1. Block diagrams can model complex systems using multiple interconnected blocks. 2. Block diagram reduction techniques simplify diagrams by combining or rearranging blocks to calculate transfer functions more easily. 3. Common techniques include combining parallel and series blocks, moving summing/pickoff points, eliminating feedback loops, and swapping summing points.

1. Block Diagram Reduction
Techniques
Prepared by,
A.Parimala Gandhi,
AP(SS)/ECE Department,
KIT/CBE
CONTROL SYSTEM ENGINEERING

2. Block diagram
Transfer Function: Ratio between transformation of output to the
transformation of input when all the initial conditions are zero.
A Block diagram is basically modelling of any simple or
complex system.
It Consists of multiple Blocks connected together to represent a system to explain
how it is functioning
)
(s
R
2
G 3
G
1
G
4
G
1
H
2
H
)
(s
Y
)
(s
G
)
(s
C
)
(s
R
G(s)=C(s)/R(s)
Simple system:
Complex System:

3. • It is normally required to reduce multiple blocks
into single block or for convenient understanding it
may sometimes required to rearrange the blocks
from its original order.
• For the calculation of Transfer function its required
to be reduced.
Need for block diagram reduction

4. Block Diagram Reduction techniques
2
G
1
G 2
1G
G
2. Moving a summing point behind a block
G G
G
1
G
2
G
2
1 G
G +
1. Combining blocks which are in cascade or in parallel

5. 5. Moving a pickoff point ahead of a block
G G
G G
G
1
G
3. Moving a summing point ahead of a block
G G
G
1
4. Moving a pickoff point behind a block

6. 6. Eliminating a feedback loop
G
H
GH
G

1
7. Swapping with two adjacent summing points
A B A
B
G
1
=
H
G
G

1

7. Example 1
Find the transfer function of the following
block diagrams
2
G 3
G
1
G
4
G
1
H
2
H
)
(s
Y
)
(s
R
(a)

8. 1. Apply the rule that Moving pickoff/takeoff point ahead of block
2. Eliminate loop I & simplify as
3
2
4 G
G
G +
B
1
G
2
H
)
(s
Y
4
G
2
G
1
H
A
B
3
G
2
G
)
(s
R
I
Solution:
2
G

9. 3. Moving pickoff point B behind block 3
2
4 G
G
G +
1
G
B
)
(s
R
2
1G
H 2
H
)
(s
Y
)
/(
1 3
2
4 G
G
G +
II
1
G
B
)
(s
R C
3
2
4 G
G
G +
2
H
)
(s
Y
2
1G
H
4
G
2
G A
3
G 3
2
4 G
G
G +

10. 4. Eliminate loop III
)
(s
R
)
(
1
)
(
3
2
4
2
1
2
1
3
2
4
1
G
G
G
H
H
G
G
G
G
G
G
+
+
+
+ )
(s
Y
)
(
)
(
1
)
(
)
(
)
(
)
(
3
2
4
1
3
2
4
2
1
2
1
3
2
4
1
G
G
G
G
G
G
G
H
H
G
G
G
G
G
G
s
R
s
Y
s
T
+
+
+
+
+
+
=
=
)
(s
R
1
G
C
3
2
4
1
2
G
G
G
H
G
+
)
(s
Y
3
2
4 G
G
G +
2
H
C
)
(
1 3
2
4
2
3
2
4
G
G
G
H
G
G
G
+
+
+
Using rule 6

11. 2
G
1
G
1
H 2
H
)
(s
R )
(s
Y
3
H
(b)

12. Solution:
1. Eliminate loop I
2. Moving pickoff point A behind block
2
2
2
1 H
G
G
+
1
G
1
H
)
(s
R )
(s
Y
3
H
B
A
2
2
2
1 H
G
G
+
2
2
2
1
G
H
G
+
1
G
1
H
)
(s
R )
(s
Y
3
H
2
G
2
H
B
A
II
I
2
2
2
1 H
G
G
+
Not a feedback loop
)
1
(
2
2
2
1
3
G
H
G
H
H
+
+

13. 3. Eliminate loop II
)
(s
R )
(s
Y
2
2
2
1
1 H
G
G
G
+
2
2
2
1
3
)
1
(
G
H
G
H
H
+
+
2
1
2
1
1
1
3
2
1
2
2
2
1
1
)
(
)
(
)
(
H
H
G
G
H
G
H
G
G
H
G
G
G
s
R
s
Y
s
T
+
+
+
+
=
=
Using rule 6

14. 2
G 4
G
1
G
4
H
2
H
3
H
)
(s
Y
)
(s
R
3
G
1
H
(c)

15. Solution:
2
G 4
G
1
G
4
H
)
(s
Y
3
G
1
H
2
H
)
(s
R
A B
3
H
4
1
G
4
1
G
I
1. Moving pickoff point A behind block 4
G
4
3
G
H
4
2
G
H

16. Solution:
2
G 4
G
1
G
4
H
)
(s
Y
3
G
1
H
2
H
)
(s
R
A B
3
H
4
1
G
4
1
G
I
1. Moving pickoff point A behind block 4
G
4
3
G
H
4
2
G
H

17. Solution:
2
G
1
G
4
H
)
(s
Y
4
3G
G
1
H
2
H
)
(s
R
B
3
H
4
1
G
4
1
G
I
1. Moving pickoff point A behind block 4
G
4
3
G
H
4
2
G
H

18. Solution:
2
G
1
G
)
(s
Y
1
H
2
H
)
(s
R
B
3
H
4
1
G
4
1
G
I
1. Moving pickoff point A behind block 4
G
4
3
G
H
4
2
G
H
2
4
4
3
4
3
1 H
G
G
G
G
+

19. 2. Eliminate loop I and Simplify
II
III
4
4
3
4
3
2
1 H
G
G
G
G
G
+
1
G
)
(s
Y
1
H
B
4
2
G
H
)
(s
R
4
3
G
H
II
3
3
2
4
4
3
4
3
2
1 H
G
G
H
G
G
G
G
G
+
+
III
4
1
4
2
G
H
G
H −
Not feedback
feedback

20. )
(s
R )
(s
Y
4
1
4
2
G
H
G
H −
3
3
2
4
4
3
4
3
2
1
1 H
G
G
H
G
G
G
G
G
G
+
+
3. Eliminate loop II & III
1
4
3
2
1
2
3
2
1
4
4
3
3
3
2
4
3
2
1
1
)
(
)
(
)
(
H
G
G
G
G
H
G
G
G
H
G
G
H
G
G
G
G
G
G
s
R
s
Y
s
T
−
+
+
+
=
=
Using rule 6

21. 3
G
1
G
1
H
2
H
)
(s
R )
(s
Y
4
G
2
G A
B
(d)

22. Solution:
1. Moving pickoff point A behind block 3
G
I
1
H
3
1
G
)
(s
Y
1
G
1
H
2
H
)
(s
R
4
G
2
G
A B
3
1
G
3
G

23. 2. Eliminate loop I & Simplify
3
G
1
H
2
G B
3
1
G
2
H
3
2G
G B
2
3
1
H
G
H
+
1
G
)
(s
R )
(s
Y
4
G
3
1
G
H
2
3
2
1
2
3
2
1 H
G
G
H
G
G
G
+
+
II

24. )
(s
R )
(s
Y
1
2
1
2
3
2
1
2
3
2
1
1 H
G
G
H
G
G
H
G
G
G
G
+
+
+
3. Eliminate loop II
1
2
1
2
3
2
1
2
3
2
1
4
1
)
(
)
(
)
(
H
G
G
H
G
G
H
G
G
G
G
G
s
R
s
Y
s
T
+
+
+
+
=
=
4
G

25. End