lec-45 control system engineering
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Block Diagram fundamentals &
reduction techniquesLect# 4-5
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Introduction
• Block diagram is a shorthand, graphicalrepresentation of a physical system, illustratingthe functional relationships among itscomponents.
OR
• Block !iagram is a shorthand pictorialrepresentation of the cause-and-effectrelationship of a system.
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Introduction
• "he simplest form of the lock diagram is the single block,with one input and one output.
• "he interior of the rectangle representing the lock usually
contains a description of or the name of the element, or thesymol for the mathematical operation to e performed onthe input to yield the output.
• "he arro$s represent the direction of information or signal
flo$.
dt
d x y
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Introduction• "he operations of addition and sutraction ha%e a special
representation.• "he lock ecomes a small circle, called a summing point,
$ith the appropriate plus or minus sign associated $ith thearro$s entering the circle.
• ny numer of inputs may enter a summing point.
• "he output is the algeraic sum of the inputs.
•&ome ooks put a cross in the circle.
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Components of a Block Diagram for
a Linear Time Invariant System• &ystem components are alternati%ely called
elements of the system.
• Block diagram has four components'( Signals
( System/ block
( Summing junction
( Pick-off/ Take-off point
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• )n order to ha%e the same signal or %ariale e an input to
more than one lock or summing point, a takeoff point isused.
• !istriutes the input signal, undiminished, to se%eraloutput points.
• "his permits the signal to proceed unaltered along se%eraldifferent paths to se%eral destinations.
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Eample!"
• *onsider the follo$ing e+uations in $hich -, ., /, are
%ariales, and a1, a2 are general coefficients or
mathematical operators.
522113 −+= xa xa x
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Eample!"
• *onsider the follo$ing e+uations in $hich -, ., /, are
%ariales, and a1, a2 are general coefficients or
mathematical operators.
522113
−+= xa xa x
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Eample!#
• *onsider the follo$ing e+uations in $hich -, .,. . . , n, are %ariales, and a1, a2,. . . , an , are general coefficients or
mathematical operators.
112211 −−++=
nnn xa xa xa x
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Eample!$
• !ra$ the Block !iagrams of the follo$ing e+uations.
1
1
2
2
2
13
1
1
12
32
11
bxdt
dx
dt
xd a x
dt x
bdt
dxa x
−+=
∫ +=
)(
)(
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Topologies
• 0e $ill no$ eamine some common topologiesfor interconnecting susystems and deri%e thesingle transfer function representation for eachof them.
• "hese common topologies $ill form the asis for
reducing more complicated systems to a single lock.
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CASCADE
• ny finite numer of locks in series may ealgeraically comined y multiplication oftransfer functions.
• "hat is, n components or blocks with transfer functions G 1 , G 2, . . . , G n, connected in cascade
are e+ui%alent to a single element 1 $ith atransfer function gi%en y
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Eample
• 2ultiplication of transfer functions iscommutative; that is,
GiGj = GjGi for any i or j .
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Cascade%
Figure:a) Cascaded Subsystems.
b) Equivalent Transfer Function.
The equivalent transfer function
is
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arallel 'orm%
• 3arallel susystems ha%e a common input and anoutput formed y the algeraic sum of the outputsfrom all of the susystems.
Figure: Parallel Subsystems.
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arallel 'orm%
Figure:
a) Parallel Subsystems.
b) Equivalent Transfer Function.
The equivalent transfer function is
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'eed(ack 'orm%• "he third topology is the feedack form. Let us deri%e the
transfer function that represents the system from its inputto its output. "he typical feedack system, sho$n in figure'
Figure: Feedback Closed !oo") Control System.
The system is said to have negative feedback if the sign at the
summing junction is negative and positive feedback if the signis positive.
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'eed(ack 'orm%
Figure:
a)Feedback Control System.
b)Sim"lified #odel or Canonical Form.
c) Equivalent Transfer Function.
The equivalent or closed$loo"
transfer function is
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Characteristic Equation% The control ratio is the closed loo" transfer function of the
system.
% The denominator of closed loo" transfer function determines the
characteristic equation of the system.
% &hich is usually determined as:
)()(
)(
)(
)(
s H sG
sG
s R
sC
±=1
01 =± )()( s H sG
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(. "en loo" transfer function
*. Feed For+ard Transfer function
,. control ratio
-. feedback ratio
. error ratio
/. closed loo" transfer function
0. characteristic equation
1. closed loo" "oles and 2eros if 34(5.
)()()(
)( s H sG
s E
s B=
)()(
)( sG
s E
sC =
)()(
)(
)(
)(
s H sG
sG
s R
sC
+=1
)()()()(
)()(
s H sG s H sG
s R s B
+=1
)()()(
)(
s H sG s R
s E
+=1
1
)()()(
)()(
s H sG sG
s R sC
+=1
01 =+ )()( s H sG
)( sG
)( s H
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Characteristic Equation
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)nity 'eed(ack System
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*eduction techniques
2G1G 21GG
(. Combining blocks in cascade
1G
2G
21 GG +
*. Combining blocks in "arallel
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*eduction techniques
,. #oving a summing "oint behind a block
G G
G
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. #oving a "ickoff "oint ahead of a block
G G
G G
G
1
G
,. #oving a summing "oint ahead of a block
G G
G
1
-. #oving a "ickoff "oint behind a block
*eduction techniques
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/. Eliminating a feedback loo"
G
H GH
G
1
0. S+a" +ith t+o neighboring summing "oints
A B A B
G
1= H
G
G
1
*eduction techniques
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Block Diagram Transformation Theorems
The letter P is used to represent any transfer function, and W, X ,
Y, Z denote any transformed signals.
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*eduction of Complicated Block Diagrams%
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Eample!+% Continue,
6o+ever in this e7am"le ste"$- does not a""ly.
6o+ever in this e7am"le ste"$/ does not a""ly.
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Eample!-% Simplify the Block Diagram,
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Eample!-% Continue,
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Eample!.% *educe the Block Diagram,
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Eample!.% Continue,
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Eample!/% *educe the Block Diagram, 0from 1ise% page!
#+#2
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Eample!/% Continue,
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Eample!3% 'or the system represented (y thefollo4ing (lock diagram determine%
. Open loop transfer function
. eed or$ard "ransfer function
/. control ratio
4. feedack ratio
5. error ratio
. closed loop transfer function
6. characteristic e+uation
7. closed loop poles and 8eros if 9:;.
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Eample!3% Continue
( irst $e $ill reduce the gi%en lock diagram to canonical
form
1+ s
K
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Eample!3% Continue
1+ s
K
s s
K
s
K
GH
G
1
1
1
1
+
+
+=
+
Eample 3% Continue
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Eample!3% Continue
(. "en loo" transfer function
*. Feed For+ard Transfer function
,. control ratio
-. feedback ratio
. error ratio
/. closed loo" transfer function
0. characteristic equation
1. closed loo" "oles and 2eros if 34(5.
)()()(
)( s H sG
s E
s B=
)()(
)( sG s E
sC =
)()(
)(
)(
)(
s H sG
sG
s R
sC
+=1
)()(
)()(
)(
)(
s H sG
s H sG
s R
s B
+= 1
)()()(
)(
s H sG s R
s E
+=1
1
)()(
)(
)(
)(
s H sG
sG
s R
sC
+= 1
01 =+ )()( s H sG
)( sG
)( s H
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• Eample!5% 'or the system represented (y the follo4ing(lock diagram determine%
. Open loop transfer function. eed or$ard "ransfer function
/. control ratio
4. feedack ratio
5. error ratio
. closed loop transfer function6. characteristic e+uation
7. closed loop poles and 8eros if 9:;;.
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Eample!"6% *educe the system to a single transfer
function, 0from 1ise%page!#+$2,
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Eample!"6% Continue,
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Eample!"6% Continue,
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Eample!""% Simplify the (lock diagram then o(tain the
close!loop transfer function C0S27*0S2, 0from 8gata%
age!+/2
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Eample!"#% *educe the Block Diagram,
R
8 9
8
91G 2G 3G
1 H
2 H
99
C
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Eample!"#%
R
8 9
8
9 1G 2G 3G
1 H
1
2
G H
99
C
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Eample!"#%
R
8 9
8
9 21GG 3G
1 H
1
2
G H
99
C
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Eample!"#%
R
8 9
8
9 21GG 3G
1 H
1
2
G H
99
C
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Eample!"#%
R
8 9
8
9121
21
1 H GG
GG
− 3G
1
2
G H
C
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Eample!"#%
R
8 9
8
9121
321
1 H GG
GGG
−
1
2
G H
C
l #
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Eample!"#%
R
8 9
232121
321
1 H GG H GG
GGG
+−
C
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Eample!"#%
R
321232121
321
1 GGG H GG H GG
GGG
++−
C
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2G1G
1 H 2 H
)( s R )( sY
3 H
Eample "$% 'ind the transfer function of the follo4ing
(lock diagrams,
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Solution:
(. Eliminate loo"
*. #oving "ickoff "oint ; behind block22
2
1 H G
G
+
1G
1 H
)( s R )( sY
3 H
B A
22
2
1 H G
G
+
2
221
G
H G+
1G
1 H
)( s R )( sY
3 H
2G
2 H
B A
II
I
22
2
1 H G
G
+
<ot a feedback loo"
)1(
2
2213
G
H G H H +
+
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,. Eliminate loo"
)( s R )( sY
22
21
1 H G
GG
+
2
2213
)1(
G
H G H H
++
21211132122
21
1 H H GG H G H GG H G
GG
s R
sY
++++
=
)(
)(
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Superposition of 9ultiple Inputs
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Eample!"+% 9ultiple Input System, Determine the
output C due to inputs R and U using the Superposition9ethod,
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Eample!"+% Continue,
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Eample!"+% Continue,
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Eample!"-% 9ultiple!Input System, Determine the
output C due to inputs R: U " and U # using the
Superposition 9ethod,
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Eample!"-% Continue,
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Eample!"-% Continue,
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Eample!".% 9ulti!Input 9ulti!8utput System, Determine
C" and C# due to R" and R#,
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Eample!".% Continue,
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Skill ;ssessment Eercise%
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;ns4er of Skill ;ssessment Eercise%