automatic cintrol by meiling chen1 lesson 6 (absolute) stability automatic control 2. analysis
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Automatic Cintrol by Meiling CHEN 1
Lesson 6
(absolute) Stability
Automatic control2. Analysis
Automatic Cintrol by Meiling CHEN 2
Stability
• Internal behavior– The effect of all characteristic roots.
• External behavior– The effect by cancellation of some transfer
function poles.
Automatic Cintrol by Meiling CHEN 3
Definition :
A system is internal (asymptotic) stable, if the zero-input response decays to zero, as time approaches infinity, for all possible initial conditions.
Asymptotic stable =>All the characteristic polynomial roots are located in the LHP (left-half-plan)
Automatic Cintrol by Meiling CHEN 4
Definition :
A system is external (bounded-input, bounded-output) stable, if the zero-state response is bounded, as time approaches infinity, for all bounded inputs..
bounded-input, bounded-output stable =>All the poles of transfer function are located in the LHP (left-half-plan)
Asymptotic stable => BIBO stableBIBO stable=> Asymptotic stable
Automatic Cintrol by Meiling CHEN 5
System response
(i) First order system response
(ii) Second order system response
(iii) High order system response
Automatic Cintrol by Meiling CHEN 6
First order
tata eyea
Abtu
a
Abty
yasas
ab
A
s
ab
A
sY
tAutrlet
yas
sRas
bsY
rbyadt
dy
00 )0()()(
)0(1
)(
)()(
)0(1
)()(
0
0
0
0
00
0
0
0
0
00
0
00
Automatic Cintrol by Meiling CHEN 7
Second order
012
1
012
01
01012
2
)0()1()0()()(
asas
yayssR
asas
bsbsY
rbdt
drbya
dt
dya
dt
yd
(a) Two characteristic roots are real and distinct.(b)Two characteristic roots are equal.(c) Two characteristic roots are complex numbers.
Three cases :
Automatic Cintrol by Meiling CHEN 8
Two characteristic roots are real and distinct.
)()()(
)(
)()(0)0()0(
321
3
2
2
1
1
21 tukekekty
s
k
ss
k
ss
ksY
tutryylet
tsts
Automatic Cintrol by Meiling CHEN 9
Two characteristic roots are equal
)()()(
)()(
)()(0)0()0(
321
3
1
22
1
1
11 tuktekekty
s
k
ss
k
ss
ksY
tutryylet
tsts
Automatic Cintrol by Meiling CHEN 10
Two characteristic roots are complex numbers
2
22
2
221
1
)(2
)(
)()(
)(
)()(0)0()0(
n
n
nn
n sRss
sY
sRs
ksY
tutryylet
Undamped natural frequencyDamping ratio
)cos1sin(1
1)( 12
2
te
ty n
tn
Automatic Cintrol by Meiling CHEN 11
Automatic Cintrol by Meiling CHEN 12
Automatic Cintrol by Meiling CHEN 13
Automatic Cintrol by Meiling CHEN 14
Higher-order system
13441)(
5281379
58)(
24321
234
2
ss
ksk
s
k
s
ksY
ssss
ssY
Dominant root nondominant root
Automatic Cintrol by Meiling CHEN 15
Automatic Cintrol by Meiling CHEN 16
Automatic Cintrol by Meiling CHEN 17
Automatic Cintrol by Meiling CHEN 18
Stability testing
Properties of the polynomial coefficients :
• Differing algebraic signs
• Zero-valued coefficients
• All of the same algebraic sign, non zero
102357 2346 sssss
173823 2456 sssss
1072368 2345 sssss
At least one RHP root
Has imaginary axis roots or RHP roots or both
No direct information
Automatic Cintrol by Meiling CHEN 19
Routh-Hurwitz testing
0111)( asasasasP n
nn
n
0
3213
3212
5311
42
s
cccs
bbbs
aaas
aaas
n
n
nnnn
nnnn
1
3121
n
nnnn
a
aaaab
1
5142
n
nnnn
a
aaaab
1
21131 b
babac nn
The number of RHP roots of P(s) is the number of algebra sign changes in the elements of the left column of the array.
Automatic Cintrol by Meiling CHEN 20
Example 1
6
11
32
63
018
3
11
3
41523
652
62532)(
0
1
2
3
4
24 3
s
s
s
s
s
sssssP
Two roots in the RHP
Automatic Cintrol by Meiling CHEN 21
Example 2
1
2
11
42
131
1432)(
0
1
2
3
4
24 3
s
s
s
s
s
sssssP
no root in the RHP
Automatic Cintrol by Meiling CHEN 22
Example 3
0
1
2
3
4
234
50
46
523
54263)(
s
s
s
s
s
sssssP
5)1(
50nB
A
5
10
55
0
1
2
s
s
s Two roots in the RHP
n 移位次數移至 0消失為止
Automatic Cintrol by Meiling CHEN 23
Example 4
0
1
2
3
4
5
245
00
123
105.2
12112
1681
12161182)( 3
s
s
s
s
s
s
ssssssP
123 2 s
factor
Automatic Cintrol by Meiling CHEN 24
12
6
123
105.2
12112
1681
12161182)(
0
1
2
3
4
5
245 3
s
s
s
s
s
s
ssssssP
ssds
d6)123( 2
no root in the RHP
Automatic Cintrol by Meiling CHEN 25
Example 5
s
k102
102 s
2
1
s
+ -)(sR )(sY
kssss
k
sR
sY
10)2)(10(
10
)(
)(2