block diagram algebra
TRANSCRIPT
Block Diagram Algebra
A graphical tool can help us to visualize the model of
a system and evaluate the mathematical
relationships between their elements, using their
transfer functions.
Using block diagram, we can solve the equations by
graphical simplification, which is often easier and
more informative than algebraic manipulation, even
though the methods are in every way equivalent
Introduction
Reminder: Component Block
Diagram
Block Diagram
• It represents the mathematical relationships between the elements of the system.
• The transfer function of each component is placed in box,
and the input-output relationships between components are indicated by lines and arrows.
)()()( 111 sYsGsU
Three basic forms
G1 G2
G2
G1 G1
G2
G1 G2 G1 G2 G1
G1
G2 1+
cascade parallel feedback
Block diagram transformations
behind a block
x1 y G
±
x2
±
x1
x2
y
G
G
Ahead a block
±
x1
x2
y
G
x1 y
G ±
x2 1/G
1. Moving a summing point to be:
2. Moving a pickoff point to be:
behind a block
G x1
x2
y G
x1
x2
y
1/G
ahead a block
G x1
x2
y
G
G x1
x2
y
3. Interchanging
Summing points
x3
x1
x2
y ＋
－
x1
x3
y ＋
－ x2
Single loop negative feedback
1
1 2
( )
( ) 1
GY s
R s G G
2 1( ) ( ) ( )Y s R s Y s G G
1 1 2( ) ( ) ( )Y s G R s G G Y s
1 2 1( ) 1 ( )Y s G G G R s
1
1 2
( )
( ) 1
GY s
R s G G
What about single loop with positive feedback? ?
Negative feedback
block diagram: reduction example
R
_ +
_
+ 1G 2G 3G
1H
1
2
G
H
+ +
C
block diagram: reduction example
R
_ +
_
+ 21GG 3G
1H
1
2
G
H
+ +
C
block diagram: reduction example
R
_ +
_
+ 21GG 3G
1H
1
2
G
H
+ +
C
block diagram: reduction example
R
_ +
_
+ 121
21
1 HGG
GG
3G
1
2
G
H
C
block diagram: reduction example
R
_ +
_
+ 121
321
1 HGG
GGG
1
2
G
H
C
block diagram: reduction example
R
_ + 232121
321
1 HGGHGG
GGG
C
block diagram: reduction example
R
321232121
321
1 GGGHGGHGG
GGG
C
Example: Find the transfer function of the system shown below.
1 2 5 1 6
1 3 1 2 41
G G G G G
G G G G G
1 2
1 3
1 2 4
1 3
1 65
21
( )( )
( ) 1
G G
G G
G G G
G G
GY sT s G
R s G
Transfer Function from Block
Diagram
Example: Find the transfer function of the system shown below.
1 2 5 1 6
1 3 1 2 41
G G G G G
G G G G G
1 2
1 3
1 2 4
1 3
1 65
21
( )( )
( ) 1
G G
G G
G G G
G G
GY sT s G
R s G
Transfer Function from Block
Diagram
Moving pickoff point
G1 G2 G3 G4
H3
H2
H1
a b
G4 1
G1 G2 G3 G4
H3
H2
H1
? 1 2
1 2
( )
( ) 1
RY s G G
R s G G H
When D(s) 0,
When R(s) 0,
2 2
1 2 1 2
( )
( ) 1 ( ) 1
DY s G G
D s G G H G G H
Example: MISO Find the response of the system Y(s) to simultaneous application of the reference input R(s) and disturbance D(s).
Example MISO
( ) ( ) ( )R DY s Y s Y s
1 2 2
1 2 1 2
( ) ( ) ( )1 1
G G GY s R s D s
G G H G G H
21
1 2
( ) ( ) ( )1
GY s G R s D s
G G H
