Automatic Cintrol by Meiling CHEN 1

Lesson 6

(absolute) Stability

Automatic control2. Analysis

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Stability

• Internal behavior– The effect of all characteristic roots.

• External behavior– The effect by cancellation of some transfer

function poles.

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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)

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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

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System response

(i) First order system response

(ii) Second order system response

(iii) High order system response

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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

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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 :

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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

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Two characteristic roots are equal

)()()(

)()(

)()(0)0()0(

321

3

1

22

1

1

11 tuktekekty

s

k

ss

k

ss

ksY

tutryylet

tsts

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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

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Higher-order system

13441)(

5281379

58)(

24321

234

2

ss

ksk

s

k

s

ksY

ssss

ssY

Dominant root nondominant root

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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

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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.

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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

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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

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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

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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

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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

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Example 5

s

k102

102 s

2

1

s

+ -)(sR )(sY

kssss

k

sR

sY

10)2)(10(

10

)(

)(2