‫بسم ا‪ ...‬الرحمن الرحيم‬
‫سیستمهای کنترل خطی‬
‫پاییز ‪1389‬‬
‫دکتر حسین بلندي‪ -‬دکتر سید مجید اسما عیل زاده‬
Recap.
• State Space Equation:
• Physical System,
• Phase Variable,
• Canonical Forms,
Control Systems
2
‫‪-3‬نمايش معادالت فضاي حالت توسط فرمهاي كانوليكال‬
‫هدف ‪ :‬با فرض مشخص بودن تابع تبديل سيستم‪ ،‬تحقق های فضای حالت که از اهميت ويژه ای بر خوردار هستند را بدست می‬
‫آوريم‪.‬‬
‫الف) فرم كانونيكي كنترلپذير‬
‫ب) فرم كانونيكي مشاهدهپذير‬
‫ج) فرم كانونيكي قطري (جردن)‬
‫اين تحقق ها عبارتند از‪:‬‬
‫تابع تبديل زير را در نظر می گيريم‪:‬‬
‫‪ ...  b n  1 s  b n‬‬
‫‪ ...  a n‬‬
‫تبديل الپالس ورودي ‪:‬‬
‫‪u (s) ‬‬
‫‪n2‬‬
‫‪n2‬‬
‫‪ b2 s‬‬
‫‪ a2 s‬‬
‫‪n 1‬‬
‫‪n 1‬‬
‫‪b 0 s  b1 s‬‬
‫‪n‬‬
‫‪s  a1 s‬‬
‫‪n‬‬
‫‪‬‬
‫تبديل الپالس خروجي ‪:‬‬
‫)‪y(s‬‬
‫‪G (s) ‬‬
‫)‪u (s‬‬
‫‪y (s) ‬‬
‫فرم کانونيکی کنترل پذير‬
‫مدل فضای حالت ‪:‬‬
‫‪0  0 ‬‬
‫‪  ‬‬
‫‪... 0   0 ‬‬
‫‪x‬‬
‫‪u‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪1‬‬
‫‪‬‬
‫‪  ‬‬
‫‪...  a 1   1 ‬‬
‫‪...‬‬
‫‪0‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪ a n2‬‬
‫‪ a n 1‬‬
‫‪ 0‬‬
‫‪‬‬
‫‪ 0‬‬
‫‪‬‬
‫‪x ‬‬
‫‪‬‬
‫‪‬‬
‫‪ a‬‬
‫‪ n‬‬
‫) ‪y  [ b n  a n b0 b n  1  a n  1 b0 ... b1  a 1 b0 ] x  b0 u ( t‬‬
‫ويژگی ها ‪:‬‬
‫‪ )1‬اين تحقق همواره کنترل پذير است‪.‬‬
‫‪ )2‬در صورتيکه تابع تبديل سيستم‪ ،‬قطب و صفر مشترکی نداشته باشند‪ ،‬اين تحقق رويت پذير خواهد بود‪.‬‬
‫فرم کانونيکی رويت پذير‬
‫مدل فضای حالت ‪:‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪u‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪ an ‬‬
‫‪ b n  a n b0‬‬
‫‪‬‬
‫‪‬‬
‫‪ a n 1 ‬‬
‫‪ b n 1  a n 1 b0‬‬
‫‪x‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪ ‬‬
‫‪‬‬
‫‪‬‬
‫‪ b a b‬‬
‫‪ a 1 ‬‬
‫‪ 1‬‬
‫‪1 0‬‬
‫) ‪1 ) x  b0 u ( t‬‬
‫‪...‬‬
‫‪...‬‬
‫‪0‬‬
‫‪...‬‬
‫‪0‬‬
‫‪0‬‬
‫‪0‬‬
‫‪0‬‬
‫ويژگی ها ‪:‬‬
‫‪ )1‬اين تحقق همواره رويت پذير است‪.‬‬
‫‪ )2‬در صورتيکه تابع تبديل سيستم‪ ،‬قطب و صفر مشترکی نداشته باشند‪ ،‬اين تحقق کنترل پذير خواهد بود‪.‬‬
‫‪0‬‬
‫‪0‬‬
‫‪‬‬
‫‪1‬‬
‫‪‬‬
‫‪x ‬‬
‫‪‬‬
‫‪0‬‬
‫‪‬‬
‫‪y ( 0‬‬
‫فرم کانونيکی قطری (جردن)‬
‫‪1   2 ...   n‬‬
‫حالت اول ‪ :‬اگر مقادير ويژه سيستم‪ ،‬حقيقی و غير تکراری باشند‪.‬‬
‫مدل فضای حالت ‪:‬‬
‫‪‬‬
‫‪1‬‬
‫‪‬‬
‫‪ ‬‬
‫‪‬‬
‫‪1‬‬
‫‪ x    u‬‬
‫‪‬‬
‫‪ ‬‬
‫‪1‬‬
‫‪‬‬
‫‪n ‬‬
‫‪ ‬‬
‫) ‪c n ) x  b 0 u (t‬‬
‫ويژگی ها ‪:‬‬
‫‪ )1‬اين تحقق همواره کنترل پذير است‪.‬‬
‫‪ )2‬در صورتيکه‬
‫‪0‬‬
‫باشند‪ ،‬اين تحقق رويت پذير خواهد بود‪.‬‬
‫‪c ‬‬
‫‪i‬‬
‫‪0‬‬
‫‪2‬‬
‫‪...‬‬
‫‪c2‬‬
‫‪ 1‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪x ‬‬
‫‪0‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪y  ( c1‬‬
‫حالت دوم ‪ :‬اگر تعدادی از مقادير ويژه سيستم‪ ،‬حقيقی و تکراری باشند‪.‬‬
‫‪1   2   3‬‬
‫‪ 4   5   6  ...   n‬‬
‫ز‬
‫مدل فضای حالت ‪:‬‬
‫‪0 ‬‬
‫‪0‬‬
‫‪‬‬
‫‪ ‬‬
‫‪‬‬
‫‪0‬‬
‫‪‬‬
‫‪0‬‬
‫‪ x   u‬‬
‫‪‬‬
‫‪1 ‬‬
‫‪‬‬
‫‪ ‬‬
‫‪1‬‬
‫‪‬‬
‫‪ ‬‬
‫‪1 ‬‬
‫‪ n ‬‬
‫‪ ‬‬
‫‪4‬‬
‫‪5‬‬
‫‪0‬‬
‫‪1‬‬
‫‪1‬‬
‫‪1‬‬
‫‪1‬‬
‫‪0‬‬
‫‪ 1‬‬
‫‪‬‬
‫‪ 0‬‬
‫‪ 0‬‬
‫‪‬‬
‫‪x ‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪ 0‬‬
‫‪‬‬
‫) ‪y  ( c 1 c 2 c 3  c 4 ... c n ) x  b 0 u ( t‬‬
‫ويژگی ها ‪:‬‬
‫‪ )1‬اين تحقق همواره کنترل پذير است اگر و فقط اگر آخرين سطر بلوکهای جردن مربوط به هر مقدار ويزه تکراری‪ ،‬در ماتريس ‪ B‬مستقل خطی (اگر تنها‬
‫يک بردار باشد‪ ،‬مخالف صفر) باشند ‪.‬‬
‫‪ )2‬اين تحقق همواره رويت پذير است اگر و فقط اگر اولين ستون بلوکهای جردن مربوط به هر مقدار ويزه تکراری‪ ،‬در ماتريس ‪ C‬مستقل خطی (اگر تنها‬
‫يک بردار باشد‪ ،‬مخالف صفر) باشند ‪.‬‬
‫بدست آوردن تابع تبديل از معادالت فضاي حالت‬
‫حالت اول ‪ :‬سيستمهای تک ورودی – تک خروجی )‪(SISO‬‬
‫) ‪ S x ( s )  x ( 0 )  Ax ( s )  Bu ( s‬‬
‫‪‬‬
‫) ‪Y ( s )  C x ( s )  Du ( s‬‬
‫) ‪Bu ( s‬‬
‫‪1‬‬
‫‪x  Ax  Bu‬‬
‫تبديل الپالس‬
‫‪y  Cx  Du‬‬
‫) ‪X ( S )  ( SI  A‬‬
‫)‪B  D ) u (s‬‬
‫‪1‬‬
‫) ‪y ( s )  ( C ( SI  A‬‬
‫يعني مقادير ويژه ماتريس ‪ A‬فيالواقع همان قطبهاي سيستم‬
‫ميباشند‪.‬‬
‫تابع تبديل‬
‫) ‪Q (S‬‬
‫‪SI  A‬‬
‫‪BD‬‬
‫‪1‬‬
‫) ‪G ( S )  C ( SI  A‬‬
‫‪G (S ) ‬‬
 x 1   5 x 1  x 2  2 u
  5  1  2 

 x    u
 x  

 3  1  5 
 x 2  3 x 1  x 2  5 u
y  x1  2 x 2
2  5
( SI  A )  
 3


S  1 
1
: ‫ تابع تبديل سيستم زير را بدست آوريد‬: ‫مثال‬
y  (1
2) x
( SI  A )
1

S 1

( S  5 ) ( S  1)  3  3
    
1
1 

S  5 
  (S 2) (S 4)
G ( S )  [1
1 S  1
2] 
  3
G (s) 
 1  2
 
S  5  5 
12 S  59
(S  2) (S  4)
‫حالت دوم ‪ :‬سيستمهای چند ورودی – چند خروجی )‪(MIMO‬‬
‫اگر بردار ورودي ‪ m ،u‬بعدي و بردار خروجي ‪ l ، y‬بعدي باشد‪ ،‬آنگاه ماتريس ‪G‬‬
‫عبارت است از ‪:‬‬
‫‪G1m (s) ‬‬
‫‪‬‬
‫‪G 2 m (s) ‬‬
‫‪‬‬
‫‪‬‬
‫‪G l m ( s ) ‬‬
‫‪...‬‬
‫‪...‬‬
‫‪...‬‬
‫) ‪ G 11 ( S‬‬
‫‪‬‬
‫) ‪ G 21 ( S‬‬
‫‪G ‬‬
‫‪‬‬
‫‪‬‬
‫)‪ G (s‬‬
‫‪ l1‬‬
‫) ‪Gi j (S‬‬
‫‪ ،‬تبديلي است كه‬
‫در واقع عنصر )‪ (i , j‬ام از تابع ‪، G‬‬
‫خروجي ‪ i‬ام را به ورودي ‪ j‬ام مربوط ميسازد‪.‬‬
‫بنابراين ‪:‬‬
‫)‪y(S )  G (s) u (s‬‬
‫‪BD‬‬
‫‪1‬‬
‫) ‪G ( S )  C ( SI  A‬‬
Outline Block Diagram
•
•
•
Terms and concepts
Canonical form of a feedback control system
Block diagram transformations
11
Block diagrams
• Block diagrams consist of unidirectional,
operational blocks that represent the
transfer function of the variables of
interest.
• The block diagram representation of a
given system often can be reduced to a
simplified block diagram with fewer blocks
than original diagram.
12
Introduction
• A graphical tool can help us to visualize the model
of a system and evaluate the mathematical
relationships between their elements, using their
transfer functions.
• In many control systems, the system of equations
can be written so that their components do not
interact except by having the input of one part be
the output of another part.
13
14
©Oxford University Press 2001
A block diagram is a shorthand, pictorial
representation of the cause-and-effect relationship
between the input and output of a physical system. It
provides a convenient and useful method for
characterizing the ‘functional relationships among the
various components of a control system.
The arrows represent the direction of information or
signal flow.
15
Component Block Diagram
16
Block Diagram
• It represents the mathematical relationships between the
elements of the system.
U 1 ( s ) G 1 ( s )  Y1 ( s )
• The transfer function of each component is placed in box, and
the input-output relationships between components are
indicated by lines and arrows.
17
Block Diagram Algebra
• We can solve the equations by graphical simplification,
which is often easier and more informative than algebraic
manipulation, even though the methods are in every way
equivalent.
• The interconnections of blocks include summing points, where
any number of signals may be added together.
18
1st & 2nd Elementary Block
Diagrams
• Blocks in series:
Y2 ( s )
U 1( s )
 G 1G 2
• Blocks in parallel with
their outputs added:
Y ( s)
U ( s)
 G1  G 2
19
Combining blocks in cascade
20
3rd Elementary Block Diagram
• Single-loop negative feedback
• Transfer function
Y( s )
R( s )
Two blocks are connected in a
feedback arrangement so that
each feeds into the other.

G1
1  G 1G 2
21
• Proof:
x
+
e
b
G
1
y

G1
x
1  G 1G 2
G
2
y
e  x  b , b  G 2 y , y  G 1e

e  x  G 2 G 1e
(1  G 1G 2 ) e  x  e 
y
1
1  G 1G 2
x
y 
G1
x
1  G 1G 2
G1
1  G 1G 2
x
+
+
G
1
G
2
G1
1  G 1G 2
n1
+
-
d1
n1 d 2
n2
n1 n 2  d 1 d 2
d2
Quarter car suspension
Series
R(s
bs  k
+
-
R(s
)
R(s
)
1
1
1
m
s
s
bs  k
+
-
ms
bs  k
ms  bs  k
2
y
y
2
Feedback
y
TF  H ( s ) 
bs  k
ms  bs  k
2
1st Elementary Principle of Block
Diagram Algebra
25
2nd Elementary Principle of Block
Diagram Algebra
26
3rd Elementary Principle of Block
Diagram Algebra
27
28
©Oxford University Press 2001
Example 1
T( s ) 
Y( s )
R( s )
2s  4
2
T( s ) 
1
s
2s  4
s
T( s ) 
2
2s  4
s  2s  4
2
29
Example 2: Find TF from U to Y:
2
s5
U
+
100
-
+
-
s 1
10
s2
s ( s  20)
+
+
• No pure series/parallel/feedback
• Needs to move a block, but which one?
Key: move one block to create pure series or
parallel or feedback!
10
So move
either left or right.
s ( s  20)
Y
U+
100 +
-
U+
-
s2
s ( s  20)
10
s5
+
Y
+
10( s  1)
s ( s  20)
s ( s  2)( s  20)
5( s  5)
10( s  1)
s ( s  20)
s ( s  2)( s  20)  10( s  1)
5( s  5)
+
1
Y
+
-
100
-
10
2
-
100 +
U+
s 1
s ( s  20)
+
1
+
Y
All the transformations can be derived
by simple algebraic manipulation of the
equations representing the blocks.
37
Ex. 3 Block diagram reduction
38
39
40
Example 4
T( s ) 
G 1G 2 G 5  G 1G 6
1  G 1G 3  G 1G 2 G 4
41
Example 5
Can use superposition:
First set D=0, find Y due to R
Then set R=0, find Y due to D
Finally, add the two component to get the overall Y
First set D=0, find Y due to R
Y1 ( s ) 
G1G 2
1  G 1G 2 H 1
R (s)
Then set R=0, find Y due to D
Y2 ( s ) 
G2
G2
1  G 1G 2 H 1
  D (s) 
Finally, add the two component to get the overall Y
Y (s) 
G1G 2
1  G1G 2 H 1
R (s) 
G2
1  G 1G 2 H 1
D (s)
Summary
• Using transfer function notations, block
relationships were obtained.
46
Signal-Flow Graph Models
Outline
•
•
•
Terms and concepts
Mason’s signal-flow gain formula
Numerical examples
Control Systems
48
A signal-flow graph
• A diagram consisting of nodes that are
connected by several directed branches.
• A graphical representation of a set of linear
relations.
Control Systems
49
The basic elements of a signalflow graph
• branch - a unidirectional path segment,
which relates the dependency of an input
and an output variable.
• nodes - the input and output points or
junctions.
• path - a branch or continuous sequence of
branches that can be traversed from one
node to another node.
Control Systems
50
• All branches leaving a node will pass the
nodal signal to the output node of each
branch ( uniderectionally ).
• The summation of all signals entering a
node is equal to the node variable.
• The relation between each variable is
written next to the directional arrow.
Control Systems
51
• A loop - a closed path that originates and
terminates on the same note, and along
the path no node is met twice.
• Two loops are said to be nontouching if
they do not have a common node.
• Two touching loops share one or more
common nodes.
Control Systems
52
Block and branch of DC motor
Control Systems
53
Two-input, two-output system
Control Systems
54
Y1(s) = G11(s) R1(s) + G12(s) R2(s)
Y2(s) = G21(s) R1(s) + G22(s) R2(s)
Gik - transfer function relating the i-th output to
the k-th input
Control Systems
55
Interconnected system
Control Systems
56
Control Systems
57
• The flow graph is a pictorial method of
writing a system of algebraic equations so
as to indicate the interdependencies of the
variables.
Control Systems
58
A set of simultaneous equations
• 1. Write the system equations in the form
X1 = A11 X1 + A12 X2 + …+ A1n Xn
X2 = A21 X1 + A22 X2 + …+ A2n Xn
….……………………………………
Xm= Am1 X1 + Am2 X2 + …+ Amn Xn
Note: An equation for X1 is not required if X1 is an input
node.
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• 2. Arrange the m or n (whichever is larger) nodes
from left to right.
• 3. Connect the nodes by the appropriate
branches A11, A12, etc.
• 4. If the desired output node has outgoing
branches, add a dummy note and unity branch.
• 5. Rearrange the nodes and/or loops in the graph
to achieve pictorial clarity.
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Set of simultaneous algebraic equations
a11 x1 + a12 x2 + r1 = x1
a21 x1 + a22 x2 + r2 = x2
r1, r2 - input variables
x1, x2 - output variables
x1 (1 - a11) + x2
(- a12) = r1
x1 ( - a21) + x2 (1 - a22) = r2
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Mason’s signal-flow gain formula
Tij)s( = ∑kPijk ∆ijk /∆
Pijk = k-th path from variable xi to variable xj
∆ = determinant of the graph
∆ijk = cofactor of the path Pijk
∑k = all possible k path from xi to xj
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Step by step construction
y2 = a12 y1 + a32 y3
y3 = a23 y2 + a43 y4
y4 = a24 y2 + a34 y3 + a44 y4
y5 = a25 y2 + a45 y4
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• The nodes representing the variables y1,y2,y3,y4
and y5 are located in order from left to right.
• The first equation states that y2 depends upon
two signals a11 y1 and a32 y3.
• The second equation states that y3 depends upon
the sum of a23 y2 and a43 y4,therefore a branch of
gain a23 is drawn from node y2 to y3 and a branch
of gain a43 is drawn from y4 to y3, with directions
of the branches indicated by arrows.
• Similarly, with consideration of the third and
fourth equation.
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Mason’s signal-flow gain formula
∆
= 1 - (sum of all different loop gains)
+ ( sum of the gain products of all
combinations of two nontouching loops)
- ( sum of the gain products of all
combinations of three nontouching loops)
+ …,
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∆ijk = cofactor of the path Pijk is the the
determinant with the loops touching the
k-th path removed.
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T(s) = Y(s)/R(s)
T)s( = Y)s(/R)s( = ∑kPk ∆k /∆
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• The path gain or transmittance Pk is defined
as continuous succession of branches that
are traversed in the direction of the arrows
and with no node encountered more than
once.
• A loop is defined as a closed path in which
no node is encountered more than ones per
traversal.
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Ex. 2.8 Interacting system
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The paths connecting the input
R(s) and output Y(s)
path 1
P1= G1 G2 G3 G4
path 2
P2= G5 G6 G7 G8
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Four self-loops
•
•
•
•
L1 = G2 H2
L2 = H3 G3
L3 = G6 H6
L4 = G7 H8
• Loops L1 and L2 do not touch L3 and
L4
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The determinant :
∆ = 1 - (L1 + L2 + L3 + L4) + ( L1L3 +L1L4 +
L2L3 + L2L4)
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The cofactor along path 1 is evaluated
by removing the loops that touch path 1
from ∆.
L1 = L 2 = 0
∆1 = 1 - (L3 +L4)
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The cofactor along path 2 is evaluated
by removing the loops that touch path 2
from ∆.
• L3 = L4 = 0
∆2 = 1 - (L1 +L2)
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The transfer function of the
system
• T(s) = Y(s)/R(s) = (P1 ∆1 + P2 ∆2 (/∆
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Ex. 2.7 Block diagram reduction
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Ex. 2.7 Mason’s signal-flow gain
P1 = G1 G2 G3 G4
L1 = - G2 G3 H2
L2 = G3 G4 H1
L3 = - G1 G2 G3 G4 H3
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• All the loops have common nodes and therefore
are all touching.
• The path P1 touches all the loops, so ∆1 = 1
T(s) = Y(s)/R(s) = P1 ∆1 /(1 - L1 - L2 - L3)
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• P1 = G1 G2 G3 G4
• L1 = - G2 G3 H2
• L2 = G3 G4 H1
• L3 = - G1 G2 G3 G4 H3
T(s) = Y(s)/R(s) = P1 ∆1 /(1 - L1 - L2 - L3)
G1 G2 G3 G4/(1+ G2 G3 H2 - G3 G4 H1G1 G2 G3 G4 H3)
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T(s) = G1 G2 G3 G4/(1 - G3 G4 H1 +
G2 G3 H2 + G1 G2 G3 G4 H3)
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Ex. 2.11
A complex system
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The forward paths:
• P1 = G1 G2 G3 G4 G5 G6
• P2 = G1 G2 G7 G6
• P3 = G1 G2 G3 G4 G8
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Ex. 2.11
A complex system
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The feedback loops:
•
•
•
•
•
•
•
•
L1 = - G2 G3 G4 G5 H2
L2 = - G5 G6 H1
L3 = - G8 H1
L4 = - G7 H2 G2
L5 = - G4 H4
L6 = - G1 G2 G3 G4 G5 G6 H3
L7 = - G1 G2 G7 G6 H3
L8 = - G1 G2 G3 G4 G8 H3
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The determinant and cofactors:
• ∆ = 1 - (L1 + L2 + L3 + L4 + L5 + L6 + L7 +
L8) + ( L5L7 + L5L4 + L3L4)
• ∆1 = ∆3 = 1
∆2 = 1 - L5 = 1 + G4H4
Loop L5 does not touch loop L7or loop L4,
and loop L3 does not touch loop L4;
but all other loops touch.
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The transfer function:
• T(s) = Y(s)/R(s) = (P1 ∆1 + P2 ∆2 + P3 ∆3 (/∆
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Important properties of SF-G
• A SF-G applies only to linear systems.
• The equations for which a SF-G is drawn must
be algebraic equations in the form of effects
as function of causes.
• Nodes are used to represent variables.
Normally, the nodes are arranged from left to
right, following a succession of causes and
effects through the system.
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• Signal travel along branches only in the direction
described by the arrows of the branches.
• The branch directing from node yk to yj represents
the dependence of variable yj upon yk, but not the
reverse.
• A signal yk traveling along a branch between
nodes yk and yj is multiplied by the gain of the
branch, akj, so that a signal akj yk is delivered at
note yj.
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In the case when the system is represented by
a set of integrodifferencial equations, we must
first transform these into Laplace transform
equations and then rearrange the latter into the
form of
Yj)s( = ∑Gkj(s) Yk(s) for k, j =1,2,…,N
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Summary
An alternative use of T(s) models in S-FG was
investigated. Mason’s SF-G formula was found to be
useful for obtaining the relationship between system
variables in complex feedback system.
Mason’s SF-G formula provides the relationship between
system variables without any reduction or manipulation
of the flow graph.
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