Nguyễn Công Phương
CONTROL SYSTEM DESIGN
State Variable Models
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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State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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3
The State Variables of a Dynamic System (1)
• The state of a system is a set of variables
whose values, together with the input signals & the equations describing the dynamics, will provide the future state & output of the system.
• The state variables describe the present
configuration of a system & can be used to determine the future response, given the excitation inputs & the equations describing the dynamics.
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The State Variables of a Dynamic System (2)
Wall friction b
M b ky t ( ) u t ( )
2 d y t ( ) 2 dt
k
y t ( ),
x t ( ) 1
x t ( ) 2
dy t ( ) dt
Mass M
dy t ( ) dt
2
y(t)
u(t)
M
bx
u t ( )
2
kx 1
dx 2 dt
x
2
C
u t ( )
i
x u x 1 k M 1 M b M dx 1 dt dx 2 dt
i c
L
2
dv C dt
x u t ( )
L
Li C
R
1 C 1 C
L
Ri
2
v C
L
Cv
( )u t
Ci
t ( )
di L dt
x x 1 1 L R L dx 1 dt dx 2 dt ov
x
i
2
v o x 1
Ri L v ,C
2
L
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Rx v t ( ) o
State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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6
The State Differential Equation (1) ...
...
...
...
x 1 x
a x 11 1 a x 21 1
a x 12 2 a x 22 2
a x 1 n n a x 2 n n
b u 11 1 b u 21 1
b u 1 m m b u 2 m m
2
...
...
n
a x 1 1 n
a x 2 2 n
a x nn n
b u 1 1 n
b u nm m
x
x 1 x
a 11 a
a 1 n a
a 12 a
x 1 x
n
d dt
b b 11 1 m b b nm n 1
u 2 u m
2 x
21 a
2 x
22 2 a a
n
n 1
nn
n
2
n
x Ax Bu y Cx Du
t
t
x
t ( )
exp(
A x t
) (0)
exp[
A
(
t
)
Bu
t
)
r d ( )
( )
d
t ( ) (0) Φ x
( Φ
Bu
0
0
1
1
X
[
s
I A x ]
(0)[
I A BU s ]
s ( )
s ( )
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The State Differential Equation (2)
x
u t ( )
2
1 C
1 C
x
x 1
2
1 L
R L
dx 1 dt dx 2 dt
L
Li C
R
Rx
Cv
v t ( ) o
2
( )u t
Ci
ov
0
x u t ( ) x
1 C 0
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1 C R L x R 1 L 0 y
The State Differential Equation (3)
q
p
1k
2k
u
f
f
u
M a 1 1
spring
damp
M2
M1
)
(
)
M p 1
u k p q ( 1
b p q 1
1b
2b
M p b p k p 1
1
1
u k q b q 1 1
(
)
(
)
M q 2
k p q 1
b p q 1
k q b q 2
2
(
(
M q 2
k 1
k q ) 2
b 1
b q ) 2
k p b p 1
1
p
,
q
x 3 x
x 1 x
p q
2
4
2
x 1 x
b 1 M
1
1
1
p u q p p q x 3 k 1 M 1 M k 1 M b 1 M
2
4
1 b 1 M
2
2
2
2
1 k 1 M sites.google.com/site/ncpdhbkhn
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k b 2 q p q p x q k 1 M b 1 M
The State Differential Equation (4)
1
1
1
p u q p p q x 3 k 1 M b 1 M 1 M k 1 M b 1 M
2
4
1 k 1 M
1 b 1 M
2
2
2
2
k b 2 q p q p x q k 1 M b 1 M
p
,
2
4
2
x
u
x
x 3
x 1
2
x 3
4
k 1 M
k 1 M
b 1 M
b 1 M
1 M
1
1
b 2
2
x
x
x
4
4
x 1
2
x 3
k 1 M
1 k k 1 M
1 b 1 M
1 b 1 M
2
2
2
2
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q x 3 x x 1 x p q x 1 x
4
2
The State Differential Equation (5) b 1 M
1
1
u x x x 3 x 3 x 1 k 1 M k 1 M b 1 M 1 M
2
4
4
2
1 k k 1 M
1 b 1 M
1 b 1 M
2
2
2
2
0
0
b 2 x x x x 1 x 3 k 1 M
A
,
B
x
,
0 0 k 1 M
0 0 k 1 M
1 0 b 1 M
0 1 b 1 M
1
1
1 M
1
k
x 1 x 2 x 3 x
2
b 2
4
p q p q
0
k 1 M
1 k 1 M
b 1 M
1 b 1 M
2
2
2
2
x Ax B u
y
p
x Cx
1 0 0 0
x 1
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The State Differential Equation (6)
q
p
1k
2k
u
M2
M1
1b
2b
x
u
x
x 3
x 1
2
x 3
4
k 1 M
k 1 M
b 1 M
b 1 M
1 M
1
1
b 2
2
x
x
x
4
4
x 1
2
x 3
k 1 M
1 k k 1 M
1 b 1 M
1 b 1 M
2
2
2
2
p
q
)
)
p
u
M2
M2
)
)
p
k p q 1( b p q 1(
2k q 2b q
k q 1( b q 1(
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State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram
Models
4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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13
Signal – Flow Graph & Block Diagram Models (1)
L
x
u t ( )
2
1 C
1 C
Li C
R
ov
( )u t
Ci
x
x 1
2
1 L
R L
dx 1 dt dx 2 dt
R L
1 L
1 s
R
1 C
Rx
v t ( ) o
2
( )U s
Cv
?
oV s ( )
1/ s
1X
2X
2
1 C
oV s ( ) U s ( )
G s ( )
( )
( )U s
1X
oV s ( )
R
1 L
R L 2X1 s
1 s
1 C
( )
1 C
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LC ) s ( ) /( R LC R L s ) / 1/(
Signal – Flow Graph & Block Diagram Models (2)
m
m
1
b m
G s ( )
,
n m
( ) Y s U s ( )
b s m n s
a
s 1 1 n s
... ...
n
b s b 1 0 a s a 1
0
1
n m
)
(
1)
1)
n
(
n m
( n
b s m
b m
b s 0
n
s 1 1 s
s a
...
b s 1 1)
n
... ( n a s 1
a s 0
1
P k
k
N
Sum of the forward-path factor 1 sum of the feedback loop factors
1
L q
q
1
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15
Signal – Flow Graph & Block Diagram Models (3)
Ex. 1
4
G s ( )
4
3
1
3
4
2
b s 0 2
Y s ( ) U s ( )
s
1
a s 3
a s a 1
0
a s 3
a s 2
a s 1
a s 0
b 0 a s 2
4
3
2
)
a s 2
s ( 4
3
d
)
d
d
)
)
(
)
a
a
/
)
u
3
2
a y b ( 0
0
a 1
d y b / 0 dt
a s a Y s ( ) 1 0 2 y b / ( 0 3 dt
b U s ( ) 0 y b / ( 0 2 dt
4X
1
1 s
1 s
1 s
1 s
0b
( )U s
( )Y s
x 1 x
a s 3 y b / ( 0 4 dt y b / 0
3X
2X
1X
2
3a
2a
1a
x 3 x
0a
4
x 1 x 2 x 3
y b / 0 y b / 0 y b / 0
4X
3X
2X
1X
( )Y s
( )U s
0b
1 s
1 s
1 s
1 s
1 s
( )
( )
3a
2a
( )
1a
0a
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16
Signal – Flow Graph & Block Diagram Models (4)
Ex. 1
4
G s ( )
4
3
1
3
4
2
b s 0 2
Y s ( ) U s ( )
s
1
a s 3
a s a 1
0
a s 3
a s 2
a s 1
a s 0
b 0 a s 2
4
3
2
d
)
d
)
d
)
)
(
a
a
/
)
u
3
2
a 1
a y b ( 0
0
/ d y b 0 dt
( y b / 0 4 dt
( y b / 0 3 dt
( y b / 0 2 dt
x 1 x
y b / 0
2
x 3 x
4
x 1 x 2 x 3
y b / 0 y b / 0 y b / 0
u
x 4
a x 0 1
a x 1 2
a x 2 3
a x 3 4
y
b x 0 1
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Signal – Flow Graph & Block Diagram Models (5)
Ex. 1
4
G s ( )
4
3
1
3
4
2
b s 0 2
Y s ( ) U s ( )
s
1
a s 3
a s a 1
0
a s 3
a s 2
a s 1
a s 0
b 0 a s 2
u
x
4
a x 0 1
a x 1 2
a x 2 3
a x 3 4
y
b x 0 1
u t ( )
x Ax B u
0 0 0 a
0 0 0 a
0 0 0 a
x 1 x 2 x 3 x
x 1 x 2 x 3 x
0 0 0 a 1
0
2
3
4
4
0 0 0 1
x 1 x
y t ( )
Cx
0 0 0
b 0
2 x 3
x 4 sites.google.com/site/ncpdhbkhn
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Signal – Flow Graph & Block Diagram Models (6)
Ex. 1
4
G s ( )
4
3
1
3
4
2
b s 0 2
Y s ( ) U s ( )
s
1
a s 3
a s a 1
0
a s 3
a s 2
a s 1
a s 0
b 0 a s 2
4X
1
1 s
1 s
1 s
1 s
0b
( )U s
( )Y s
3X
2X
1X
3a
2a
1a
0a
P k
k
G s ( )
N
Y s ( ) U s ( )
Sum of the forward-path factor 1 sum of the feedback loop factors
1
L q
q
1
4X
3X
2X
1X
( )Y s
( )U s
0b
1 s
1 s
1 s
1 s
( )
( )
3a
2a
( )
1a
0a
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19
Ex. 2
2
1
3
2
3
4
G s ( )
1
4
4
2
Y s ( ) U s ( )
1
s
b s 2
b s 3
Signal – Flow Graph & Block Diagram Models (7) b s 0 3 a s 0
b s b 1 0 a s a 1
b s 2 a s 2
b s 3 a s 3
b s 1 a s 1
2 a s 2
0
P k
k
G s ( )
N
( ) Y s U s ( )
Sum of the forward-path factor 1 sum of the feedback loop factors
1
L q
3 a s 3
q
1
3b
2b
4X
1b
1
1 s
1/ s
1/ s
1/ s
0b
( )U s
( )Y s
3X
2X
1X
3a
2a
1a
0a
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Ex. 2
1
2
3
3
4
2
G s ( )
1
4
4
2
Y s ( ) U s ( )
1
s
b s 2
b s 3
Signal – Flow Graph & Block Diagram Models (8) b s 0 3 a s 0
b s b 1 0 a s a 1
b s 2 a s 2
b s 3 a s 3
b s 1 a s 1
2 a s 2
3 a s 3
0
3b
2b
4X
1b
1
1 s
1/ s
1/ s
1/ s
0b
( )U s
( )Y s
3X
2X
1X
3a
2a
1a
0a
X
1
X
2
X
X s / 2 X s / 3 X s / 4
3
X
) /
s
U a X (
a X 0
1
2
Y
4
3
4 b X 0
1
3 b X 1
2
a X 2 b X 2
3
a X 1 b X 3
4
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Ex. 2
2
1
3
2
3
4
G s ( )
1
4
4
2
Y s ( ) U s ( )
1
s
b s 2
b s 3
Signal – Flow Graph & Block Diagram Models (9) b s 0 3 a s 0
b s b 1 0 a s a 1
b s 2 a s 2
b s 3 a s 3
b s 1 a s 1
2 a s 2
3 a s 3
0
X
X
sX
1
2
1
X
X
sX
2
3
2
X
X
sX
X s / 2 X s / 3 X s / 4
3
4
3
X
sX
) /
s
)
U a X (
U a X (
a X 0
1
2
a X 0
1
2
4
3
4
3
3 b X 1
2
a X 2 b X 2
3
a X 1 b X 3
4
1
4 b X 0
1
3 b X 1
2
a X 2 b X 2
3
a X 1 b X 3
4
4 Y b X 0
Y
0
1
0
0
0
0
0
1
x 1 x
x 1 x
u t ( )
x
2
0
0
1
0
d dt
2 x 3
2 x 3
a
a
a
x
x
0
a 1
3
4
4
0 0 0 1
x 3 x
4
a x 0 1
2 x 1 x
y t ( )
b 0
b 1
b 2
b 3
x 1 x 2 x 3 u a x 3 4 b x 1 2
b x 0 1
a x 2 3 b x 2 3
a x 1 2 b x 3 4
x 4 y
2 x 3
x
4
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22
Ex. 2
2
3
G s ( )
4
Signal – Flow Graph & Block Diagram Models (10) Y s ( ) U s ( )
s
b s 2
b s 3
b s b 1 0 a s a 1
2 a s 2
3 a s 3
0
3
2
G s ( )
.
4
Y s ( ) U s ( )
( ) Z s Z s ( )
s
b s 3
b s 2
3 a s 3
2 a s 2
b s b 0 1 a s a 1
0
y
3
2
b 3
b 2
b 1
b z 0
(
)
b s 3
b s 2
b s b Z s ( ) 1 0
4
3
2
(
s
)
a s 3
a s 2
a s a Z s ( ) 1 0
Y s ( ) U s ( )
a
a
3
2
a 1
a z 0
dz dt
3 d z 3 dt 4 d z 4 dt
2 d z 2 dt 3 d z 3 dt
dz dt 2 d z 2 dt
u
x
2
z
2
x 3 x
4
a x 0 1
x 3 x
z z z
4
x 1 x 2 x 3
x 1 x
x 1 x 2 x 3 u a x 3 4 b x 1 2
b x 0 1
a x 2 3 b x 2 3
a x 1 2 b x 3 4
x 4 y
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Ex. 2
3
2
3b
G s ( )
4
Signal – Flow Graph & Block Diagram Models (11) Y s ( ) U s ( )
s
b s 2
b s 3
b s b 1 0 a s a 1
2 a s 2
3 a s 3
0
2b
4X
1b
1
1 s
1/ s
1/ s
1/ s
0b
x
( )U s
2
( )Y s
3X
2X
1X
3a
2a
1a
x 3 x
4
0a
a x 0 1
phase variable canonical form
x 1 x 2 x 3 u a x 3 4 b x 1 2
b x 0 1
a x 2 3 b x 2 3
a x 1 2 b x 3 4
x 4 y
3b
2b
1b
4X
3X
2X
1X
( )Y s
( )U s
0b
1 s
1 s
1 s
1 s
( )
( )
3a
2a
( )
1a
( )
0a
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Ex. 2
3
2
3b
G s ( )
4
Signal – Flow Graph & Block Diagram Models (12) Y s ( ) U s ( )
s
b s 2
b s 3
b s b 1 0 a s a 1
2 a s 2
3 a s 3
0
2b
4X
1b
1
1 s
1/ s
1/ s
1/ s
0b
( )U s
( )Y s
1X
3X
2X
3a
2a
1a
0a
phase variable canonical form
3b
2b
1x
2X
1X
1
1/ s
1/ s
1/ s
1b
1
( )Y s
( )U s
1/ s
1
1
4X
3X
0b
4x
3x
2x
2a
1a
3a
0a
input feedforward canonical form
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25
Ex. 2
3
2
G s ( )
4
Signal – Flow Graph & Block Diagram Models (13) Y s ( ) U s ( )
s
b s 2
b s 3
b s b 1 0 a s a 1
2 a s 2
3 a s 3
0
x
3
2 x 3
2
x u t ( )
x
b u 3 b u 2 b u 1
0
a x 3 1 a x 2 1 a x 1 1 a x 0 1
4 b u 0
x d dt
x 1
x 1 x x 3 x 4 y
3b
2b
1x
2X
1X
1
1/ s
1/ s
1/ s
1b
1
( )Y s
( )U s
1/ s
1
1
4X
3X
0b
4x
3x
2x
2a
1a
3a
0a
y t ( ) x a a 2 a 1 a 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 b 3 b 2 b 1 b 0 0 ( ) u t
input feedforward canonical form
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26
Ex. 2
3
2
G s ( )
4
Signal – Flow Graph & Block Diagram Models (14) Y s ( ) U s ( )
s
b s 2
b s 3
b s b 1 0 a s a 1
2 a s 2
3 a s 3
0
3b
2b
1x
2X
1X
1/ s
1
1/ s
1/ s
1b
1
( )Y s
( )U s
1/ s
1
1
4X
3X
0b
4x
3x
2x
2a
1a
3a
0a
3b
2b
1b
4x
3x
2x
1x
4X
3X
2X
1X
( )U s
( )Y s
0b
1 s
1 s
1 s
1 s
( )
( )
( )
( )
3a
2a
1a
0a
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27
Ex. 3
( )U s
( )Y s
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (15) Y s ( ) 1)( 3)( U s ( )
T s ( )
( )
2
G 1 G
3
3
s 2 s 6
3
1
X1
X2
s
6
s
s 3 1 10
9
6 s s 2 s 21
G
( )
H
2X
1X
1
G GH
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3 s 10 1 9 6 s 21 s 2
Ex. 3
( )U s
( )Y s
2
1
3
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (15) 1)( 3)(
3
( )
3
9
1
1/ s
1/ s
1/ s
6
( )U s
( )Y s
1X
3X
2X
10
21
6
1
2
3
4
T s ( )
1
4
2
3
1
b s 3 a s 3
b s 2 a s 2
b s 1 a s 1
b s 0 a s 0
3b
2b
4X
1b
1
1 s
1/ s
1/ s
1/ s
0b
( )U s
( )Y s
3X
2X
1X
3a
2a
1a
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0a
T s ( ) 1 s 6 s s 3 1 10 9 6 s s 2 s 21
Ex. 3
( )U s
( )Y s
2
3
1
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (16) 1)( 3)(
3
( )
3
9
1
1/ s
1/ s
1/ s
6
( )U s
( )Y s
1X
3X
2X
10
21
6
3
9
3X
2X
1X
( )Y s
( )U s
6
1 s
1 s
1 s
( )
10
21
( )
6
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30
T s ( ) 1 s 6 s s 3 1 10 9 6 s s 2 s 21
Ex. 3
( )U s
( )Y s
1
2
3
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (17) 1)( 3)(
3
( )
3
9
1
1/ s
1/ s
1/ s
6
( )U s
( )Y s
1X
3X
2X
10
21
6
0
1
0
x 1 x
0
0
1
x 1 x
u t ( )
x
2
d dt
6 21
10
2 x 3
2 x 3
0 0 1
10
21
x
6
x 1 x 2 u
x 1
2
x 1
y t ( )
x
6
9
x
3
x 3
6 9 3
2
x 1
x 3
x 3 x 3 y
2 x 3
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31
T s ( ) 1 s 6 s s 3 1 10 9 6 s s 2 s 21
Ex. 3
( )U s
( )Y s
3
2
1
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (18) 1)( 3)(
3
( )
3
9
1
1/ s
1/ s
1/ s
6
( )U s
( )Y s
1X
3X
2X
10
21
6
3
2X
1X
2x
1x
3X
1/ s
9
1/ s
1
1
( )Y s
( )U s
1/ s
1
6
3x
21
6
10
x
u 3
10
10 1 0
21 0 1
T s ( ) 1 s 6 s s 3 1 10 9 6 s s 2 s 21
x
u t ( )
9
u
21
2 x 3
2
x d dt
6
6
u 6
x 1 x 1
x 1
y t ( )
x
1 0 0
0 0
3 9 6 0 ( ) u t
x 1
x 1 x x 3 y
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32
Ex. 3
( )U s
( )Y s
1
2
3
G s ( )
s 3( s s (
s s
2) 4)
Signal – Flow Graph & Block Diagram Models (19) 1)( 3)(
3
( )
3
9
1
1/ s
1/ s
1/ s
6
( )U s
( )Y s
1X
3X
2X
10
21
6
3
2X
1X
2x
1x
3X
1/ s
9
1/ s
1
1
( )Y s
( )U s
1/ s
1
6
3x
21
6
10
3
9
3x
2x
1x
2X
1X
3X
( )U s
( )Y s
6
1 s
1 s
1 s
( )
( )
( )
10
21
6
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33
T s ( ) 1 s 6 s s 3 1 10 9 6 s s 2 s 21
State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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34
Alternative Signal – Flow Graph & Block Diagram Models (1)
Ex. 1
Field voltage
Field current
Velocity
( )Y s
( )R s
5
( )U s
I s ( )
1 2s
6 3s
Motor & load
s 1 s 5 Controller
x
x
6 2 0
0 20 5
0 5 ( ) r t 1
x
( ) y t
3 0 0 1 0 0
5
2X
1/ s
1X
1/ s
1
6
1
3X
( )Y s
( )R s
5
1/ s
( )U s
I s ( )
1
2
5
3
5
3X
1X
( )U s
I s ( )
( )R s
( )Y s
5
6
2X
1 s
1 s
1 s
( )( )
( )( )
( )( )
5
3
2
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35
Alternative Signal – Flow Graph & Block Diagram Models (2)
Ex. 1
Field voltage
Field current
Velocity
( )Y s
( )R s
5
( )U s
I s ( )
1 2s
6 3s
Motor & load
s 1 s 5 Controller
T s ( )
1X
20
1/ s
1X
1 s
( )( )
1
5
20
( )
5
( )
2X
2X
1
( )R s
( )Y s
( )R s
( )Y s
10
10
1/ s
1 s
( )( )
30
1
2
2 1/ s
3X
Y s ( ) R s ( ) 30( s 5)( 1) 2)( s ( s 3) s 20 s 5 10 s 2 30 s 3
Diagonal canonical form
3X
30
1 s
3
( )( )
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36
3
Alternative Signal – Flow Graph & Block Diagram Models (3)
Ex. 1
Field voltage
Field current
Velocity
( )Y s
( )R s
5
( )U s
I s ( )
1 2s
6 3s
5
r t ( )
Motor & load
s 1 s 5 Controller
x 1 x
r t ( )
2
2
2
r t ( )
3
x 1 x x 3 ( ) y t
10
x
30
x 3 x 20 1
x 3
2
1X
20
0
5
0
1 s
( )( )
2
0
x
0
x
5
( )
0
( )
2X
( )R s
( )Y s
10
20
3 10 30
1 s
0 ( ) y t
1 1 ( ) r t 1 x
( )( )
2
3X
Diagonal canonical form
30
1 s
( )( )
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37
3
x
2
x 1
u t ( ) 1 u t ( ) 2
2
2
x 1
d dt
u t ( ) 1 u t ( ) 2
x
0 0 0
2
Ex. 2 x 1 x x 3
x 1
x 1 x 2 x 3
x 1 x 2 x 3
Alternative Signal – Flow Graph & Block Diagram Models (4) 1 0 x 0 1 0 0
( )
1U
1X
1 s
1X
1/ s
1
( )
1U
1/ s
3X
3X
1 s
1/ s
1
2U
2X
2X
2U
1 s
( )( )
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38
Ex. 3
m
u t
Alternative Signal – Flow Graph & Block Diagram Models (5) my My ml
( ) 0
0
2
mgl
mly ml
mg
l
(
,
,
)
,
)
y y ( , ,
x x x x , 1 2
3
4
( )u t
u t
( ) 0
M
0
2
lx
x
Mx mlx 4 gx 3
2
4
( )y t
x
2
u t ( )
x
2
x 3
mg M
1 M
u
x
d dt
x 3
x 1 x 2 x 3 x
x 1 x 2 x 3 x
0 1 0 0 0 0 0 0
0 mg M / 0 g l /
0 0 1 0
0 M 1/ 0 Ml 1/(
)
4
4
u t ( )
x
4
x 3
4 g l
1 Ml
x 1
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39
State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State
Equation
6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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40
The Transfer Function from the State Equation (1)
x Ax B u y Cx D u
CX
AX s ( ) s ( )
B U s ( ) D U s ( )
X s ( ) s ( ) Y s
)
s ( )
B
U s ( )
I A X s (
1
X
s ( )
I A B s ]
[
U s ( )
( ) s U s ( ) Φ B
Y s ( )
[
U s ] ( )
CΦ B D s ( )
G s ( )
CΦ B D s ( )
Y s ( ) U s ( )
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41
L
Li C
R
0
ov
( )u t
x u u Ax B x Cv
Ci
1 C 0
R y
The Transfer Function from the State Equation (2) 1 C R L x Cx
1 L 0
Ex.
0
s
[
]
I A s
s 0
0 s
s
1 L
1 C R L
1 L
1 C R L
s s
1
]
2
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42
R L 1 C R L 1 C s ( ) [ s Φ I A 1 s ( ) s s s s 1 R L 1 LC 1 L 1 L
L
Li C
R
0
ov
( )u t
x u u Ax B x Cv
Ci
1 C 0
R y
The Transfer Function from the State Equation (3) 1 C R L x Cx
1 L 0
Ex.
G s ( )
0
R
1 C 0
1 C s ( ) s s ( )
R L s ( ) 1 L s ( )
s
R L 1 C s s ( ) Φ 1 s ( )
s
)
2
s
s
G s ( )
1 L
CΦ B D ( ) s
R LC /( R L
1 LC
Y s ( ) U s ( )
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43
State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using Control
Design Software
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44
The Time Response & the State Transition Matrix (1)
t
x
t ( )
exp(
A x t
) (0)
exp[
A
(
t
)
Bu
r d ( )
0
t
t
)
( )
d
t ( ) (0) Φ x
( Φ
Bu
0
Φ(t): the state transition matrix
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45
L
The Time Response & the State Transition Matrix (2) Li C
R
0
ov
( )u t
u u x Ax B x Cv
Ci
1 C 0
y R 1 L 0 1 C R L x Cx
Ex.
0 2
1 0
R 3, L 1, C 0.5 A , B , C
s
s
3
3
1
s ( )
[
]
1 3 2 0
Φ
s I A
2
1 s ( )
1
2 s
1
2 s
1 s 3
s
2
t
2
t
t
2
t
(2
e
e
)
e
2
e
)
( 2
t ( )
Φ
t
2
t
t
2
t
(
e
e
)
(
e
2
e
)
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46
L
The Time Response & the State Transition Matrix (3) Li C
R
0
ov
( )u t
u u x Ax B x Cv
Ci
1 C 0
R y 1 C R L x Cx 1 L 0
Ex.
t
2
t
t
2
t
(2
e
e
)
e
2
e
)
( 2
t ( )
Φ
t
2
t
t
2
t
(
e
e
)
(
e
2
e
)
t
x
t ( )
t
)
( )
d
t ( ) (0) Φ x
( Φ
Bu
0
2
t
e
(0)
x
u t
( ) 0
t ( )
(0) 1,
Φ
x 1
2
2
t
x t ( ) 1 x t ( ) 2
e
1 1
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47
Ex.
L
2
t
e
The Time Response & the State Transition Matrix (3) Li C
R
t ( )
Φ
2
t
x t ( ) 1 x t ( ) 2
e
1 1
ov
( )u t
Ci
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
) t (
) t (
) t (
1
1
2
0.5
0.5
0.5
x
x
x
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
0
0.5
1.5
2
0
0.5
1.5
2
0.2
0.4
0.6
0.8
1
0
1 t
1 t
x1(t)
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48
Cv
State Variable Models
1. The State Variables of a Dynamic System 2. The State Differential Equation 3. Signal – Flow Graph & Block Diagram Models 4. Alternative Signal – Flow Graph & Block
Diagram Models
5. The Transfer Function from the State Equation 6. The Time Response & the State Transition
Matrix
7. Analysis of State Variable Models Using
Control Design Software
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49
Analysis of State Variable Models Using Control Design Software
Ex.
2
T s ( )
3
Y s ( ) R s ( )
2 s
s
6
s 3 10
s 6 9 21 s
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50
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