof. Mithun Mondal Reading time ≈ 45 min
i What you'll learn
  • The four building blocks of any diagram: block, summing point, take-off point, and signal arrow.
  • The three basic interconnections — cascade, parallel, and feedback — and their combined transfer functions.
  • How to move a summing point or a take-off point across a block without changing the system.
  • A systematic reduction procedure that collapses any single-input diagram to one block.
  • How to handle multiple inputs (reference and disturbance) using superposition.
  • When to abandon reduction and switch to the signal-flow-graph method of Chapter 5.
Section 4-1

The Four Elements of a Block Diagram

Every block diagram, however large, is assembled from just four symbols. A block holds a transfer function and multiplies the signal passing through it. A signal arrow carries a single Laplace-domain quantity in one direction. A summing point (circle) adds or subtracts the signals meeting at it, the sign marked beside each arrow. A take-off (branch) point taps a signal and sends identical copies to two or more places without changing it.

ElementSymbolOperation
BlockRectangle with \(G(s)\)Output \(= G(s)\times\) input
SignalDirected arrowCarries one variable, one way
Summing pointCircle with \(\pm\)Algebraic sum of inputs
Take-off pointDot on a lineCopies a signal unchanged
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A block multiplies; a summing point adds
\[ Y(s) = G(s)X(s) \quad\text{(block)}, \qquad E(s) = R(s) - B(s) \quad\text{(summing point)} \]

Keep these two operations straight and every reduction is just bookkeeping. The golden rule: a take-off point copies a signal without dividing it — the same value goes everywhere it branches.

Section 4-2

Cascade (Series) Connection

When the output of one block feeds straight into the next, the blocks are in cascade. Their transfer functions simply multiply — a direct consequence of the convolution-becomes-product property from Chapter 3.

R G₁ G₂ C ≡ G₁G₂
Cascade: transfer functions multiply
Cascade rule
\[ \frac{C(s)}{R(s)} = G_1(s)\,G_2(s)\cdots G_n(s) \]
One caveat. Multiplying cascaded blocks assumes each stage does not load the previous one — that connecting them draws no power that changes the earlier block's output. For idealised transfer functions this holds; for real circuits, buffer between stages or model the loading explicitly.
Section 4-3

Parallel Connection

When several blocks share the same input and their outputs meet at a summing point, they are in parallel. Their transfer functions add (with the summing-point signs):

R G₁ G₂ + + C ≡ G₁+G₂
Parallel: transfer functions add
Parallel rule
\[ \frac{C(s)}{R(s)} = G_1(s) \pm G_2(s) \pm \cdots \pm G_n(s) \]
Section 4-4

Feedback Connection

The feedback loop — derived in Chapter 1 — is the third basic combination and the one that defines control. A forward block \(G\) with a feedback block \(H\) collapses to the master formula:

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Standard feedback reduction
\[ \frac{C(s)}{R(s)} = \frac{G(s)}{1 \mp G(s)H(s)} \]

Use \(+\) in the denominator for negative feedback (the usual case) and \(-\) for positive feedback. When \(H=1\) the loop is unity-feedback and the formula becomes \(G/(1+G)\). Spotting this single-loop pattern inside a larger diagram is the heart of every reduction.

Section 4-5

Moving Summing & Take-off Points

Real diagrams rarely present a clean single loop. The trick is to rearrange the diagram — sliding a summing point or take-off point past a block — until a basic pattern emerges. Each move comes with a compensating block so the signals stay mathematically identical. The two most useful moves:

take-off after G G GR GR ⇒ take-off before G (add G) G G
Move a take-off point ahead of a block by inserting a matching G in the branch

The principle is conservation of signal: whatever value flowed along each branch before the move must still flow after it. A take-off point moved against the signal direction past \(G\) needs a \(G\) added to the branch; moved with the signal, a \(1/G\). Summing points follow the mirror-image rule.

Section 4-6

The Reduction Rules at a Glance

Collected together, the algebra of block diagrams is this short table. Apply the moves until only cascade, parallel, and feedback patterns remain, then collapse each in turn.

#OperationRule
1Blocks in cascadeMultiply: \(G_1G_2\)
2Blocks in parallelAdd: \(G_1\pm G_2\)
3Feedback loop\(\dfrac{G}{1\pm GH}\)
4Move take-off past a block (downstream)Add \(1/G\) in the branch
5Move take-off past a block (upstream)Add \(G\) in the branch
6Move summing point past a block (downstream)Add \(G\) in the branch
7Move summing point past a block (upstream)Add \(1/G\) in the branch
8Interchange two adjacent summing pointsAllowed (sum is associative)
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The reduction recipe
Combine cascades → combine parallels → collapse innermost loop → repeat

Always work the innermost loop first and move only the points that block a basic pattern. Check at every step that no signal has been created or lost — that single discipline prevents almost every reduction error.

Section 4-7

Multiple Inputs: Superposition

Real loops have more than one input — a reference \(R\) and a disturbance \(D\) entering at the plant. Because the system is linear, superposition applies: find the output due to each input alone (setting the others to zero), then add. The two transfer functions usually differ.

Reference and disturbance response
\[ C(s) = \underbrace{\frac{G_1G_2}{1+G_1G_2H}}_{\text{from } R}\,R(s) \;+\; \underbrace{\frac{G_2}{1+G_1G_2H}}_{\text{from } D}\,D(s) \]
Why this split matters. The reference transfer function tells us how well the system tracks; the disturbance transfer function tells us how well it rejects. Good design pushes the first toward 1 and the second toward 0 — and the shared denominator \(1+G_1G_2H\) is exactly the loop whose roots decide stability for both.
Section 4-8

Worked Examples

1 Cascade and parallel mix

Problem. \(G_1\) is in cascade with the parallel combination of \(G_2\) and \(G_3\). Find the overall transfer function.

Solution. Add the parallel pair, then multiply by the cascade block:

Working
\[ \frac{C}{R} = G_1\,(G_2 + G_3) \]
2 Single loop

Problem. A unity-feedback loop has forward path \(G(s)=\dfrac{20}{s(s+4)}\). Find \(C/R\).

Solution. Apply the feedback rule with \(H=1\):

Working
\[ \frac{C}{R} = \frac{G}{1+G} = \frac{20}{s^2+4s+20} \]

Poles at \(s=-2\pm j4\): a stable, lightly damped pair.

3 Loop with non-unity feedback

Problem. Forward path \(G_1G_2\), feedback \(H\). Reduce to one block.

Solution. Combine the cascade, then apply the feedback rule:

Working
\[ \frac{C}{R} = \frac{G_1 G_2}{1 + G_1 G_2 H} \]
4 Moving a take-off point

Problem. A feedback signal is tapped after block \(G\), but the loop pattern needs it tapped before \(G\). What compensation is required?

Solution. Moving the take-off upstream past \(G\) means the branch previously carried \(GX\) but now carries \(X\); restore the value by inserting \(G\) in the branch:

Working
\[ \text{branch signal } GX \;\to\; X \cdot G = GX \quad\checkmark \]

The diagram is now in standard form and the feedback rule applies directly.

5 Nested (two) loops

Problem. An inner loop \(G_2\) with feedback \(H_2\) sits inside an outer loop closed by \(H_1\), preceded by \(G_1\). Reduce it.

Solution. Collapse the inner loop first, then the outer:

Working
\[ G_{\text{inner}} = \frac{G_2}{1+G_2H_2}, \qquad \frac{C}{R} = \frac{G_1 G_{\text{inner}}}{1 + G_1 G_{\text{inner}} H_1} \]

Substituting \(G_{\text{inner}}\) gives \(\dfrac{G_1G_2}{1+G_2H_2+G_1G_2H_1}\) — the innermost-loop-first discipline keeps the algebra clean.

6 Disturbance rejection by superposition

Problem. With \(R=0\), find the output caused by a disturbance \(D\) entering between \(G_1\) and \(G_2\) in a loop with feedback \(H\).

Solution. Set \(R=0\); only \(G_2\) lies in the forward path from \(D\), but the full loop still closes:

Working
\[ \frac{C}{D} = \frac{G_2}{1 + G_1 G_2 H} \]

Large loop gain \(G_1G_2H\) shrinks this ratio — the system rejects the disturbance, exactly the benefit promised in Chapter 1.

Review

Chapter Summary

✓ Four elements

Block (multiplies), arrow (carries), summing point (adds), take-off point (copies unchanged).

✓ Cascade

Series blocks multiply: \(G_1G_2\cdots\), assuming no inter-stage loading.

✓ Parallel

Blocks sharing an input and summed at the output add: \(G_1\pm G_2\).

✓ Feedback

\(\dfrac{G}{1\pm GH}\); innermost loops collapse first.

✓ Moving points

Slide summing/take-off points past blocks, inserting \(G\) or \(1/G\) so no signal changes.

✓ Multiple inputs

Superpose: solve for each input alone, then add. Shared denominator \(1+GH\) governs stability.

Practice

Problems

For each diagram, identify the innermost basic pattern first, reduce it, and repeat. Where a point blocks a pattern, move it with the correct compensating block. Difficulty rises down the list.

  1. Three blocks \(G_1, G_2, G_3\) are in cascade. Write the overall transfer function.
  2. \(G_1\) and \(G_2\) are in parallel with their outputs subtracted (\(G_1-G_2\)). Write \(C/R\).
  3. A unity-feedback loop has \(G(s)=\dfrac{10}{s+1}\). Find \(C/R\) and its pole.
  4. For forward path \(G\) and feedback \(H=\dfrac{1}{s}\), with \(G=\dfrac{5}{s+2}\), find \(C/R\).
  5. Reduce a loop in which the forward path is \(G_1G_2\) and feedback \(H\) is tapped after \(G_1\) only (not the full output).
  6. An inner loop \((G_2, H_2)\) is nested inside an outer loop \((H_1)\) with cascade \(G_1\) before it. Find \(C/R\).
  7. A take-off point sits before block \(G\) but must be moved after it. State the compensating block and justify it.
  8. Two summing points are adjacent with signs \(+,-\) then \(+\). Show they may be interchanged and write the result.
  9. A loop has forward \(G_1G_2\), an inner feedback \(H_1\) around \(G_2\), and an outer unity feedback. Reduce to one block.
  10. With a reference \(R\) and a disturbance \(D\) entering after \(G_1\), write \(C(s)\) as the superposition of both contributions.
  11. Reduce the diagram with forward path \(G_1, G_2, G_3\) in cascade, feedback \(H_1\) around \(G_2\), and feedback \(H_2\) around \(G_2G_3\).
  12. A multi-loop diagram resists reduction because two feedback paths cross. Explain why, and state which method (Chapter 5) handles it more cleanly.
Tip: when loops overlap or feedback paths cross, block-diagram reduction turns into a bookkeeping nightmare. That is precisely the signal to switch to Mason's gain formula in Chapter 5, which reads the answer off the diagram in one pass — no moving of points required.