This document discusses block diagram reduction techniques. It provides 10 rules for combining blocks in a control system diagram to simplify the overall transfer function. The rules cover combining blocks in series and parallel, eliminating feedback loops, shifting summing and take-off points, and applying associative laws. The goal is to obtain a single transfer function to represent the overall system. Examples are provided to demonstrate applying the rules to shift elements in the diagram while maintaining equivalent outputs.

1. BLOCK DIAGRAM
REDUCTION TECHNIQUES
PARMAR GAURAVSINH
(170410117023)
PATEL AADARSH
(170410117025)

2. BLOCK DIAGRAM REDUCTION
• A complex control system consists of several blocks. Each of them has its
own transfer function. But overall transfer function of the system is the
ratio of transfer function of final output to transfer function of initial input
of the system. This overall transfer function of the system can be obtained
by simplifying the control system by combining this individual blocks, one
by one.
The technique of combining of these blocks is referred to as block
diagram reduction technique. For successful implementation of
this technique, some rules for block diagram reduction to be
followed.

3. BASIC CLOSED LOOP TRANSFER FUNCTION
• Consider a simple closed loop system,
• R(s) = Laplace of input signal
• C(s) = Laplace of output signal
• E(s) = Laplace of error signal
• B(s) = Laplace of feedback signal
• G(s) = forward transfer function
• H(s) = feedback transfer function

4. for open loop transfer function, as no feedback present,
transfer function can be given by:
Open loop transfer function = G(s).H(s)

5. BLOCK DIAGRAM REDUCTION RULES
• Rule 1: Blocks In Series
Any finite specific number of blocks arranged in series can be combined
together by multiplication as shown below:
The above blocks shown can be combined together and replaced
with single block as
Output C(s) = G1 x G2 x R(s)

6. RULE 2: BLOCKS IN PARALLEL
• When the blocks are connected in parallel combination, they get added algebraically
(considering the sign of the signal)
• The above blocks can be replaced with single block as
• C(s) = R(s)G1 + R(s)G2 – R(s)G3 C(s) = R(s) (G1 + G2 –
G3)

7. RULE 3: ELIMINATION OF FEEDBACK LOOP
• We can use Closed loop transfer function to eliminate the feedback
loop present.

8. RULE 4: ASSOCIATIVE LAW FOR SUMMING POINT
• This can be better explained by taking below diagram
Y= R(s) – B1
C(s) = y – B2 = R(s) – B1 – B2
This law is applicable only to summing points which are connected directly to each other.

9. RULE 5: SHIFTING OF A SUMMING POINT
BEFORE A BLOCK
• When we shift summing point before a block, we need to do transformation in
order to achieve the same result. Please refer the diagram
C(s) = GR(s) + X After shifting the summing point, we will get
C(s) = [R +(X/G) ] G = GR + X which is same as output in the first case.
• Hence to shift a summing point before a block, we need o add another block of
transfer function ‘1/G’ before the summing point as shown in figure.

10. RULE 6: SHIFTING OF THE SUMMING POINT
AFTER A BLOCK
• When we generally shift summing point after any block, we required to do the
transformation to attain the same (required) result. Please refer the below diagram .
• C(s) = (R + X)G After shifting the summing point, we will get
• C(s) = (R +X) G = GR + XG which is same as output in the first case.
• Hence to shift a summing point before a block, we need to add another block having
same transfer function at the summing point as shown in fig

11. RULE 7: SHIFTING OF TAKE-OFF POINT AFTER A
BLOCK
• Here we want to shift the take – off point after a block, as shown in diagram
• Here we have X = R and C = RG (initially)
• In order to achieve this we need to add a block of transfer function ‘1/G’ in series with
signal taking off from that point.

12. RULE 8 : SHIFTING OF TAKE-OFF POINT BEFORE A
BLOCK
• Here we want to shift the take – off point before a block, as shown in
diagram
• Here we have X = R and C = RG (initially)
• In order to achieve this we need to add a block of transfer function ‘G’
in series with X signal taking off from that point.

13. RULE 9 : SHIFTING A TAKE-OFF POINT AFTER A
SUMMING POINT
• can be transformed to (refer both the diagrams)
• Before shifting take-off point, initially we have:
• C(s)=R±Y
• and Z = R ± Y (initially)
• Hence if we want to shift a take off point after a summing point, one more summing
point needs to be added in series with take-off point.

14. RULE 10: SHIFTING A TAKE-OFF POINT BEFORE A
SUMMING POINT
• Suppose if we want to shift take off point before a summing point, then initially we have
• C(s)=R±Y
• and Z = R ± Y (initially)
• this can be transformed to (refer both the diagrams)
• In order to satisfy this condition, we need to add a summing point in series with the
take-off point.

15. EXAMPLE:-