Automatic Control Systems
Lecture-3
Block Diagram Reduction
Emam Fathy
Department of Electrical and Control Engineering
email: emfmz@yahoo.com
1
Introduction
• A Block Diagram is a shorthand pictorial representation of
the cause-and-effect relationship of a system.
• The interior of the rectangle represent the mathematical
operation to be performed on the input to yield the output.
• The arrows represent the direction of information or signal
flow.
x
d
dt
y
Introduction
• The operations of addition and subtraction have a special
representation.
• The block becomes a small circle, called a summing point, with
the appropriate plus or minus sign associated with the arrows
entering the circle.
• The output is the algebraic sum of the inputs.
• Any number of inputs may enter a summing point.
• Some books put a cross in the circle.
Introduction
• In order to have the same signal or variable be an input
to more than one block or summing point, a takeoff
point is used.
• This permits the signal to proceed unaltered along
several different paths to several destinations.
Example-1
• Consider the following equations in which x1, x2, x3, are variables,
and a1, a2 are general coefficients or mathematical operators.
x3  a1 x1  a 2 x2  5
Canonical Form of A Feedback Control System
Characteristic Equation
• The control ratio is the closed loop transfer function of the system.
C( s )
G( s )

R( s ) 1  G( s )H ( s )
• The denominator of closed loop transfer function determines the
characteristic equation of the system.
• Which is usually determined as:
1  G( s )H ( s )  0
Example-4
G(s )
H (s )
B( s )
 G( s )H ( s )
E( s )
1. Open loop transfer function
2. Feed Forward Transfer function
3. closed loop transfer function
4. characteristic equation
C( s )
 G( s )
E( s )
C( s )
G( s )

R( s ) 1  G( s )H ( s )
1  G( s )H ( s )  0
In order to analyze the system, we want to
represent multiple subsystems as a single transfer
function.
Reduction techniques
1. Combining blocks in cascade
G2
G1
G1G2
2. Combining blocks in parallel (feed-forward)
G1
G2
G1  G2
3. Eliminating a feedback loop
G
𝑮
𝟏 ∓ 𝑮𝑯
H
OR
A
B
H
𝑨
𝑩 ∓ 𝑨𝑯
4. Moving a pickoff point behind a block
G
G
1
G
5. Moving a pickoff point ahead of a block
G
G
G
6. Moving a summing point ahead of a block
G
G
1
G
7. Moving a summing point behind a block
G
G
G
Note
A
B
B
A
Example 1
H2
_
R
+_
+
+
G1
+
H1
C
G2
G3
Example
H2
G1
_
R
+_
+
+
+
C
G1
H1
G2
G3
Example
H2
G1
_
R
+_
+
+
C
+
G1G2
H1
G3
Example
H2
G1
_
R
+_
+
C
+
G1G2
+
H1
G3
Example
H2
G1
_
R
+_
+
G1G2
1  G1G2 H1
C
G3
Example
H2
G1
_
R
+_
+
G1G2G3
1  G1G2 H1
C
Example
R
+_
G1G2G3
1  G1G2 H1  G2G3 H 2
C
Example
R
G1G2G3
1  G1G2 H1  G2G3 H 2  G1G2G3
C
Example 2
Find the transfer function of the following block diagrams
G4
R (s )
Y (s)
G1
G2
G3
H2
H1
I
G4
R(s)
B
G1
G2
A
G3
H2
H1
G2
Solution:
1. Moving pickoff point A ahead of block
2. Eliminate loop I & simplify
B
G4  G2G3
G2
Y (s)
G4
R(s)
G1
A G2 G3
GG4 
G
B
2
Y (s)
3
H2
H 1G2
G4  G2G3
3. Moving pickoff point B behind block
II
R(s)
G1
B
G4  G2G3
H2
H 1G2
1 /(G4  G2G3 )
C
Y (s)
4. Eliminate loop III
R(s)
G1
GG4 4GG
2G
3 3
2G
1  H 2 (GH4 2 G2G3 )
C
C
Y (s)
G2 H1
G4  G2G3
R(s)
G1 (G4  G2G3 )
1  G1G 2 H1  H 2 (G4  G2G3 )
Y (s)
G1 (G4  G2G3 )
Y ( s)
T ( s) 

R( s) 1  G1G 2 H1  H 2 (G4  G2G3 )  G1 (G4  G2G3 )
Example 3
Find the transfer function of the following block diagrams
R(s)
G1
G2
H1
H2
H3
Y (s)
Solution:
1. Eliminate loop I
R(s)
G1
A
H1
G2
G2
1  GH2 H
2 2
I
B
Y (s)
B
Y (s)
H3
G2
1  G2 H 2
2. Moving pickoff point A behind block
R(s)
G1
H1
G2
1  G2 H 2
A
1  G2 H 2
G2
H3
II
1  G2 H 2
H 3  H1 (
)
G2
Not a feedback loop
3. Eliminate loop II
R(s)
G1G2
1  G2 H 2
H3 
Y (s)
H1 (1  G2 H 2 )
G2
Y ( s)
G1G2
T ( s) 

R( s) 1  G2 H 2  G1G2 H 3  G1H1  G1G2 H1H 2
Example 4
Find the transfer function of the following block diagrams
H4
R(s)
Y (s)
G1
G2
G3
H3
H2
H1
G4
Solution:
G4
1. Moving pickoff point A behind block
I
H4
R(s)
Y (s)
G1
G2
G3
H3
H2
H3
G4
H2
G4
H1
1
G4
1
G4
A
G4
B
2. Eliminate loop I and Simplify
R(s)
G2G3G4
1  G3G4 H 4
G1
II
Y (s)
B
H3
G4
H2
G4
III
H1
II
feedback
G2G3G4
1  G3G4 H 4  G2G3 H 3
III
Not feedback
H 2  G4 H1
G4
3. Eliminate loop II & IIII
R(s)
G1G2G3G4
1  G3G4 H 4  G2G3 H 3
Y (s)
H 2  G4 H1
G4
G1G2G3G4
Y ( s)
T ( s) 

R( s) 1  G2G3 H 3  G3G4 H 4  G1G2G3 H 2  G1G2G3G4 H1
Example 5
Find the transfer function of the following block diagrams
H2
R(s)
G2
G1
H1
G4
A
G3
Y (s)
B
Solution:
1. Moving pickoff point A behind block
G3
I
H2
R(s)
G2
G1
A
1
G3
H1
H1
G4
G3
1
G3
B
Y (s)
2. Eliminate loop I & Simplify
H2
G3
G2
H1
R(s)
G2G3
B
H1
 H2
G3
1
G3
G1
II
G2G3
1  G2 H1  G2G3 H 2
H1
G3
G4
Y (s)
B
3. Eliminate loop II
R(s)
G1G2G3
1  G2 H1  G2G3 H 2  G1G2 H1
Y (s)
G4
G1G2G3
Y (s)
T ( s) 
 G4 
R( s )
1  G2 H1  G2G3 H 2  G1G2 H1
End of Lec 3
Example-5: Continue.
Example-6: Reduce the Block Diagram.
Example-6: Continue.
Example-7: Reduce the Block Diagram. (from Nise: page-242)
Example-7: Continue.
Example-8: Reduce the system to a single transfer function.
(from Nise:page-243).
Example-9: Simplify the block diagram then obtain the closeloop transfer function C(S)/R(S). (from Ogata: Page-47)
Example-10: Multiple Input System. Determine the output C
due to inputs R and U using the Superposition Method.
Example-11: Continue.
Example-11: Continue.
Example-19: Multiple-Input System. Determine the output C
due to inputs R, U1 and U2 using the Superposition Method.
Example-19: Continue.
Example-19: Continue.
Example-20: Multi-Input Multi-Output System. Determine C1
and C2 due to R1 and R2.
Example-20: Continue.
Example-20: Continue.
When R1 = 0,
When R2 = 0,
Example-5: Reduce the Block Diagram to Canonical Form.
Example-5: Continue.
However in this example step-4 does not apply.
However in this example step-6 does not apply.
Example-6
• For the system represented by the following block diagram
determine:
1.
2.
3.
4.
5.
6.
7.
8.
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=10.
Example-6
– First we will reduce the given block diagram to canonical form
K
s 1
Example-6
K
s 1
K
G
 s 1
K
1  GH
1
s
s 1
Example-7
• For the system represented by the following block diagram
determine:
1.
2.
3.
4.
5.
6.
7.
8.
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=100.
Block Diagram of Armature Controlled D.C Motor
Ra
La
c
Va
ia
eb
T
J

La s  Ra I a(s)  K b(s)  Va(s)
Js  c (s)  K m aI a(s)
Block Diagram of Armature Controlled D.C Motor
La s  Ra I a(s)  K b(s)  Ea(s)
Block Diagram of Armature Controlled D.C Motor
Js  c (s)  K m a I a(s)
Block Diagram of Armature Controlled D.C Motor
Block Diagram of liquid level system
h1  h2
q1 
R1
h2
q2 
R2
dh1
C1
 q  q1
dt
dh2
C2
 q1  q2
dt
Block Diagram of liquid level system
L
h1  h2
q1 
R1
H1 ( s )  H 2 ( s )
Q1 ( s ) 
R1
L
h2
q2 
R2
H 2 (s)
Q2 ( s ) 
R2
dh1
C1
 q  q1
dt
L
C1sH1( s )  Q( s )  Q1( s )
dh2
C2
 q1  q2
dt
L
C2 sH 2 ( s )  Q1( s )  Q2 ( s )
Block Diagram of liquid level system
H1 ( s )  H 2 ( s )
Q1 ( s ) 
R1
H 2 (s)
Q2 ( s ) 
R2
C1sH1( s )  Q( s )  Q1( s )
C2 sH 2 ( s )  Q1( s )  Q2 ( s )
Block Diagram of liquid level system