Nguyễn Công Phương
CONTROL SYSTEM DESIGN
Mathematical Models of Systems
Contents
Introduction
I. II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems
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Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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3
Differential Equations of Physical Systems (1)
i
v
a through-variable an across-variable
• Current i: • Voltage v:
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Differential Equations of Physical Systems (2)
System
Variable through element
Integrated through- variable
Variable across element
Integrated across- variable
Electrical
Current, i
Charge, q
Voltage, v
Flux linkage, λ
Mechanical translational
Force, F
Velocity, v
Displacement, y
Translational momentum, P
Mechanical rotational
Torque, T
Angular momentum, h
Angular velocity, ω
Angular displacement, θ
Fluid
Volume, V
Pressure, P
Pressure momentum, γ
Fluid volumetric rate of flow, Q
Thermal
Heat flow rate, q
Heat energy, H
Temperature, T
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Differential Equations of Physical Systems (3) Inductive storage
L
i
2
v
L
E
Li
21
di dt
1 2
1v
2v
Electrical inductance: v: voltage i: current L: inductance
2
k
v
E
21
1 dF k dt
1 2
F k
F 1v
2v
Translational spring: v: translational velocity F: force k: translational stiffness
k
E
21
F 1
1 dT k dt
2
21 T k 2
Rotational spring: ω: angular velocity T: torque k: rotational stiffness
QI
2
I
E
P 21
dQ dt
1 IQ 2
1P
2P
Fluid inertia: P: Pressure Q: fluid volumetric flow rate I: fluid inertance
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Differential Equations of Physical Systems (4) Capacitive storage
C
i
E
Cv
i C
2 21
21dv dt
1 2
1v
Electrical capacitance: v: voltage; i: current C: capacitance
2v
2
F M
E
M
v 1 const
2dv dt
F 2v
F k
1 2
Translational mass: v: translational velocity; F: force M: mas; k: translational stiffness
T
J
E
2 J 2
J
1 const
2d dt
1 2
T 2
Rotational mass: ω: angular velocity; T: torque J: moment of inertia
Q
E
C P
Q C
f
2 21
fC
dP 21 dt
1 2 f
Fluid capacitance: P: Pressure; Cf: fluid capacitance Q: fluid volumetric flow rate
2P
1P
q
tC
E C T 2t
q C t
T 1 const
2T
dT 2 dt
Thermal capacitance: P: Pressure; q: heat flow rate; Ct: thermal capacitance
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Differential Equations of Physical Systems (5) Energy dissipators
iR
i
P
1v
2v
21v R
2 21v R
Electrical resistance: v: voltage; i: current R: resistance
b
F
F bv
P bv
21
2 21
1v
2v
Translational damper: v: translational velocity; F: force b: viscous friction
b
T
T
b
P b
21
2 21
1
2
Rotational damper: ω: angular velocity; T: torque b: viscous friction
QfR
Q
P
P 21 R
2 P 21 R
f
f
1P
2P
Fluid resistance: P: Pressure; Rf: fluid resistance Q: fluid volumetric flow rate
qtR
q
P
T 21 R t
T 21 R t
Thermal resistance: P: Pressure; q: heat flow rate; Rt: thermal resistance
1T
2T
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Differential Equations of Physical Systems (6)
C
L
R
( )r t
k
Wall friction b
Ex. 1
+–
y
Mass M
i t ( )
t
Ri t ( )
L
i t dt ( )
r t ( )
Force r(t)
0
di t ( ) dt
1 C
M
b
ky t ( )
r t ( )
( ) dy t dt
2 d y t ( ) 2 dt
t
M
bv t ( )
k
v t dt ( )
r t ( )
0
( ) dv t dt
v t ( )
dy t ( ) dt
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Differential Equations of Physical Systems (7)
1Ri
v t 1( )
2 ( ) v t Li
1R
1Ci
2Ri
2Ci
1L
k
2R
2C
1C
( )r t
Friction b2
Velocity v2(t)
M2
i
r t ( )
i
1
R
2
Friction b1
i C i
1 R
R 1
2
L
Velocity v1(t)
M1
v
0 v 1
2
r t ( )
C 1
Force r(t)
i v 1 R 2
R 1
t
v
2
C
0
2
v dt 2
0
dv 2 dt
1 L
M
r t ( )
1
b v 1 1
b v 2 1
b v 1 2
dv 1 dt
r t ( )
C 1
v 1
v 1
t
M
k
0
2
b v 1 2
b v 1 1
v dt 2
t
0
dv 2 dt
C
v
0
2
2
v 1
v dt 2
0
dv 2 dt
v 2 R 1 1 L
1 R 2 1 R 1
1 R 1 1 R 1
i C dv 1 dt v 1 R 1 dv 1 dt
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Ex. 2
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical
Systems
3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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11
y
Linear Approximations of Physical Systems (1)
( )y t
( )x t
y mx
0
x
( )y t
( )x t
y t 1( )
x t 1( )
System
y t 2 ( )
x t 2 ( )
ky t ( )
kx t ( )
System System
System
y t ( ) 1
y t ( ) 2
x t ( ) 1
x t ( ) 2
System
System
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Superposition Homogeneity
Linear Approximations of Physical Systems (2)
y mx b
y 1y
( )y t
( )x t
y
0y
x
0
0x
1x
x
b
)
b
x
y mx 1 1
y 0
b
y m x 0(
y mx 0 0
y m x
System
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Linear !!!
y
g x ( )
y
Linear Approximations of Physical Systems (3)
( )y t
( )x t
0y
dg dx x x 0
y t ( )
g x t
[ ( )]
0
0x
x
2
(
x
)
(
x
)
)
...
g x ( 0
dg dx
x 0 1!
x 0 2!
2 d g 2 dx
x x 0
x x 0
)
(
x
)
g x ( 0
x 0
dg dx x x 0
(
)
)
y
m x (
)
y (
y m x 0
x 0
x 0
0 y m x
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System
Linear Approximations of Physical Systems (4)
( )y t
( )x t
y
(
,
,...,
x
g x x 1
2
)n
g x (
,
x
,...,
x
)
(
)
20
10
n
0
x 1
x 10
g x 1
x x 0
(
x
x
)
(
x
x
)
...
2
20
n
n
0
g x
g x
2
n
x x 0
x x 0
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System
Linear Approximations of Physical Systems (5)
T mgL
sin
)
T mgL
(
0
0
sin
0
0;
0
0
T 0
o
T mgL
o 0 )
(cos 0 )(
mgL
T
2
http://www.ic.sunysb.edu/Class/phy141md/ doku.php?id=phy141:labs:lab1
0
2
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16
Ex.
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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17
The Laplace Transform (1)
The Laplace transformation of f(t):
F s ( )
st t e dt ( )
f
0
j
st
The inverse Laplace transform of F(s):
f
t ( )
F s e ds
( )
j
j
1 2
at
cos at
t
sin at
ate
te
( )u t
( )t
f(t)
1
2
2
2
2
2
F(s)
1 2 s
1 s
1 s a
(
)
s
s
1 s a
a a
s a
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The Laplace Transform (2)
Property
f(t)
F(s)
t ( )Af
AF s ( )
1. Magnitude scaling
2. Addition/subtraction
f
t ( )
f
t ( )
1
2
F s ( ) 1
3. Time scaling
F
f at (
)
f
0
4. Time shifting
)]
ase L f [
F s ( ) 2 s 1 a a ase F s ( ) t a (
),
0
5. Frequency shifting
(
)
F s a
t a u t a a ) ( ( ), t u t a a f ( ) ( ate
t ( )
f
n
n
2
1
n
1
1
n
6. Differentiation
n s F s ( )
s
f
(0)
s
f
(0) ...
o s f
(0)
n d f
t dt ( ) /
n
n
n
7. Multiplication by t
nt
f
t ( )
( 1)
d F s ds ( ) /
8. Division by t
F
d ( )
f
t ( ) /
t
s
t
9. Integration
f
d ( )
F s
( ) /
s
0
t
10. Convolution
f
f
t ( ) *
f
t ( )
f
(
t
( )
d )
1
1
2
2
F s F s ( ) ( ) 1
2
0
11. Final value
0
t lim ( ) f t
sF s lim ( ) s
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The Laplace Transform (3)
M
b
ky t ( )
r t ( )
dy t ( ) dt
2 d y t ( ) 2 dt
r t ( )
R s ( )
ky t ( )
kY s ( )
y t ( )
Y s ( )
b
b sY s ( )
(0 )
y
( ) dy t dt
M
2 M s Y s ( )
sy
(0 )
(0 )
dy dt
2 d y t ( ) 2 dt
2 M s Y s ( )
sy
(0 )
(0 )
b sY s ( )
(0 )
y
kY s ( )
R s ( )
dy dt
( )
(0 )
(0 )
by
(0 )
R s M sy
Ex. 1
? ( )y t
Y s ( )
2
p s ( ) q s ( )
Ms
k
dy dt bs
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The Laplace Transform (4)
1.5
Y s ( )
1
s 1)( s
2)
(
s
4(
3)
0.5
0
K 1 s 1
K 2 s 2
i
t r a P y r a n g a m
I
-0.5
8
K 1
-1
s 3) 4( 2) s (
s 1)
(
s
2)
4(
3) s (
s
1
s
1
-1.5
-3
-2.5
-2
-1.5
-0.5
0
0.5
1
K
4
-1 Real Part
2
s 4( s (
2)
(
s
s 4( 1) ( s
3)
4
3) 1) s
2
s
3.5
3
Y s ( )
2.5
s
1
2
2 8
2
1.5
Ke
at
4 s K s a
1
0.5
t
2
t
4
0
0
1
2
3
4
5
e ( ) 8 e y t sites.google.com/site/ncpdhbkhn
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Ex. 1
The Laplace Transform (5)
15
10
5
Y s ( )
2
0
i
s 6 s
265
s
4
76
t r a P y r a n g a m
I
-5
-10
s
s
j 16
j 16
K 1 3
K 2 3
-15
-30
-25
-20
-15
-10
-5
0
5
10
Real Part
6
K 1
76 4 s s ( 16) j
(
s
j 16)
3
3
s
j
16
3
4
5.66e-3tcos(16t - 45o) 5.66e-3t -5.66e-3t
2
j
2 2.83
2
o45
0
t 3
t cos(16
o 45 )
( ) 2 2.83 e y t
-2
-4
5.66
te 3
t cos(16
o 45 )
-6
0
0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8
2
1 Time
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Ex. 2
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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23
The Transfer Function of Linear Systems (1) ( )Y s
( )R s
G s ( )
Output Input
Y s ( ) R s ( )
n
n
n
2
1
1
p
p
...
...
q n
q y 0
n
p r 0
n
2
1
1
n
n
r 2
n
y 1
r 1
n d y n dt
d dt
d dt n 2
1
p
p
n
p 0
Y s G s R s ( )
( )
( )
R s ( )
R s ( )
p s ( ) q s ( )
d dt n s 1 n s
s n 2 1 n s
...
q n
... q 0
1
R s ( )
Y s ( ) 1
Y s ( ) 2
Y s ( ) 3
m s ( ) q s ( )
( ) p s q s ( )
( ) m s q s ( )
p s n s ( ) ( ) q s d s ( ) ( )
y t ( )
y t ( ) 3 steady-state reponse
y t y t ( ) ( ) 1 2 transient response
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System
Given
4 ( ), where
r t
2
3
y
y
t
0. Find ( ) ?
y t
( ) 1, r t
(0) 0;
(0) 1;
The Transfer Function of Linear Systems (2) dy dt
dy dt
2 d y 2 dt
4
2 s Y s ( )
sy
(0)
(0)
3
sY s ( )
y
(0)
Y s 2 ( )
R s 4 ( )
1 s
dy dt
2[ s Y s ( )
s
sY s
Y s ( ) 1] 2 ( )
4 /
s
] 3[
Y s ( )
2
2
K 3 s
K 1 s 1
K 2 s 2
K 4 s 1
K 5 s 2
2
s
s s (
2)
s
3 3 s
4 s 3
s
1
s
2
2 s
s
1
s
2
2
1
4
2
t
2
t
2
t
t
y t ( )
(2
e
e
)
(2
e
4
e
) 2
2
t
t
e
2
e
2
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Ex. 1
The Transfer Function of Linear Systems (3)
R1
R2
i2
vi
ii
vo
Ex. 2 Find the transfer function?
– +
ii
i 2
v
v
0
v i
v i
v i
i i
R 1
R 1
R 1
v i R 1
v
v
0
v o
v o
i 2
R 2
R 2
v o R 2
v o R 2
v i R 1
v o R 2
v o v i
R 2 R 1
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The Transfer Function of Linear Systems (4)
k
Friction b2
M
(
r t ( )
1
b 1
b v ) 2 1
b v 1 2
Velocity v2(t)
M2
t
M
0
k
2
b v 1 2
v dt 2
b v 1 1
0
dv 1 dt dv 2 dt
Friction b1
M sV s ( )
(
R s ( )
b V s ( ) ) 2 1
1
1
b 1
b V s ( ) 1 2
Velocity v1(t)
M1
M sV s ( )
k
0
2
2
Force r(t)
V s ( ) 1
(
b V s ( ) 1 2 ( M s b 1 1
( ) V s 2 b V s ( ) 1 1 s ( ) / ) M s b k s R s 2 1 b M s b )( k s / ) 2 2 1
2 b 1
2
G s ( ) v
V s ( ) 1 R s ( )
k
)
(
M
M s 2 b M s )( 2 2
k b s 1 2 b s 1
2 b s 1
1
s
b 1
2
s
G s ( ) x
X s ( ) 1 R s ( )
V s ( ) / 1 R s ( )
G s ( ) v s
k
)
]
s M [(
b s M s 1 2 2 b M s )( 2 2
k b s 1
2 b s 1
b 1
1
s
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Ex. 3 Find the transfer function?
The Transfer Function of Linear Systems (5)
Angle θ
t ( )
T m
K i 1 a
Inertia load
t ( )
K K i 1 f
f
t i ( ) a
http://www.electrical4u.com/permanent- magnet-dc-motor-or-pmdc-motor/
(
i )
t ( )
t ( )
K K I 1 f a
f
K i m f
s ( )
T s ( ) m
K I m f
T s ( ) L
T s ( ) d
T s ( ) L
Js
s ( )
s ( )
2
bs
(
R
s ( )
LT s ( ) V s ( ) f
f
sL I ) f
f
m
G s ( )
K s Js b L s R ( )(
)
s ( ) V s ( ) f
f
f
K
/(
JL
)
f
)(
/
/
s R L f
f
m ) s s b J ( sites.google.com/site/ncpdhbkhn
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Ex. 4 Find the transfer function? fK i f
The Transfer Function of Linear Systems (8)
s I ( )
s ( )
T s ( ) m
K K I 1
f
f
a
If the armature current Ia is constant
m
G s ( )
K s Js b L s R ( )(
)
( ) s V s ( ) f
f
f JL /(
)
K
K
/(
bR
)
f
f
/
)(
/
)
m s s b J (
s
s
s
1)
(
1)(
m
s R L f
f
f
L
dT s ( )
( )
s ( )
( )mT s
fI
( )s
fV s ( )
( )s1
mK
R
1
1 s
LT s ( )
Js b Load
L s f s Field
Field – controlled DC motor sites.google.com/site/ncpdhbkhn
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Ex. 4 Find the transfer function?
The Transfer Function of Linear Systems (9)
s I ( )
s ( )
T s ( ) m
K K I 1
f
f
a
s ( )
I
)
(
s ( )
If the field current If is constant
K K I 1
T s ( ) m
a
f
f
K I m a
(
s ( )
V s ( ) a
R a
sL I ) a
a
V s ( ) b
I
s ( )
a
V s ( ) a R a
V s ( ) b sL a
K
V s ( ) b
s ( ) b
Js
s ( )
s ( )
2
bs
LT s ( )
G s ( )
)
K K
]
K m
( ) s V s ( ) a
a
b m
s Js b L s R [( )( a Armature – controlled DC motor sites.google.com/site/ncpdhbkhn
30
Ex. 4 Find the transfer function?
The Transfer Function of Linear Systems (10)
s I ( )
s ( )
T s ( ) m
K K I 1
f
f
a
s ( )
T s ( ) m
K I m a
K
s ( )
b
s ( )
I
a
If the field current If is constant V s ( ) b
Js
s ( )
s ( )
2
bs
V s ( ) b sL a
G s ( )
)
K K
]
T s ( ) L K m
V s ( ) a R a s ( ) V s ( ) a
s Js b L s R [( )( a
a
b m
dT s ( ) ( )
( )mT s
( )s
( )s1
aV s ( )
1 s
LT s ( )
( )
Js b Load
K m L s R a a Armature
bK
Back electromotive force
Armature – controlled DC motor sites.google.com/site/ncpdhbkhn
31
Ex. 4 Find the transfer function?
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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32
Block Diagram Models (1) ( )s
fV s ( )
m
m
G s ( )
G s ( )
K s Js b L s R ( )(
)
K s Js b L s R ( )(
)
s ( ) V s ( ) f
f
f
f
f
( )
( )
( )
( )
11
12
2
System
( )
( )
( )
( )
21
22
2
2
Y s G s R s G s R s ( ) 1 1 Y s G s R s G s R s ( ) 1
1( )R s R s 2 ( )
Y s 1( ) Y s 2 ( )
Y s 1( ) Y s 2 ( )
1( )R s R s 2 ( )
System
JR s ( )
IY s ( )
11
12
G s G s ( ) G s G s ( )
( ) ( )
Y GR
22
21 ( )
G s G s ( )
I
1
I
2
G s ( ) 1 J G s ( ) J 2 G s ( ) IJ
R s ( ) 1 R s ( ) 2 ( ) R s J
Y s ( ) 1 Y s ( ) 2 ( ) Y s I
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Block Diagram Models (2)
( )
( )
( )
( )
11
12
2
System
( )
( )
( )
( )
22
21
2
2
Y s G s R s G s R s ( ) 1 1 Y s G s R s G s R s ( ) 1
1( )R s R s 2 ( )
1( ) Y s Y s 2 ( )
1( )R s
Y s 1( )
R s 2 ( )
Y s 2 ( )
G11(s)
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G22(s)
Block Diagram Models (3)
X1 X2 X3 X1 X3
or
G1(s) G2(s) G1G2
X1 X3
Combining blocks in cascade
Moving a summing point behind a block
G2G1
X1 X3 X1 X3
( )
( )
G G
X2
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G X2
Block Diagram Models (4)
X1 X2 X2 X1
G G
G
Moving a pickoff point ahead of a block
Moving a pickoff point behind a block
X2 X2
X2 X1 X1 X2
G G
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36
X1 X2 1/G
Block Diagram Models (5)
X1 X3 X1 X3
( )
( )
G G
1/G X2
Moving a summing point ahead of a block
Eliminating a feedback loop
X2
2X
1X
X1 X2
1
G GH
( )
G
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37
H
Block Diagram Models (6)
Z(s)
U(s)
Y(s)
R(s)
Ea(s)
(–)
B(s)
Process G(s) Controller Gc(s) Actuator Ga(s)
R s ( )
B s ( )
R s H s Y s ( )
( )
( )
aE s ( )
Y s G s U s ( )
( )
( )
( )
( )
G s G s G s E s ( )
( )
( )
( )
G s G s Z s ( ) a
c
a
a
( )[
( )
( )
( )
( )
( )
( )]
Y s G s G s G s R s H s Y s c
a
Y s ( ) R s ( )
1
G s G s G s ( ) ( ) ( ) c a G s G s G s H s ( ) ( ) ( ) ( )
a
c
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38
Sensor H(s)
Block Diagram Models (7)
( )
Ex. H2
R(s) Y(s)
( )
G1 G2 G3 G4
H1
H3
X1 X2 X1 X2
G G
Moving a pickoff point behind a block
( )
X1 X2 1/G
R(s) Y(s)
( )
G1 G2 H2/G4 G3 G4
H1
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39
H3
( )
Ex.
R(s) Y(s)
G1
Block Diagram Models (8) H2/G4 G3
( )
G2 G4
H1
H3
X1 X2 X3 X1 X3
Combining blocks in cascade
G1(s) G2(s) G1G2
( )
H2/G4
R(s) Y(s)
( )
G1 G2
G3G4 H1
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40
H3
Ex.
Block Diagram Models (9) H2/G4
( )
R(s) Y(s)
( )
G1 G2
G3G4 H1
2X
1X
H3
1
G GH
( )
X1 X2 G
Eliminating a feedback loop
H
( )
H2/G4
R(s) Y(s)
1
G G 3 4 G G H 4
3
1
( )
G1 G2
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41
H3
Ex.
Block Diagram Models (10) H2/G4
( )
R(s) Y(s)
1
G G 3 4 G G H 4
3
1
( )
G1 G2
H3
( )
H2/G4
Y(s) R(s)
1
G G G 2 3 4 G G H 4
3
1
( )
G1
H3
2
Y(s) R(s)
1
G G G 3 4 G G H G G H 1
3
2
3
4
2
( )
G1
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42
H3
Block Diagram Models (11)
( )
Ex. H2
R(s) Y(s)
( )
G1 G2 G3 G4
H1
H3
2
Y(s) R(s)
1
G G G 4 3 G G H G G H 1
4
3
3
2
2
( )
G1
1
Y s ( ) R s ( )
1
3 G G H G G H G G G G H
G G G G 4 2
1
2
3
4
3
4
2
3
1
2
3
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43
H3
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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44
Signal – Flow Graph Models (1)
R(s)
Y(s)
Y s G s R s ( )
( )
( )
R(s)
Y(s)
1( )R s
Y s 1( )
G s 11( )
R1(s)
Y1(s)
G(s)
G s 12 ( )
G s 21( )
R s 2 ( )
Y s 2 ( )
R2(s)
Y2(s)
G11(s)
G s 22 ( )
( )
( )
( )
( )
11
12
2
( )
( )
( )
22
21
2
2
Y s G s R s G s R s ( ) 1 1 Y s G s R s G s R s ( ) ( ) 1 sites.google.com/site/ncpdhbkhn
45
G22(s)
11a
1
1R
1X
x 1 x
Signal – Flow Graph Models (2) a x 12 2 a x 22 2
r 1 r 2
2
a x 11 1 a x 21 1
12a
21a
(1
)
2R
a x 12 2 x )
a
(1
2X
a x 11 1 a x 21 1
22
2
r 1 r 2
1
22a
1
22
x 1
r 1
r 2
(1
r 1 a
) a 22 )(1
a r 12 2 )
a
a 12
a a 12 21
1
x
2
r 2
r 1
(1
r 2 a
a 11
a 21
(1 a 11 ) a (1 11 )(1 a 11
22 a r 21 1 ) a a 12 21
22
(1
)(1
a
)
a
1
a 11
a a 12 21
22
a 11
22
a a 11 22
a a 12 21
N
...
1
L n
L L n m
L L L n m p
n
,
1
n m , nontouching
n m p , nontouching
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46
N
...
1
L L L n m p
L L n m
L n
Signal – Flow Graph Models (3)
n
,
1
n m , nontouching
n m p , nontouching
= 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) + ... 11a
1L
1
1R
1X
;
;
a
L 1
a 11
L 2
a a 12 21
L 3
22
2L
21a
12a
2R
2X
1
3L
22a
(sum of all different loop gains) = L1 + L2 + L3 = a11 + a12a21 + a22 (sum of the gain products of all combinations of two nontouching loops) = L1L3 = a11a22
1 (
a
)
a 11
22
a a 12 21
a a 11 22
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N
1
Signal – Flow Graph Models (4) L L ... n m
L L L n m p
L n
n
,
1
n m , nontouching
n m p , nontouching
P k
k
G s ( )
( ) Y s R s ( )
k
• Pk: gain of kth path from input to output • Δk (cofactor): the determinant Δ with the loop(s)
touching the kth path removed.
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Signal – Flow Graph Models (5) 3H
2H
P k
k
G s ( )
2L
1L
Y s ( ) R s ( )
k
1G
4G
2G
3G
• Pk: gain of kth path from input
( )R s
( )Y s
to output
6G
7G
8G
5G
• Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.
3L
4L
;
P G G G G 1 4 1
3
2
P G G G G 2 8 5
6
7
6H
7H
Δ = 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) + ...
;
;
L G H L G H L G H L G H ; 1 3
3
4
7
2
6
6
3
2
2
7
1 (
)
(
)
L 1
L 2
L 3
L 4
L L 1 3
L L 1 4
L L 2 3
L L 2 4
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49
Ex. 1
Signal – Flow Graph Models (6) 3H
2H
P k
k
G s ( )
2L
1L
Y s ( ) R s ( )
k
1G
4G
2G
3G
3
2
( )R s
( )Y s
6G
7G
)
P G G G G 1 1 4 P G G G G 2 5 8 1 (
7
8G
5G
(
)
L 3
6 L 1 L L 1 3
L 2 L L 1 4
L 4 L L 2 3
L L 2 4
3L
4L
Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.
6H
7H
1 (
)
1
L 3
L 4
0
L L 2 1
1 (
)
2
L 1
L 2
0
L L 4 3
5
P 2
2
G s ( )
)] )
P 1 1
3
L 3
L )] 4 ( )
L 1
G G G G 1 4 2 1 ( L L 2 1
[1 ( L 3
L 4
L L 1 3
[1 ( G G G G 6 8 7 L L L L 2 3 1 4
L 2 L L 2 4
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Ex. 1
Signal – Flow Graph Models (7)
dT s ( ) ( )
( )mT s
( )s
( )s1
aV s ( )
P k
k
G s ( )
1 s
LT s ( )
k
( )
s ( ) V s ( ) a
Js b Load
K m R L s a a Armature
bK
Back electromotive force
Δ = 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
G s ( )
combinations of three nontouching loops) + ...
)
K K
]
K m
s ( ) V s ( ) a
s Js b L s R [( )( a
a
b m
L
G G H 2
1
dT s ( )
1
( )s
aV s ( )
( )mT s
LT s ( )
1
1L
1
1
1
1G
2G
3G
1
L
• •
G G H 2 Pk: gain of kth path from input to output Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.
;
;
;
H K
G 1
G 2
G 3
b
P G G G 1 1 3
2
1 s
H 1 Js b
1
1
K m L s a 1 s
R a
G s ( )
P 1 1
L 0 s ( ) V s ( ) a
G G G 1 3 2 1 G G H 1 2
1
K
1
b
R a
R a K 1 m Js b L s a K 1 m Js b L s a sites.google.com/site/ncpdhbkhn
51
Ex. 2
Signal – Flow Graph Models (8)
H2
( )
Y(s)
R(s)
P k
k
G s ( )
G1
G2
G3
G4
Y s ( ) R s ( )
k
( )
H1
H3
Δ = 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
1
combinations of three nontouching loops) + ...
( ) Y s R s ( )
3 G G H G G H G G G G H
1
G G G G 4 2
3
1
3
1
3
4 G G G G H 3
4
1
2
3
2 3 2 1 L 1
2 L 2
4 L 3
L 1
G G H 3
2
2
L G G H 3
2
4
1
L 3
• •
Pk: gain of kth path from input to output Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.
P G G G G 1 4 1
2
3
1
1
0
3
1
G s ( )
1
3 G G H G G H G G G G H
G G G G 4 2
L L L , , 2 1 Y s ( ) R s ( )
P 1 1
1
2
3
4
2
3
3
4
1
2
3
1 G G G G 4 2 3 1 L L 1 3 2
L 1
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52
Ex. 3
Signal – Flow Graph Models (9)
7G
8G
P k
k
1
2G
6G
1G
4G
5G
G s ( )
( )Y s
( )R s
k
3G
4H
1H
2H
Δ = 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) + ...
3H
L 1
G G G G H 4
3
2
5
2;
L 2
G G H 6
5
1;
L 3
G H 8
1;
L 4
G G H 7
2
2;
L 5
G H 4
L 6
G G G G G G H 4
2
3
6
1
5
L 7
G G G G H 6
2
1
7
L 8
G G G G G H 3
2
4
8
1
3;
1 (
)
(
3; )
4;
3; L 8
L 1
L 2
L 3
L 4
L 5
L 6
L 7
L L 5 7
L L 5 4
L L 3 4
• •
Pk: gain of kth path from input to output Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.
P G G G G G G 1 2
1
5
3
4
6;
P G G G G 2 1
2
7
6;
P G G G G G 3 2
1
3
4
8;
1
1
2
L 5
3
1
all loops except L are zero
all loops are zero
5
1
P 3
3
G s ( )
1
2
4
)
2
( ) Y s R s ( )
P P 2 1 1 2
G G G G G G 6 3 L 2
1 L 1
5 L 3
1 L 4
G G G G (1 7 6 2 L L L 5 7 6
L 5 L 8
G G G G G 3 1 4 8 L L L L 5 7 5 4
1 L L 3 4
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53
Ex. 4 s ( ) V s ( ) a
Mathematical Models of Systems
1. Differential Equations of Physical Systems 2. Linear Approximations of Physical Systems 3. The Laplace Transform 4. The Transfer Function of Linear Systems 5. Block Diagram Models 6. Signal – Flow Graph Models 7. The Simulation of Systems Using Control
Design Software
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54
The Simulation of Systems Using Control Design Software (1)
k
Wall friction b
y
Mass M
M
b
ky t ( )
r t ( )
dy t ( ) dt
2 d y t ( ) 2 dt
Force r(t)
y
(0)
nt
y t ( )
e
sin
1
t
n
2
Ex. 1
2
1
;
;
cos
1
n
k M
b kM
2
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55
The Simulation of Systems Using Control Design Software (2)
Y s ( )
2
s 6 s
265
s
4
76
• zplane • roots • poly • conv • polyval • tf • series • parallel • feedback
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56
Ex. 2
The Simulation of Systems Using Control Design Software (3)
H2
( )
Y(s)
R(s)
G s ( ) 1
G1
G2
G3
G4
5
s
( )
H1
G s ( ) 2
s
H3
G s ( ) 3
4
7
G s ( ) 4
H s ( ) 1
1 1 1 2 2 s 2 4 s 1 6 1 2
s s s s s ( ) 3
( ) 1
H s 2 H s 3
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57
Ex. 3
The Simulation of Systems Using Control Design Software (4)
H2
( )
Y(s)
R(s)
G1
G2
G3
G4
( )
H1
H3
Ex. 3
X1 X2 X1 X2
G G
Moving a pickoff point behind a block
( )
X1 X2 1/G
R(s) Y(s)
( )
G1 G2 H2/G4 G3 G4
H1
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58
H3
( )
Ex. 3
R(s) Y(s)
The Simulation of Systems Using Control Design Software (5) H2/G4 G3
( )
G1 G2 G4
H1
H3
X1 X2 X3 X1 X3
Combining blocks in cascade
G1(s) G2(s) G1G2
( )
H2/G4
R(s) Y(s)
( )
G1 G2
G3G4 H1
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H3
The Simulation of Systems Using Control Design Software (6) H2/G4
( )
Ex. 3
R(s) Y(s)
( )
G1 G2
G3G4 H1
2X
1X
H3
1
G GH
( )
X1 X2 G
Eliminating a feedback loop
H
( )
H2/G4
R(s) Y(s)
1
G G 3 4 G G H 4
3
1
( )
G1 G2
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60
H3
The Simulation of Systems Using Control Design Software (7) H2/G4
( )
Ex. 3
R(s) Y(s)
1
G G 3 4 G G H 4
3
1
( )
G1 G2
H3
( )
H2/G4
R(s) Y(s)
1
G G G 2 3 4 G G H 4
3
1
( )
G1
H3
2
R(s) Y(s)
1
G G G 3 4 G G H G G H 1
4
2
3
3
2
( )
G1
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H3
The Simulation of Systems Using Control Design Software (8)
dT s ( ) ( )
( )s
d s ( )
3( )G s 500
0.5
2
1( )G s 10 1s
2 ( )G s 1 s
( )
( )
0.1
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62
Ex. 4
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