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Linear Control Systems (LCS)
Engr.Irshad Memon Lecturer
Isra University, Pakistan email: irshad.memon@isra.edu.pk
URL :http://electronics106.jimdo.com/
Lecture-06
Block Diagram Representation of Control Systems
Introduction
• A Block Diagram is a shorthand pictorial representation of
the cause-and-effect relationship of a system.
• The interior of the rectangle representing the block usually
contains a description of or the name of the element, gain,
or the symbol for the mathematical operation to be
performed on the input to yield the output.
• The arrows represent the direction of information or signal
flow.
dt
dx 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 (or
pickoff) 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 � , � , � , are
variables, and � , � are general coefficients or mathematical
operators.
522113 xaxax
Example-1
522113 xaxax
Example-2
• Draw the Block Diagrams of the following equations.
1
1
2
2
2
13
1
1
12
32
11
bxdt
dx
dt
xdax
dtxbdt
dxax
)(
)(
Canonical Form of A Feedback Control System
Characteristic Equation
• The control ratio is the closed loop transfer function of the system.
• The denominator of closed loop transfer function determines the
characteristic equation of the system.
• Which is usually determined as:
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
01 )()( sHsG
Example-3 1. Open loop transfer function
2. Feed Forward Transfer function
3. control ratio
4. feedback ratio
5. error ratio
6. closed loop transfer function
7. characteristic equation
8. Open loop poles and zeros if 9. closed loop poles and zeros if K=10.
)()()(
)(sHsG
sE
sB
)()(
)(sG
sE
sC
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
)()(
)()(
)(
)(
sHsG
sHsG
sR
sB
1
)()()(
)(
sHsGsR
sE
1
1
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
01 )()( sHsG
)(sG
)(sH
Reduction techniques
2G
1G
21GG
1. Combining blocks in cascade
1G
2G
21GG
2. Combining blocks in parallel
3. Eliminating a feedback loop
G
HGH
G
1
G
1H
G
G
1
Example-4: Reduce the Block Diagram to Canonical Form.
Example-4: Continue.
Example-5 • For the system represented by the following block diagram
determine:
1. Open loop transfer function
2. Feed Forward Transfer function
3. control ratio
4. feedback ratio
5. error ratio
6. closed loop transfer function
7. characteristic equation
8. closed loop poles and zeros if K=10.
Example-5
– First we will reduce the given block diagram to canonical form
1s
K
Example-5
1s
K
ss
Ks
K
GH
G
11
1
1
Example-5 (see example-3) 1. Open loop transfer function
2. Feed Forward Transfer function
3. control ratio
4. feedback ratio
5. error ratio
6. closed loop transfer function
7. characteristic equation
8. closed loop poles and zeros if K=10.
)()()(
)(sHsG
sE
sB
)()(
)(sG
sE
sC
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
)()(
)()(
)(
)(
sHsG
sHsG
sR
sB
1
)()()(
)(
sHsGsR
sE
1
1
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
01 )()( sHsG
)(sG
)(sH
Example-6 • For the system represented by the following block diagram
determine:
1. Open loop transfer function
2. Feed Forward Transfer function
3. control ratio
4. feedback ratio
5. error ratio
6. closed loop transfer function
7. characteristic equation
8. closed loop poles and zeros if K=100.
Reduction techniques
4. Moving a summing point behind a block
G G
G
G G
G
1
5. Moving a summing point ahead a block
7. Moving a pickoff point ahead of a block
G G
G G
G
1
G
6. Moving a pickoff point behind a block
8. Swap with two neighboring summing points
A B AB
Example-7
R
_ +
_
+ 1
G 2G 3
G
1H
2H
+ +
C
• Reduce the following block diagram to canonical form.
Example-7
R
_ +
_
+ 1
G 2G 3
G
1H
1
2
G
H
+ +
C
Example-7
R
_ +
_
+ 21
GG 3G
1H
1
2
G
H
+ +
C
Example-7
R
_ +
_
+ 21
GG 3G
1H
1
2
G
H
+ +
C
R
_ +
_
+ 121
21
1 HGG
GG
3G
1
2
G
H
C
Example-7
R
_ +
_
+ 121
321
1 HGG
GGG
1
2
G
H
C
Example-7
R
_ + 232121
321
1 HGGHGG
GGG
C
Example-7
Example 8
Find the transfer function of the following block diagram
2G 3
G1
G
4G
1H
2H
)(sY) ( s R
1. Moving pickoff point A ahead of block 2
G
2. Eliminate loop I & simplify
324GGG
B
1G
2H
)(sY4
G
2G
1H
AB3
G
2G
)(sR
I
Solution:
3. Moving pickoff point B behind block 324
GGG
1G
B)(sR
21GH 2
H
)(sY
)/(1324
GGG
II
1G
B)(sR
C
324GGG
2H
)(sY
21GH
4G
2G A
3G 324GGG
4. Eliminate loop III
)(sR
)(1
)(
3242121
3241
GGGHHGG
GGGG
)(sY
)()(
)(
)(
)(
32413242121
3241
1 GGGGGGGHHGG
GGGG
sR
sY
)(sR
1G
C
324
12
GGG
HG
)(sY324
GGG
2H
C
)(13242
324
GGGH
GGG
2G
4G1
G
4H
2H
3H
)(sY)(sR
3G
1H
Example 9
Find the transfer function of the following block diagrams
Solution:
2G
4G1
G
4H
)(sY
3G
1H
2H
)(sR
A B
3H
4
1
G
4
1
G
I1. Moving pickoff point A behind block
4G
4
3
G
H
4
2
G
H
2. Eliminate loop I and Simplify
II
III
443
432
1 HGG
GGG
1G
)(sY
1H
B
4
2
G
H
)(sR
4
3
G
H
II
332443
432
1 HGGHGG
GGG
III
4
142
G
HGH
Not feedback feedback
)(sR )(sY
4
142
G
HGH
332443
4321
1 HGGHGG
GGGG
3. Eliminate loop II & IIII
143212321443332
4321
1 HGGGGHGGGHGGHGG
GGGG
sR
sY
)(
)(
Example-10: Reduce the Block Diagram.
Example-10: Continue.
Example-11: Simplify the block diagram then obtain the close-
loop transfer function C(S)/R(S). (from Ogata: Page-47)
Example-11: Continue.
Superposition of Multiple Inputs
Example-12: Multiple Input System. Determine the output C
due to inputs R and U using the Superposition Method.
Example-12: Continue.
Example-12: Continue.
Example-13: Multiple-Input System. Determine the output C
due to inputs R, U1 and U2 using the Superposition Method.
Example-13: Continue.
Example-13: Continue.
Example-14: Multi-Input Multi-Output System. Determine C1
and C2 due to R1 and R2.
Example-14: Continue.
Example-14: Continue.
When R1 = 0,
When R2 = 0,
Block Diagram of Armature Controlled D.C Motor
Va
ia T
Ra La
J
c
eb
(s)IK(s)cJs
(s)V(s)K(s)IRsL
am
abaaa
Block Diagram of Armature Controlled D.C Motor
(s)V(s)K(s)IRsL abaaa
Block Diagram of Armature Controlled D.C Motor
(s)IK(s)cJs ama
Block Diagram of Armature Controlled D.C Motor
END OF LECTURES-06
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URL :http://electronics106.jimdo.com/
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