lecture-3 block diagram reduction - aastmt | homepage
TRANSCRIPT
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Automatic Control Systems
Lecture-3Block Diagram Reduction
1
Emam FathyDepartment of Electrical and Control Engineering
email: [email protected]
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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 mathematicaloperation to be performed on the input to yield the output.
• The arrows represent the direction of information or signal flow.
dt
dx y
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Introduction• The operations of addition and subtraction have a special
representation.
• The block becomes a small circle, called a summing point, withthe appropriate plus or minus sign associated with the arrowsentering 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.
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Introduction
• In order to have the same signal or variable be an inputto more than one block or summing point, a takeoffpoint is used.
• This permits the signal to proceed unaltered alongseveral different paths to several destinations.
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Example-1• Consider the following equations in which x1, x2, x3, are variables,
and a1, a2 are general coefficients or mathematical operators.
522113 xaxax
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Canonical Form of A Feedback Control System
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Characteristic Equation
• The control ratio is the closed loop transfer function of the system.
• The denominator of closed loop transfer function determines thecharacteristic equation of the system.
• Which is usually determined as:
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
01 )()( sHsG
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Example-4
)(sG
)(sH
1. Open loop transfer function
2. Feed Forward Transfer function
3. closed loop transfer function
4. characteristic equation
)()(
)(sG
sE
sC
)()(
)(
)(
)(
sHsG
sG
sR
sC
1
01 )()( sHsG
)()()(
)(sHsG
sE
sB
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In order to analyze the system, we want to
represent multiple subsystems as a single transfer
function.
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Reduction techniques
2G1G21GG
1. Combining blocks in cascade
1G
2G21 GG
2. Combining blocks in parallel (feed-forward)
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3. Eliminating a feedback loop
G
H
B
A
H
OR
𝑮
𝟏 ∓ 𝑮𝑯
𝑨
𝑩 ∓ 𝑨𝑯
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5. Moving a pickoff point ahead of a block
G G
G G
G
1
G
4. Moving a pickoff point behind a block
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6. Moving a summing point ahead of a block
G G
G
1
7. Moving a summing point behind a block
G G
G
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Note
A B AB
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Example 1
R
_+
_
+1G 2G 3G
1H
2H
++
C
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Example
R
_+
_
+1G 2G 3G
1H
1
2
G
H
++
C
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Example
R
_+
_
+21GG 3G
1H
1
2
G
H
++
C
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Example
R
_+
_
+21GG 3G
1H
1
2
G
H
++
C
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Example
R
_+
_
+121
21
1 HGG
GG
3G
1
2
G
H
C
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Example
R
_+
_
+121
321
1 HGG
GGG
1
2
G
H
C
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Example
R
_+232121
321
1 HGGHGG
GGG
C
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Example
R
321232121
321
1 GGGHGGHGG
GGG
C
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Example 2
Find the transfer function of the following block diagrams
2G 3G1G
4G
1H
2H
)(sY)(sR
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1. Moving pickoff point A ahead of block2G
2. Eliminate loop I & simplify
324 GGG B
1G
2H
)(sY4G
2G
1H
AB3G
2G
)(sR
I
Solution:
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3. Moving pickoff point B behind block324 GGG
1GB
)(sR
21GH 2H
)(sY
)/(1 324 GGG
II
1GB
)(sRC
324 GGG
2H
)(sY
21GH
4G
2G A3G 324 GGG
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4. Eliminate loop III
)(sR
)(1
)(
3242121
3241
GGGHHGG
GGGG
)(sY
)()(1
)(
)(
)()(
32413242121
3241
GGGGGGGHHGG
GGGG
sR
sYsT
)(sR
1GC
324
12
GGG
HG
)(sY324 GGG
2H
C
)(1 3242
324
GGGH
GGG
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2G1G
1H 2H
)(sR )(sY
3H
Example 3
Find the transfer function of the following block diagrams
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Solution:
1. Eliminate loop I
2. Moving pickoff point A behind block
22
2
1 HG
G
1G
1H
)(sR )(sY
3H
BA
22
2
1 HG
G
2
221
G
HG
1G
1H
)(sR )(sY
3H
2G
2H
BA
II
I
22
2
1 HG
G
Not a feedback loop
)1
(2
2213
G
HGHH
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3. Eliminate loop II
)(sR )(sY
22
21
1 HG
GG
2
2213
)1(
G
HGHH
21211132122
21
1)(
)()(
HHGGHGHGGHG
GG
sR
sYsT
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2G4G1G
4H
2H
3H
)(sY)(sR
3G
1H
Example 4
Find the transfer function of the following block diagrams
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Solution:
2G4G1G
4H
)(sY
3G
1H
2H
)(sR
A B
3H4
1
G
4
1
G
I1. Moving pickoff point A behind block
4G
4
3
G
H
4
2
G
H
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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 feedbackfeedback
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)(sR )(sY
4
142
G
HGH
332443
4321
1 HGGHGG
GGGG
3. Eliminate loop II & IIII
143212321443332
4321
1)(
)()(
HGGGGHGGGHGGHGG
GGGG
sR
sYsT
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3G1G
1H
2H
)(sR )(sY
4G
2GA
B
Example 5
Find the transfer function of the following block diagrams
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Solution:
1. Moving pickoff point A behind block3G
I
1H3
1
G
)(sY1G
1H
2H
)(sR
4G
2GA B
3
1
G
3G
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2. Eliminate loop I & Simplify
3G
1H
2G B
3
1
G
2H
32GG B
2
3
1 HG
H
1G)(sR )(sY
4G
3
1
G
H
23212
32
1 HGGHG
GG
II
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)(sR )(sY
12123212
321
1 HGGHGGHG
GGG
3. Eliminate loop II
12123212
3214
1)(
)()(
HGGHGGHG
GGGG
sR
sYsT
4G
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End of Lec 3
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Example-5: Continue.
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Example-6: Reduce the Block Diagram.
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Example-6: Continue.
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Example-7: Reduce the Block Diagram. (from Nise: page-242)
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Example-7: Continue.
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Example-8: Reduce the system to a single transfer function. (from Nise:page-243).
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Example-9: Simplify the block diagram then obtain the close-loop transfer function C(S)/R(S). (from Ogata: Page-47)
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Example-10: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method.
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Example-11: Continue.
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Example-11: Continue.
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Example-19: Multiple-Input System. Determine the output C due to inputs R, U1 and U2 using the Superposition Method.
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Example-19: Continue.
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Example-19: Continue.
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Example-20: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2.
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Example-20: Continue.
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Example-20: Continue.
When R1 = 0,
When R2 = 0,
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Example-5: Reduce the Block Diagram to Canonical Form.
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Example-5: Continue.
However in this example step-4 does not apply.
However in this example step-6 does not apply.
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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=10.
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Example-6– First we will reduce the given block diagram to canonical form
1s
K
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Example-6
1s
K
ss
Ks
K
GH
G
11
1
1
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Example-7• 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.
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Block Diagram of Armature Controlled D.C Motor
Va
iaT
Ra La
J
c
eb
(s)IK(s)cJs
(s)V(s)K(s)IRsL
ama
abaaa
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Block Diagram of Armature Controlled D.C Motor
(s)E(s)K(s)IRsL abaaa
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Block Diagram of Armature Controlled D.C Motor
(s)IK(s)cJs ama
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Block Diagram of Armature Controlled D.C Motor
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Block Diagram of liquid level system
11
1 qqdt
dhC
1
211
R
hhq
212
2 qqdt
dhC
2
22
R
hq
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Block Diagram of liquid level system
)()()( sQsQssHC 111
1
211
R
sHsHsQ
)()()(
2
22
R
sHsQ
)()( )()()( sQsQssHC 2122
11
1 qqdt
dhC
1
211
R
hhq
212
2 qqdt
dhC
2
22
R
hq
L
L
L
L
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Block Diagram of liquid level system
)()()( sQsQssHC 111
1
211
R
sHsHsQ
)()()(
2
22
R
sHsQ
)()( )()()( sQsQssHC 2122
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Block Diagram of liquid level system