LINEAR CONTROL SYSTEMS

LINEAR CONTROLLINEAR CONTROL SYSTEMS SYSTEMS

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

2 Ali Karimpour Apr 2009

Lecture 24

Lecture 24

Nyquist stability criteria.

Topics to be covered include:

Nyquist stability criteria.

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Lecture 24

Mapping نگاشت

2( )f s s

b'b

dc

a a'

c'

d'

2 4

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Lecture 24

44

Mappingنگاشت

1( )

2

sf s

s

s=0.5

1-2

0.2

map of

byencircledarethatsfofzerosofnumberZ )(0

)(0 sfbymadeorigintheofntsencirclemeofnumberN

000

000 NPZ

byencircledarethatsfofpolesofnumberP )(0

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Lecture 24

55

Mappingنگاشت

1( )

2

sf s

s

r=0.5

1

-2

map of

101

000 NPZ

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Lecture 24

66

Mappingنگاشت

1( )

2

sf s

s

1-2

r=0.5

map of

110

000 NPZ

-5 7

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Lecture 24

7

1-2

1( )

2

sf s

s

map of

Mappingنگاشت

011

000 NPZ

r=2.5

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Lecture 24

1+kf(s)=0

RHPتعداد قطبهای : ، 1+kf(s)

1 ( )kf s

Nyquist path

map of

Nyquist fundamental

011 NPZ

1P1Z تعداد صفرهای : RHP ، 1+kf(s)

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Lecture 24

9

( )kf s

Nyquist path1 ( )kf s

( )kf s

Nyquist fundamental

111 NPZ

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Lecture 24

10

( )kf s

Nyquist path Nyquist plot

111 NPZ

000 NPZ

Nyquist fundamental

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Lecture 24

11

-1

Example 1: Check the stability of the following system by Nyquist method. : پایداری سیستم را توسط روش 1مثال

نایکوئیست بررسی کنید.)25(

40

ss

)(sCk

+

-

)(sR

0)25(

401

ss

k

)25(

40)(

sssf

0025

40

25

40)(

ssf

454525

40

25

40)(

ssf

909025

40

25

40)(

ssf

)25(

40)(

jjsf

180)90(

4040)(

22

ssf

90)45(

4040)(

22

ssf

0)0(

4040)(

22

ssf

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Lecture 24

12

0k 111 NPZ

0k

111 PNZ 01 PN 00 0 0for Stable k

111 NPZ 111 PNZ 01 PN 01 1 0for Unstable k

)25(

40)(

sssf

zeroRHPOne

Example 1: Check the stability of the following system by Nyquist method. : پایداری سیستم را توسط روش 1مثال

نایکوئیست بررسی کنید.

)25(

40

ss

)(sCk

+

-

)(sR

-1

system above theofstability or the

0)25(

401

of roots RHP theChecking

ss

k

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Lecture 24

13

( )kf s

Nyquist path Nyquist plot

0111 kNPZ000 NPZ

Nyquist fundamental

0111 kNPZ

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Lecture 24

14

More study مطالعه بیشتر

0)25(

40

of roots RHP theChecking

ss

k

000 NPZ 000

)25(

40

ss

)(sCk

+

-

)(sR

)25(

40)(

sssf

-1

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Lecture 24

15

-1

)10)(5(

375

sss

)(sCk

+

-

)(sR

0)10)(5(

3751

sss

k

)10)(5(

375)(

ssssf

0050

375

50

375)(

ssf

454550

375

50

375)(

ssf

909050

375

50

375)(

ssf

270)90(

375375)(

33

ssf

135)45(

375375)(

33

ssf

0)0(

375375)(

33

ssf

Ver

y im

por

tan

t

Ver

y im

por

tan

t

Example 2: Check the stability of following system by Nyquist method. : پایداری سیستم را توسط روش 2مثال

نایکوئیست بررسی کنید.

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Lecture 24

16

Example 2: Check the stability of following system by Nyquist method.

پایداری سیستم را توسط روش نایکوئیست بررسی کنید.

)10)(5(

375

sss

)(sCk

+

-

)(sR

0)10)(5(

3751

sss

k

)10)(5(

375)(

ssssf

)10)(5(

375)(

jjjjf

)( jf

0

90

5

16295.0

07.7

1805.0

10

19824.0

20

23004.0

-1

Very important

How to ca

lculate?

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Lecture 24

17

)10)(5(

375)(

ssssf

-1

0k 111 NPZ111 PNZ 01 PN 1N

2k0for Stable

2for Unstable k

system above ofstability or

0)10)(5(

3751

of roots RHP theChecking

sss

k

2

0

rootsRHPTwo

Example 2: Check the stability of following system by Nyquist method.

پایداری سیستم را توسط روش نایکوئیست بررسی کنید.

)10)(5(

375

sss

)(sCk

+

-

)(sR

18 Ali Karimpour Apr 2009

Lecture 24

18

)10)(5(

375)(

ssssf

-1

0k 111 NPZ 111 PNZ 01 PN 01 1 0for Unstable k

system above ofstability or

0)10)(5(

3751

of roots RHP theChecking

sss

k

rootRHP1

Example 2: Check the stability of following system by Nyquist method.

پایداری سیستم را توسط روش نایکوئیست بررسی کنید.

)10)(5(

375

sss

)(sCk

+

-

)(sR

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Lecture 24

19

)10)(5(

375

sss

)(sCk

+

-

)(sR

)10)(5(

375)(

ssssf

-1

More study مطالعه بیشتر

0)10)(5(

375

of roots RHP theChecinkg

sss

k

000 NPZ 000

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Lecture 24

)(

1

sd

)(sCk

+

-

)(sR

3ٍExample 3: Check the stability of following system from the given Nyquist plot. : پایداری سیستم را با توجه به منحنی نایکوئیست داده 3مثال

شده بررسی کنید.

000 NPZ 10 0 P 10 1 PP

d(s) is a polynomial.

01.06

1

60 k 111 NPZ 011 Z 11 Z System is unstable( 1 RHP zero)

6 10k 111 NPZ 1 0z System is stable

10k 111 NPZ 1 2z 0k 111 NPZ 1 1z

111 Z

111 Z011 Z

System is unstable( 2 RHP zero)

System is unstable( 1 RHP zero)

)(

1

sd

21 Ali Karimpour Apr 2009

Lecture 24

)(

1

sd

)(sCk

+

-

)(sR

3ٍExample 3: Check the stability of following system from the given Nyquist plot. : پایداری سیستم را با توجه به منحنی نایکوئیست داده 3مثال

شده بررسی کنید.

000 NPZ 10 0 P 10 1 PP

d(s) is a polynomial.

01.06

1

60 k

111 NPZSystem is unstable( 1 RHP zero)

6 10k System is stable

10k

0k 111 NPZ 1 1z 011 Z

System is unstable( 2 RHP zero)

System is unstable( 1 RHP zero)

)(

1

sd

11 PN

10

11

11

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Lecture 24

Exercises تمرینها

1- Assume the following control system

Find the value of k which make the system stable. Answer: k>1

)2( sk)3(

102 ss

+

-

+

-

R(s) C(s)

2- Consider the following control system

Find the value of k which make the system stable.

+G(s)

- K

)1(

3)(

ss

ssG

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Lecture 24

Exercises تمرینها

3- Polar plot of an open loop transfer function is shown in the following:

W=0

-1+j0

Discuss about the stability of system.

Answer: It is unstable and it has 4 unstable poles.

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Lecture 24

Exercises تمرینها

4- Discuss the stability of following system by Nyquist criteria..

)3)(1(

)2(4

ss

s++

-