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Block Diagrams
u
x  Ax  Bu , x 0
y  Cx  Du
state space
representation
y
u
a n s  a n 1 s
n
H (s) 
n 1
b m s  b m 1 s
m
m 1
 ...  a1 s  a 0
y
 ...  b1 s  b0
transfer function
representation
Assume that we are only interested in the input-output relation:
transfer function
Then, a system can be represented by a block with an input and output.
One can represent a large system as an interconnection of block diagrams of
subsystems.
... gives an insight both in analysis and synthesis.
X (s)
Y (s)
G (s)
X 1(s)
+
G (s) X (s)  Y (s)
X 3 (s)  X 1(s)  X 2 (s)
X 3 (s)
X 3 (s)  X 1(s) -X 2 (s)
-/+
X 2 (s)
Y (s)
U (s)
+
Y (s)
X (s)
G (s)
-/+
X (s)  U (s)  Y (s) U (s)
Y (s)  G (s) X (s)

Y (s)
U (s)
G (s)
1  G (s)

G (s)
1+ G (s )
U (s)
Y (s)
+
Y (s)
X (s)

U (s)
G (s)
G (s)
1  H ( s )G ( s )
-
X ( s )  U ( s )  H ( s )Y ( s )
-/+
Y (s)  G (s) X (s)
H (s)
Y (s)
U (s)

G (s)
+ H ( s....
1.........
)G ( s )
N (s)
R (s)
+
+
X (s)
G1 ( s )
-
+
Y (s)
G2 (s)
H (s)
Y (s) 
G2 (s)
1  H ( s )G1 ( s )G 2 ( s )
G 1 ( s ) R ( s )  N ( s ) 
Reduction Rules of Block Diagrams
A
+
AB
+
-
+
B
A
ABC
C
+
AC
+
C
+
ABC
B
C
C
+
A
+
+C
AB
+
A
-
+
AB
+
ABC
-
B
B
A
A
G1
G1
AG 1
AG 1
AG 1G 2
G2
AG 1G 2
G2
A
G2
A
G 1G 2
AG 2
AG 1G 2
AG 1G 2
G1
A
G1
AG 1
+
AG 1  AG 2
A
G1  G 2
A (G1  G 2 )
+
G2
A
G
AG 2
AG
AG 1  B
+
B
-
A+
A
B
G
AG  B
G
1
G
B
A
AB
+
G (s)
( A  B )G
A
AG
G
-
-
B
B
A
A
AG  BG
+
G
BG
G
G
AG
AG
AG
G
AG
H
R(s)+
-
+
G1
+
+
2
+
G2
G3
H1
Find the transfer function
C(s)
by reducing the block diagram!
R(s)
C(s)
A block diagram of a circuit is
given. Find the transfer
function for each block and
obtain the transfer function of
the block diagram Vd3/Vk!
Signal-Flow Diagram
A diagram equivalent to block diagrams.
Suitable to apply Mason’s rule – a formula that gives the transfer
function.
The signal-flow diagram is a directed graph (digraph)
G  {V , E }
set of
vertices
set of
edges
with weights on edges, which represent gains (transfer functions)
between components inputs and outputs.
Reduction of Signal-Flow Diagrams
a
x1
ax 1  x 2
x2
a1
a1 a2
a2
x1
x2
x1
x3
ax 1 x 2  x 3
x3
b
a+b
a
x1
( a  b ) x1  x 2
x1
x2
x2
a
a
ac
x2
b
x1
x2
c
x3
bc
x4
x1
acx 1  bcx 2  x 4
x4
c
ab
ab
a
x1
x2
a
x1
a
e
x1
d
x3
1 c
e
x1
x3
b
x3
c
b
x1
x3
b
d
1 c
1  bc
bc
x3
Block Diagrams vs. Signal-Flow Diagrams
R(s)
G(s)
C(s)
G(s)
R(s)
C(s)
-1
R(s) +
-
E(s)
G(s)
C(s)
1
R(s)
G(s)
E(s)
C(s)
Mason’s Rule
Aim:
To find the transfer function directly from the diagram without
any reduction.
Definitions:
Path: A subgraph GP of the graph G is called a path if
• it contains n edges and n+1 vertices,
• one can label its edges as e1, e2, ...,en and its vertices as v1,v2,
....,vn+1 such that the edge ek points from vk to vk+1.
Path gain: The product of gains along a path.
Loop: A subgraph GL of the graph G is called a loop if
• it contains n edges and n vertices,
• one can label its edges as e1, e2, ...,en and its vertices as
v1,v2, ....,vn such that the edge ek points from vk to vk+1(mod n).
Loop gain: The product of gains along a loop.
Determinant of a (sub)graph:
 loop gains
  products of loop gains for disjoint loop pairs
  products of loop gains for disjoint loop triples
  1
Mason’s Rule
For a signal-flow diagram the transfer function between the
input and output vertices is given by
1
P 
Pk  k
 k

Gain of the k.
path
Determinant of the whole diagram
Determinant of the
subgraph obtained by
deleting the k. path
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