Transcript PowerPoint 演示文稿

Three basic forms
cascade
G1
G1 G2
G2
parallel
feedback
G1
G1
G2
G2
G1 G2
G1
1+ G1 G2
2.6 block diagram models (dynamic)
2.6.2.2 block diagram transformations
1. Moving a summing point to be:
behind a block
x1
G
±
x2
y
y
x1
G
x2
±
G
Ahead a block
x1
y
G
y
x1
±
±
x2
G
1/G
x2
2.6 block diagram models (dynamic)
2. Moving a pickoff point to be:
behind a block
x1
x1
y
G
x2
y
G
x2
1/G
ahead a block
x1
G
x2
x1
y
x2
G
G
y
2.6
block diagram models (dynamic)
3. Interchanging the neighboring—
Summing points
x3
＋ y
x1
x3
x1
－
＋
y
－
x2
x2
Pickoff points
y
x1
y
x2
x1
x2
2.6 block diagram models (dynamic)
4. Combining the blocks according to three basic forms.
Notes:
1. Neighboring summing point and pickoff point can not be
interchanged！
2. The summing point or pickoff point should be moved to the
same kind！
3. Reduce the blocks according to three basic forms！
Examples:
Moving pickoff point
G1
H2
G3
G2
G4
H3
H1
Example 2.17
G1
1
G4
H2
G3
G2
a
H3
H1
G4
b
Moving summing point
Move to the same kind
G3
G1
G2
H1
Example 2.18
G3
G1
G2
G1
H1
Disassembling the
actions
G4
G1
G2
G3
H3
H1
G4
G1
Example 2.19
G2
G3
H1
H3
H1
H3
Chapter 2 mathematical models of systems
2.7 Signal-Flow Graph Models
Block diagram reduction ——is not convenient to a complicated
system.
Signal-Flow graph —is a very available approach to determine
the relationship between the input and output variables of a system, only needing a Mason’s formula without the complex reduction procedures.
2.7.1 Signal-Flow Graph
only utilize two graphical symbols for describing the relationship between system variables。
Nodes, representing the signals or variables.
G
Branches, representing the relationship and gain
Between two variables.
2.7 Signal-Flow Graph Models
Example 2.20:
f
x1  ax0  bx1  cx2
x2  dx1  ex3
x3  fx 0  gx2
x4  hx 3
x0
c
a x1
b
d
x3 h
x2 g
x4
e
2.7.2 some terms of Signal-Flow Graph
Path — a branch or a continuous sequence of branches traversing
from one node to another node.
Path gain — the product of all branch gains along the path.
2.7 Signal-Flow Graph Models
Loop —— a closed path that originates and terminates on the
same node, and along the path no node is met twice.
Loop gain —— the product of all branch gains along the loop.
Touching loops —— more than one loops sharing one or more
common nodes.
Non-touching loops — more than one loops they do not have a
common node.
2.7.3 Mason’s gain formula
G( s) 
C ( s)

R( s )
m
 Pk  k
k 1

  1   L1   L2   L3    
2.7 Signal-Flow Graph Models
m
G( s) 
C ( s)

R( s )
 Pk  k
k 1

  1   L1   L2   L3    
pk  k-th forward path gain
 k  cofactor of pk : make the b ranch gains, which
touch the k-th forward path, are zero in Δ.
 L1  sum of all different loop gains.
 L2  sum of the gain products of all combination of
2 non-touching loops.
 L3 
sum of th e gain products of all combination of
3 non-touching loops.

2.7 Signal-Flow Graph Models
Example 2.21
x0
f
x1
a
c
d
x2
b
x3
g
h
x4
e
C ( s ) k 1
G( s) 

R( s )

  1  (b  cd  ge )  (bge )
P1  adgh
1  1
  1   L1   L2   L3    
P2  fh ;
m
 Pk  k
 2  1  (b  cd )
x4 adgh  fh(1  b  cd )
G

x0
1  b  cd  ge  bge
2.7 Signal-Flow Graph Models
2.7.4 Portray Signal-Flow Graph based on Block Diagram
Graphical symbol comparison between the signal-flow graph
and block diagram:
Block diagram
Signal-flow graph
and
G(s)
G(s)
2.7 Signal-Flow Graph Models
Example 2.22
R(s) E(s)
G1
－
H1
X
1
－
－
G2
X
H2
G3
X3
C(s)
G4
2
H3
R(s)
1
E(s) G1
X1
G2 X 2 G3
－H2
－H3
－H1
X3 G4 C(s)
2.7 Signal-Flow Graph Models
－H1
R(s)
1
E(s) G1
X 1 G2
X 2 G3
G4
X3
1
C(s)
－H2
－H3
  1  (G1G2G3G4 H 3  G2G3 H 2  G3G4 H1 )
P1  G1G2G3G4 ;
1  1
G1G2G3G4
C ( s)
G

R( s ) 1  G1G2G3G4 H 3  G2G3 H 2  G3G4 H1
2.7 Signal-Flow Graph Models
Example 2.23
R(s)
－
－
X1
+
E(s)
－
－X
2
－
-1
R(s) 1
Y1
G1
X1
+
+
G2
Y2
-1
G1
Y1
1
-1
E(s)
1
X2
-1
C(s)
G2
1
-1
Y2
1
C(s)
2.7 Signal-Flow Graph Models
R(s) 1
E(s)
-1
1
-1
X
G
1
1
X
G
2
2
-1
7 loops:
3 ‘2 non-touching loops’ :
Y
1
-1
1
Y
2
-1
1
1
C(s)
2.7 Signal-Flow Graph Models
R(s) 1
E(s)
-1
1
-1
X
G
1
1
X
G
2
2
Then:
1
-1
-1
Y
1
Y
1
C(s)
1
2
-1
Δ  1  2G2  4G1G2
p1  (  1)  G1  1
p2  (  1)  G1  (  1)  G2  1
p3  1  G2  1
4 forward paths:
p4  1  G2  1  G1  1
Δ1  1  G2
Δ2  1
Δ3  1  G1
Δ4  1
2.7 Signal-Flow Graph Models
We have
C ( s)
G( s) 
R( s )
pk  k



G2  G1  2G1G2

1  2G2  4G1G2