Nguyễn Công Phương
CONTROL SYSTEM DESIGN
The Stability of Linear Feedback 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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The Stability of Linear Feedback Systems
1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design
Software
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The Concept of Stability (1)
• Stability is of the utmost importance. • A close – loop feedback system that is unstable
is of little value.
• A stable system is a dynamic system with a bounded response to a bounded input.
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The Concept of Stability (2)
http://www.ctc.org.uk/cyclists-library/bikes-and-other- cycles/cycle-styles/city-bike
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The Concept of Stability (4)
M
N
M
N
t
t
i
k
Y s ( )
y t ( ) 1
sin(
t
)
A e i
D e k
k
k
2
1 s
s
A i
k s
s
)
i
B s C k 2 ( 2 k
k
2 k
i
k
i
k
1
1
1
1
j
1
1
10
1
0
0
0
0
-1
-1
-1
-10
5
10
5
10
5
10
5
10
0
0
0
0
1
1
10
1
0
0
0
0
-1
-1
-1
-10
0
0
0
5
10
5
10
5
10
5
10
0
1
1
2
10
0.5
0.5
1
5
0
0
0
0
0
0
0
5
10
5
10
5
10
0
5
10
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The Stability of Linear Feedback Systems
1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design
Software
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The Routh – Hurwitz Stability Criterion (1)
n
n
n
2
1
n
n
2
0
1
ns
na
2na
4na
a
n
2
n
n
a a n n
3
a 1
,
b n
1
a a
a n a
1 a
2 a
n
n
3
1
n
n
1
1
1ns
1na
3na
5na
2ns
3nb
5nb
1nb
n
4
,
b n
3
a a
a n a
1 a
3ns
n
n
5
1
n
1
3nc
5nc
1nc
3
1
,
c n
1
0s
1nh
a n b n
a n b n
3
1
1 b n
1
The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array
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q s ( ) a s a s 0 ... a s n a s a 1
The Routh – Hurwitz Stability Criterion (2)
n
n
n
2
1
n
n
2
0
1
q s ( ) a s s a 0 ... a s n
ns
na
2na
4na
zero.
2. There is a zero in the 1st column,
1ns
1na
3na
5na
2ns
3nb
5nb
1nb
but some other elements of the row containing the zero in the 1st column are nonzero.
3ns
3nc
5nc
1nc
3. There is a zero in the 1st column, and the other elements of the row containing the zero are also zero. 4. Repeated roots of the characteristic
0s
1nh
equation on the jω – axis.
The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array
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a s a 1 1. No element in the 1st column is
The Routh – Hurwitz Stability Criterion (3)
2
ns
na
2na
4na
Ex. 1 ( )q s
2s
2s
1ns
2a
0a
2a
0a
1na
3na
5na
1s
1s
2ns
0
0
1a
1a
3nb
5nb
1nb
0s
0s
3ns
0
0
1b
0a
3nc
5nc
1nc
n
2
b n
1
b 1
a 0
a a
a n a
a 0 0
1 a
0s
n
n
3
1
n
1
a 1 2 aa 1 1
1nh
The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array
The system is stable if a2, a1 & a0 are all positive or all negative
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a s 2 a s a 1 0
The Routh – Hurwitz Stability Criterion (4)
3
2
ns
na
2na
4na
s s 2 s 50
Ex. 2 q s ( )
1ns
3s
3s
1na
3na
5na
1
2
1
2
2ns
2s
50
2s
3nb
5nb
1nb
50
1
1
3ns
1s
1s
3nc
5nc
1nc
0
48
1b
0b
0s
0s
50
0
1c
0c
0s
1nh
0,
48,
50,
0
b 0
b 1
c 1
c 0
1 0 1 0
2 1 1 50
1 48
50 0
0 1 48 0
1 1
1 1
1 48
1 48
The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array
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The Routh – Hurwitz Stability Criterion (5)
3
2
Ex. 3 ( )q s
0
3s
3a
1a
2s
0a
2a
a a 2 1
a a 0 3
,
b 1
c 1
a 0
a
2
1s
0
1b
0s
0
1c
0
0
0
a 3 a
0
a 3 a
0
0
2
2 a a 2 1
a a 0 3
0
0
0
a a 0 3
a
0
0
2 a a 2 1
a 0
a 3 a b 1 c 1
2 0
a 0
3
2
q s ( )
s
2
s
6
s
10,
2 6 10 1 2
a a 2 1
a a 0 3
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a s 3 a s 2 a s a 1
3
4
2
5
The Routh – Hurwitz Stability Criterion (6) s 11
10 2 4 s s s s 2
Ex. 4 q s ( )
5s
5s
2
11
1
2
11
1
4s
4s
10
10
4
2
4
2
3s
3s
6
0
0
6
0
2s
2s
10
0
10
0
1c
1c
1s
1s
0
0
0
0
1d
1d
0s
0s
10
10
12
6
4
10
c 1
,
4
6 10
c 1
d 1
12
c 1
Two sign changes two roots with positive real part the system is unstable
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The Routh – Hurwitz Stability Criterion (7)
4
3
2
s s s s K
Ex. 5 ( )q s
4s
4s
K
K
1
1
1
1
3s
3s
0
0
1
1
1
1
2s
2s
K
K
0
0
0
1s
1s
0
0
0
0
1c
1c
1s
1s
K
K
K
1
c 1
K
One sign change one root with positive real part the system is unstable for all values of K
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2
3
3
4
3
2
5
The Routh – Hurwitz Stability Criterion (8) s s ( s 3
21)( 3) s s 24 63 4 s s s s
Ex. 6 q s ( )
5s
3
4
1
5
4
2
2
3
4s
63
24
1
5
s s 24 s s 3 63
s
21
s 3s 3 2s s
4
3s
0
20
60
s
s
s 3
63
4
s
2s
63
0
21
3
s
s 3
63
224 s 23 s 221 s
1s
0
0
0
3
s
s 3
2
2
U s
s
3)
( ) 21 s
63 21(
221 s
63
221 s
4 s 33 s 3
0
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63
3
2
3
4
3
2
5
The Routh – Hurwitz Stability Criterion (9) s s ( s 3
21)( 3) s s 24 63 4 s s s s
Ex. 6 q s ( )
5s
3
1
4
4s
63
1
24
3s
1
1
3s
0
20
60
2s
21
1
2s
63
0
21
1s
0
20
1s
0
0
0
0s
0
21
Two sign changes two roots with positive real part the system is unstable
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The Routh – Hurwitz Stability Criterion (10)
Ex. 7
( )R s
( )Y s
)
)
G s ( )
1 2)(
s s (
s
3)
K s a ( 1 s
1 2)(
s s (
s
3)
K s a ( 1 s
( )
T s ( )
4
3
1
G s ( ) G s ( )
s
6
s
6)
K s a ( ) 2 ( K s 11
s Ka
K
60
4
3
2
0
Ka
6 b K ( 3
Ka
4s
11
1
0
6) 6 b 3
3s
6
6K
0
Ka
0
2s
Ka
0
3b
60
K
0
1s
0
0
3c
(60
6)
1s
Ka
)( K K K 36
a
Ka
60
K
b K ( 3
,
b 3
c 3
6
6) 6 b 3
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q s ( ) s 6 s s 11 ( K 6) s Ka
The Stability of Linear Feedback Systems
1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design
Software
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18
The Stability of State Variable Systems (1)
1
3
3
x
K
1
1
( )U s
X s 1( )
1/ s
x
x 1
2 Kx Ku
1/ s
2
1
2
Ex. 1 x 1 x
2X
X s 2 ( ) 1
2
1
1
s
,
s 3
,
Ks
L 1
L 2
L 3
N
...
1
L n
L L n m
L L L n m p
n
,
1
n m , nontouching
n m p , nontouching
1
2
1 2
s
(
K
3)
s
1 (
)
(
)
L 1
L 2
L 3
L L 1 2
2
q s ( )
s
2
3)
s K (
K
K
3
3 0
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a
x
x d dt
u 1 u 2
1 0 0 1 0 0
0 0 0
The Stability of State Variable Systems (2)
Ex. 2
0
0
a
0
0
det(
0
0
0
I A
) det
0
0
0
0
2
2
)
)]
[
(
(
2
2
)
)]
q ( )
[
(
(
0
2
0 sites.google.com/site/ncpdhbkhn
20
The Stability of Linear Feedback Systems
1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design
Software
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21
System Stability Using Control Design Software (1)
Ex. 1
( )R s
( )Y s
1 2)(
s s (
s
3)
s 10( 2) s 1
( )
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System Stability Using Control Design Software (2)
3
2
s 2 s
Ex. 2 q s ( )
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s K 3
System Stability Using Control Design Software (3) Ex. 3 Given a characteristic equation q(s) = s4 + 8s3 + 17s2 + (K + 10)s + aK = 0. Find a & K such that the system is stable.
K
aK
4s
17
1
[(
K
10) 8 17]
b 3
1 8
126 8
3s
10
8
K
0
aK
)
aK
(1 0 8
b 1
2s
0
1b
3b
1 8
1s
(126
10)
0
0
3c
aK 8
10)]
c 3
b [8 1
b K ( 3
K K )( 8
1 b 3
1s
3d
K
d
(
)
aK
0
3
b 3
b c 1 3
b 1
0
1 c 3
0
126 8 (126
10)
aK 8
0
0
K K )( 8
0
3
b 3 c 3 d
aK
0
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