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Lecture -5: Signal Flow Graphs 1 hour
By Mr S WijewardanaPhD student QMUL 20-04-2013
Learning Objectives:
1. Elements of signal flow graphs.
2. Block diagrams vs signal flow graphs.3. Maisons formula.
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Signal Flow Graphs(SFG)
The other way of representing the control systemsdynamics(Pl. See Lecture notes on Lecture-5) is with theuse of Signal Flow Graphs. Even though the Block diagramrepresentation and the signal flow graph methods are
quite similar, there are some notable differences betweenthem.
Signal flow graph method is easier to construct.
System equation is readily available than the Blockdiagram method.
SFG method gives a visual representation of the systemequation and hence logical reduction is much easier.
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Elements of a SFG
A SFG is a network of directed branches which
connect at Nodes.
The Nodes are connected by line segments
and these line segments are called branches.
Node: is defined as a junction point to
represent variables in a SFG.
The branches have associated branch gains
and directions
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Consider a linear system described by a set of N algebraic
equations as shown below:
N
kkkjj NjyaY 1 ,....,2,1
These N equations can be written in the form of causeand
effect relationship as shown below:
inputgainoutput
causekjtokfromgaineffectjN
k
thth
1
)()(
Equation-1
Equation-1 can be written in Laplace domain as shownbelow:
N
k
kkjj NjsYsGsY1
,...,2,1)()()(
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Input Node:
An input node is a node that has only out going branches.
Output Node:
Output node is a node that has only incoming branches.The small circles represent nodes(e.g: y1,y2,x4, y3,y4) The arrows represent the
flow of data from one node to the other. The arrow head indicates the gain of the
signal.
In signal flow graphs we cannot normally find the condition which satisfies
according to the definition of an output node. Because, in SFG as shown in fig
below does not have the output node which satisfies all the above condition.
Therefore we draw a similar node which is equivalent to the output node with a
UNITY GAIN.
x4
y1a12 y2 a23 y3 a34
y4
a42
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a12 y2 a23 y3
To derive the equation from graph is easy:
y2= a12y1 + a24y4
x4
y1a12 y2 a23 y3 a34 y4
a42
Y3
B1
Unity gain
SFG with a unity gain: y3=y3(this is not
possible for an input node)
y4
y1
a24
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G(s) C(s)
R(s)
x2
C(s)
a
x3x1 x1
R(s)
G2(s)
C(s)
G1(s)
+
+
G1(s)
G2(s)
G(s)
C(s)R(s) R(s)
+
b
+
a
b
x2
x3
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x1
x
h
Y = gx + hz ( addition)
x2
x1
b
a
c
x2
x3
a
x3
-
c
+
b
y
z
g
x
h
y
z
-g
Y = hzgx (subtraction)
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Y= gx (Multiplication)
Y= -x (negative unit
transmittance)
Y= x/g (division)
Z=x (Unit
transmittance)
x
1/g
y
x
g
x
y
z
1
xy
-1
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R(s)
G(s)
E(s)
E(s)
C(s)
-
H(s)
+R(s)
1 G(s) C(s) 1 C(s)
x1
b
-H(s)
y
x2
x3a
Y = ax1 + bx2cx3 +dy
-c
d
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Construct the signal flow diagram for two simultaneous equations given below :
t11x + t12y + t13z = 0 -----------------eq1
t21x+ t22y + t23z = 0 -----------------eq2
From eq1 and 2 we can write:
xt
tzt
ty
yt
tz
t
tx
22
21
22
23
11
12
11
13
x
z
y
11
13
t
t
11
12
t
t
x
z
y
22
23
t
t
22
21
t
t
Example-1:
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Combining both SFGs:
z
y
x
11
13
t
t
11
12
t
t
22
23
t
t
22
21
t
t
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Gain Formula for Signal Flow Graphs
Maisons gain formula gives us an easy
method to find the input-output relationship.
Input-output relationship of any control
system is important for stability analysis.
Transmittance:
Is defined as the overall gain between an input
node an output node.
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Maisons gain formula:
kkPP1
Pk = Path gain or Transmittance of kth
forward path.
= determinant of graph
= 1(sum of all individual loop gains)
+ ( sum of gain products of all possible combinations of two non-
touching loops)
- (sum of gains of all combinations of three non touching loops)+ (sum of ........) - .......
cb fed
fedca
a
a LLLLLL, ,,
...1
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a
aL
cb
caLL,
= Sum of gain products of all possible combinations of two
non-touching loops
fed
fed LLL,,
= Sum of gain products of all possible combinations of
three-non-touching loops.
= Sum of all individual loop gains
k = Cofactor of kth forward path is determined from the graph with the loopstouching the kth forward path removed. That is, the cofactor k is obtained from by removing the loops that touch the path Pk .
= or we can say -loop gains touching the kth forward path.
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R(s)
G(s)
E(s)
E(s)
C(s)
-
H(s)
+R(s)
1 G(s) C(s) 1 C(s)
Example-2:
Determine the TF of the control system by using (Maisons Gain Formula) MGF
-H(s)
(i) From the SFG shown above there is only one forward path
between R(s) and C(s) and the forward path gain is
P1= G(s)
(ii) There is only one loop in the SFG.
E(s)-G(s) through H(s) and E(s)
Hence: L1 = -G(s).H(s)
Solution:
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(iii) There are no non-touching loops since there is only one loop. The
forward path R(s)-E(s)-C(s)-C(s) is in touch with the only loop.
Therefore 1 = 1
= 1- L1= 1 + G(s).H(s)
(iv) Using gain formula now we can write the C.L .T.F.:
)()(1
)(
)(
)( 11
sHsG
sGP
sR
sC
R(s) +
Example-3: Convert the block diagram into SFG
and use MGF to find the TF or C/R gain ratio. G4
G2G1
H1
+
C(s)
G3
H2
+-
+
SFG f th b bl k di i h b l
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R 1 a1 G1 a2 G2 a3 G3 a4 1 a5 1 C
SFG for the above block diagram is shown below:
R 1 a1
G1
a2
G2
a3
G3
a4
1 a5
1 C
R 1 a1 G1 a2 G2 a3 a4 1 a5 1 C
Path 1:
Path 2:
R(s) +
-H1
C(s)
H2
G4
There are two forward paths that
can be drawn as shown below:
G4
Gain for two forward paths:
P1 = G1G2G3
P2 = G1G2G4
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(ii) There are three non-touching feedback loops in the SFG as shown below:
G4
a1 G1 a2 G2 a3
H2
a1 G1 a2 G2 a3 G3 a4 1 a5
-H1
-H1
a1 G1 a2 G3 a3 a4 1 a5
Three non-touching feedback loops.
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Loop Gain:
L1 = G2 G2 H2
L2 = -G1G2G3H1
L3= -G1G2G4H1
(III) There are no two non-touching loops in the SFG.Therefore
= 1-(G1G2H2-G1G2G3H1-G1G2G4H1)
(iv)
As P 1 and P2 are both touching the feedback loops
1 = 1, 2=1Hence the required gain is:
)(1
)(
1
1.1.
4113221
4321
13211321221
421321
2211
GHHGHGG
GGGG
R
C
HGGGHGGGHGG
GGGGGG
PP
R
C
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