Bloock Diagram Reduction Summary in control system.pdf

block diagram reduction technique

1. Block Diagram Reduction: A Summary

The block diagram is a pictorial representation of system where
blocks represent the transfer functions of different elements of
system.

In addition to blocks, block diagram also possess summing points and
take-off points. Summing points are used to add the different signals;
however, the take-off points are utilized to take-off the signals.

2. A simple or canonical form of block diagram consists of one block in
the forward path, one block in the feedback path, one summing point
and one take-off point. This canonical form is generally termed as
closed-loop system, closed-loop feedback system or feedback system.

The closed-loop feedback system is shown in following figure. The
transfer function of closed loop feedback system is given as

 

   1
C s G s
R s G s H s


(1)
R s G s
H s

C s

Figure: The closed-loop feedback block diagram.

3. Rules for reducing the block diagram:
The complicated block diagram is converted into transfer functions with
the help of block diagram reduction rules. The rules for reducing the
block diagram are summarized in following table.

4. Table: Rules for reducing the block diagram.
1. Consecutive
summing
points
The two or
more
consecutive
summing
points can be
interchanged
directly
R

C
x
y


R

C
y
x


Not only,
interchanging
can be done,
but also, a
single
summing
point can be
used for
R

C
x
y


R

C
x
y


5. representing
the
consecutive
summing
points
2. Blocks in
cascade
Transfer
functions of
such blocks
get multiplied
2
G
C
3
G
1
G
R

C
1 2 3
G G G
R

3. Blocks in
parallel
Transfer
functions of
such blocks
get added
algebraically
R
2G
3
G
1
G

C

R C
1 2 3
G G G 

6. 4. Shifting a
summing
point
behind the
block
Add a block
of transfer
function
equal to
reciprocal of
block
R
G
C
y

R
y
1/G
G
C

5. Shifting a
summing
point
beyond the
block
Add a block
of transfer
function
same as the
block
R
G
C
y

R
G
C
y
G

6. Shifting a
take off
point
behind the
block
Add a block
of transfer
function
same as the
block
C
R
G
y

C
y
R
G
G

7. 7. Shifting a
take off
point
beyond the
block
Add a block
of transfer
function
equal to
reciprocal of
block

C
R
G
y

C
y
R
G
1/G

8. Removing
minor
feedback
loop
Replace the
feedback
loop with its
equivalent
transfer
function

R
G
H

C

1
G
GH
R C

8. 9. Deriving
the transfer
function of
MIMO
systems
Obtain
transfer
functions
considering
one output
and one input
at a time
MIMO
System
1
R
2
R
p
R

1
C
2
C
q
C


Obtain p q transfer
functions as given
below:
1
1
C
R
,
1
2
C
R
, ,
1
p
C
R
,
2
1
C
R
,
2
2
C
R
, ,
2
p
C
R
,

1
q
C
R
,
2
q
C
R
, ,
q
p
C
R

9. 10. Shifting
take-off
point after a
summing
point
Add a new
summing
point to
compensate
for the shift
R C

yx

R C
z

y

x

11. Shifting
take off
point before
a summing
point
Add a
summing
point to
compensate
for a shift of
take-off point
R C

yx

R C
z

yx


10. Sequence of rules for reducing the block diagram:
1. Reduce the blocks in series.
2. Reduce the blocks in parallel.
3. Reduce minor feedback loops.
4. Try to shift take-off point beyond the block to the output side, if
possible.
5. Try to shift summing point behind the block to the input side, if
possible.
6. Interchange the positions of summing point and take-off point, only
if it is must to interchange.
7. Repeat the steps 1-5 until the transfer function of overall system is
obtained.