Lecture Control system design: State variable models - Nguyễn Công Phương
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Nguyễn Công Phương
CONTROL SYSTEM DESIGN
State Variable Models
Contents
I. Introduction
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.
2.
3.
4.
The State Variables of a Dynamic System
The State Differential Equation
Signal – Flow Graph & Block Diagram Models
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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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)
d 2 y (t )
dy (t )
M
b
ky (t ) u (t )
2
dt
dt
dy (t )
x1 (t ) y (t ), x2 (t )
dt
Wall
friction
b
dx1
x2
dx
dt
M 2 bx2 kx1 u (t )
dt
dx2 b x k x 1 u
2
1
dt
M
M
M
ic C
dvC
u ( t ) iL
dt
di
L L RiL vC
dt
vo RiL (t )
x1 vC , x2 iL
1
1
dx1
x
u (t )
2
dt
C
C
dx2 1 x R x
1
2
dt
L
L
u (t )
v (t ) Rx
o
2
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vC
k
Mass
M
y(t)
iL
C
u(t)
L
vo
R
iC
5
State Variable Models
1.
2.
3.
4.
The State Variables of a Dynamic System
The State Differential Equation
Signal – Flow Graph & Block Diagram Models
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)
x1 a11 x1 a12 x2 ... a1n xn b11u1 ... b1m um
x a x a x ... a x b u ... b u
2
21 1
22 2
2n n
21 1
2m m
xn an1 x1 an 2 x2 ... ann xn bn1u1 ... bnm um
x1 a11 a12
d x2 a21 a22
dt
xn an1 an 2
a1n x1
b11 b1m u2
a 2 n x2
bn1 bnm um
ann xn
x Ax Bu
y Cx Du
x (t ) exp( At )x (0)
t
t
0
0
exp[A(t )Bu(r)d Φ(t )x(0) Φ(t )Bu( )d
X ( s ) [ sI A ]1 x (0)[ sI A ]1 BU( s )
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The State Differential Equation
(2)
1
1
dx1
x
u (t )
2
dt
C
C
dx2 1 x R x
1
2
dt
L
L
v (t ) Rx
o
2
1
1
0 C
x
x C u (t )
1 R
0
L
L
y 0 R x
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u (t )
vC
iL
C
L
vo
R
iC
8
The State Differential Equation
(3)
q
k1
k2
M 1a1 u f spring f damp
M 1
p u k1 ( p q ) b1 ( p q )
M2
b2
p
u
M1
b1
M 1
p b1 p k1 p u k1q b1q
M 2 q k1 ( p q) b1 ( p q ) k2 q b2 q
M 2 q ( k1 k2 ) q (b1 b2 ) q k1 p b1 p
x3 x1 p
x4 x2 q
b1
k1
1
k1
b1
x
p
p
p
u
q
q
3
M1
M1
M1
M1
M1
x4 q k1 k2 q b1 b2 q k1 p b1 p
M2
M2
M2
M2
x1 p
,
x2 q
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The State Differential Equation
(4)
b1
k1
1
k1
b1
x
p
p
p
u
q
q
3
M1
M1
M1
M1
M1
x4 q k1 k2 q b1 b2 q k1 p b1 p
M2
M2
M2
M2
x1 p
,
x2 q
x3 x1 p
x4 x2 q
k1
k1
b1
b1
1
x
x
x
x
x
u
1
2
3
4
3
M1
M1
M1
M1
M1
x k1 x k1 k2 x b1 x b1 b2 x
2
3
4
4 M 2 1
M2
M2
M2
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The State Differential Equation
(5)
k1
k1
b1
b1
1
x
x
x
x
x
u
1
2
3
4
3
M1
M1
M1
M1
M1
x4 k1 x1 k1 k2 x2 b1 x3 b1 b2 x4
M2
M2
M2
M2
0
0
x1 p
x q
k
x 2 , A 1
M1
x3 p
k1
x4 q
M
2
0
0
k1
M1
k1 k2
M2
1
0
b
1
M1
b1
M2
0
0
, B 1
M
1
b1 b2
0
M2
0
1
b1
M1
x Ax Bu
y p x1 1 0 0 0 x Cx
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The State Differential Equation
(6)
q
p
k1
k2
M2
u
M1
b1
b2
k1
k1
b1
b1
1
x
x
x
x
x
u
1
2
3
4
3
M1
M1
M1
M1
M1
x k1 x k1 k2 x b1 x b1 b2 x
2
3
4
4 M 2 1
M2
M2
M2
p
q
k2 q
b2 q
M2
k1 ( q p )
k1 ( p q)
b1 ( q p )
b1 ( p q )
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M2
u
12
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)
?
1
1
dx1
dt C x2 C u (t )
dx2 1 x R x
1
2
dt
L
L
v (t ) Rx
o
2
vC
u (t )
1
C
C
vo
R
iC
1
L
1
s
L
iL
R
L
R
U ( s)
X1
Vo ( s )
R /( LC )
G( s)
2
U ( s ) s ( R / L) s 1/( LC )
U ( s) 1
C
( )
Vo ( s )
1/ s
X2
1
C
R
L
( )
1 X1 1
L
s
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1
C
1
s
X2
R
14
Vo ( s )
Signal – Flow Graph
& Block Diagram Models (2)
Y ( s ) bm s m bm 1s m 1 ... b1s b0
G( s)
n
, nm
n 1
U ( s)
s an 1s ... a1s a0
bm s ( n m ) bm 1s ( n m 1) ... b1s ( n 1) b0 s n
s an 1s 1 ... a1s ( n 1) a0 s n
P
1 L
k
k
N
q 1 q
Sum of the forward-path factor
1 sum of the feedback loop factors
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Ex. 1
Signal – Flow Graph
& Block Diagram Models (3)
b0
b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3 s 1 a2 s 2 a1s 3 a0 s 4
( s 4 a3 s 3 a2 s 2 a1s a0 )Y ( s ) b0U ( s )
d 4 ( y / b0 )
d 3 ( y / b0 )
d 2 ( y / b0 )
d ( y / b0 )
a
a
a
a0 ( y / b0 ) u
3
2
1
4
3
2
dt
dt
dt
dt
1
1
1
1
x1 y / b0
b0
1
s X4
s
s
s
U ( s)
x2 x1 y / b0
X3
X2
X1
a
3
x3 x2
y / b0
x4 x3
y / b0
a2
Y ( s)
a1
a0
U ( s)
( )
1
( ) s
X4
1
s
X3
1
s
X2
1
s
X1
1
s
b0
Y ( s)
a3
( )
( )
a2
a1
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a0
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Ex. 1
Signal – Flow Graph
& Block Diagram Models (4)
b0
b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3s 1 a2 s 2 a1s 3 a0 s 4
d 4 ( y / b0 )
d 3 ( y / b0 )
d 2 ( y / b0 )
d ( y / b0 )
a
a
a
a0 ( y / b0 ) u
3
2
1
dt
dt 4
dt 3
dt 2
x1 y / b0
x2 x1 y / b0
x3 x2
y / b0
x4 x3
y / b0
x 4 a0 x1 a1 x2 a2 x3 a3 x4 u
y b0 x1
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Ex. 1
Signal – Flow Graph
& Block Diagram Models (5)
b0
b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3s 1 a2 s 2 a1s 3 a0 s 4
x4 a0 x1 a1 x2 a2 x3 a3 x4 u
y b0 x1
x1 0
x 0
2
x3 0
x 4 a0
0
0
0
a1
y (t ) Cx b0
0
0
0
a2
0 x1 0
0 x2 0
u (t ) x Ax Bu
0 x3 0
a3 x4 1
x1
x
0 0 0 2
x3
x4
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Ex. 1
Signal – Flow Graph
& Block Diagram Models (6)
b0
b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3 s 1 a2 s 2 a1s 3 a0 s 4
1
s
1
X4
1
s
1
s
1
s
b0
U ( s)
Y ( s)
a3
P
L
Y ( s)
G( s)
U ( s) 1
k
k
N
( )
X2
a2
X1
a1
a0
q 1 q
U ( s)
X3
Sum of the forward-path factor
1 sum of the feedback loop factors
1
( ) s
X4
1
s
X3
1
s
X2
1
s
X1
b0
Y ( s)
a3
( )
( )
a2
a1
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a0
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Ex. 2
Signal – Flow Graph
& Block Diagram Models (7)
b3 s 3 b2 s 2 b1s b0
b3s 1 b2 s 2 b1s 3 b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3s 1 a2 s 2 a1s 3 a0 s 4
P
L
Y ( s)
G( s)
U ( s) 1
k
k
N
q 1 q
Sum of the forward-path factor
1 sum of the feedback loop factors
b3
1
1
s
b2
X4
U ( s)
a3
1/ s
1/ s
X3
a2
1/ s
X2
b1 b
0
Y ( s)
X1
a1
a0
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Ex. 2
Signal – Flow Graph
& Block Diagram Models (8)
b3 s 3 b2 s 2 b1s b0
b3s 1 b2 s 2 b1s 3 b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3s 1 a2 s 2 a1s 3 a0 s 4
b3
1
1
s
b2
X4
U ( s)
a3
1/ s
1/ s
X3
a2
1/ s
X2
b1 b
0
Y ( s)
X1
a1
a0
X1 X 2 / s
X X / s
2
3
X3 X4 / s
X 4 (U a3 X 4 a2 X 3 a1 X 2 a0 X 1 ) / s
Y b0 X 1 b1 X 2 b2 X 3 b3 X 4
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Ex. 2
Signal – Flow Graph
& Block Diagram Models (9)
b3 s 3 b2 s 2 b1s b0
b3s 1 b2 s 2 b1s 3 b0 s 4
Y ( s)
G( s)
4
3
2
U ( s ) s a3 s a2 s a1s a0 1 a3s 1 a2 s 2 a1s 3 a0 s 4
X1 X 2 / s
X 2 X 3 / s
X 3 X 4 / s
X (U a X a X a X a X ) / s
3 4
2 3
1 2
0 1
4
Y b0 X 1 b1 X 2 b2 X 3 b3 X 4
x2 x1
x3 x2
x4 x3
x u a x a x a x a x
3 4
2 3
1 2
0 1
4
y b0 x1 b1 x2 b2 x3 b3 x4
sX 1 X 2
sX 2 X 3
sX 3 X 4
sX (U a X a X a X a X )
3 4
2 3
1 2
0 1
4
Y b0 X 1 b1 X 2 b2 X 3 b3 X 4
1
0
x1 0
0
1
d x2 0
dt x3 0
0
0
x4 a0 a1 a2
x1
x
y (t ) b0 b1 b2 b3 2
x3
x4
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0 x1 0
0 x2 0
u (t )
1 x3 0
a3 x 4 1
22
Ex. 2
Signal – Flow Graph
& Block Diagram Models (10)
b3 s 3 b2 s 2 b1s b0
Y ( s)
G( s)
4
U ( s ) s a3 s 3 a2 s 2 a1s a0
b3 s 3 b2 s 2 b1s b0
Y ( s)
Z ( s)
G( s)
4
.
U ( s ) s a3 s 3 a2 s 2 a1s a0 Z ( s )
d 3z
d 2z
dz
y
b
b
b
b0 z
3
2
1
3
2
Y ( s ) (b3 s 3 b2 s 2 b1s b0 ) Z ( s )
dt
dt
dt
4
3
2
4
3
2
U ( s ) ( s a3 s a2 s a1s a0 ) Z ( s )
u d z a d z a d z a dz a z
3
2
1
0
dt
dt 4
dt 3
dt 2
x1 z
x x z
2
1
z
x3 x2
x4 x3
z
x2 x1
x3 x2
x4 x3
x u a x a x a x a x
3 4
2 3
1 2
0 1
4
y b0 x1 b1 x2 b2 x3 b3 x4
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Ex. 2
Signal – Flow Graph
& Block Diagram Models (11)
b3 s 3 b2 s 2 b1s b0
Y ( s)
G( s)
4
U ( s ) s a3 s 3 a2 s 2 a1s a0
b3
1
s
1
x2 x1
U ( s)
x3 x2
x4 x3
x u a x a x a x a x
3 4
2 3
1 2
0 1
4
y b0 x1 b1 x2 b2 x3 b3 x4
b2
X4
1/ s
1/ s
a3
X3
a2
1/ s
X2
b1 b
0
Y ( s)
X1
a1
a0
phase variable canonical form
b3
b2
b1
U ( s)
( )
1
( ) s
X4
1
s
X3
1
s
X2
1
s
X1
b0
Y ( s)
a3
( )
( )
a2
a1
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a0
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Ex. 2
Signal – Flow Graph
& Block Diagram Models (12)
b3 s 3 b2 s 2 b1s b0
Y ( s)
G( s)
4
U ( s ) s a3 s 3 a2 s 2 a1s a0
b3
1
s
1
b2
X4
U ( s)
a3
1/ s
1/ s
X3
a2
b1 b
0
1/ s
X2
Y ( s)
X1
a1
a0
phase variable canonical form
b3
b2
b1
U ( s)
b0 x4
1/ s
1/ s X 2
1/ s
X4
1
x3
X3
1
x2
a1
1
a2
a0
x1 1/ s X 1
1
Y ( s)
a3
input feedforward canonical form
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