LINEAR CONTROLLINEAR CONTROL SYSTEMS SYSTEMS

Ali KarimpourAssistant Professor

Ferdowsi University of Mashhad

2

Lecture 27

Ali Karimpour Apr 2009

Lecture 27

Topics to be covered include: Nyquist chart.

Constant M loci. Constant N loci.

Nichols chart. Constant gain loci. Constant phase loci. Nichols chart specification.

Effect of adding poles and zeros on loop transfer function.

Frequency domain charts

3

Lecture 27

Ali Karimpour Apr 2009

Closed loop transfer functions

)(sG)(sC

)(sGc

+-

)(sR

)(11

)()(11)(

)(1)(

)()(1)()()()()()(

sLsGsGsS

sLsL

sGsGsGsGsTsGsGsL

cc

cc

Frequency (rad/s)

|| T

|| S

|| L

sMPM

p

BW

4

Lecture 27

Ali Karimpour Apr 2009

)(sG)(sC

)(sGc

+-

)(sR

)10)(5(150)()(Let

ssssGsGc

-2.5 -2 -1.5 -1 -0.5 0 0.5 1-3

-2.5

-2

-1.5

-1

-0.5

0

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

1

2

3

7 20

45

Closed loop values from Nyquist chart

1

jjGjG c 8.286.0)1()1(

?)1( jT

jj

jLjLjT

8.286.018.286.0

)1(1)1()1(

2004.1)1( jT

?)2( jT

?pM

?)1( jT

?)3( jT ?)( jT

5

Lecture 27

Ali Karimpour Apr 2009

M circles (constant magnitude of T)

6

Lecture 27

Ali Karimpour Apr 2009

N circles (constant phase of T)

7

Lecture 27

Ali Karimpour Apr 2009

Closed loop values from Nichols chart

-270 -225 -180 -135 -90-40

-30

-20

-10

0

10

20Nichols Chart

Open-Loop Phase (deg)

Ope

n-Lo

op G

ain

(dB)

-270 -225 -180 -135 -90-40

-30

-20

-10

0

10

20Nichols Chart

Open-Loop Phase (deg)

Ope

n-Lo

op G

ain

(dB)

1

3

68

20

2

45

)(sG)(sC

)(sGc

+-

)(sR

)()( jGjGc

43.9)()(log20 jGjGc

1

107)()( jGjGc

10793.2)1()1( jGjG c

?)1( jT

10793.2110793.2

)1(1)1()1(

jL

jLjT

2004.1)1( jT?)2( jT

?pM

?)1( jT)10)(5(

150)()(Let

sss

sGsGc

?)3( jT ?)( jT

8

Lecture 27

Ali Karimpour Apr 2009

Constant gain and phase loci in Nichols chart

M circles and N circles on Nichols chart

GC

cG+-

R

9

Lecture 27

Ali Karimpour Apr 2009

Nichols chart specification

How to plot |T| versus frequency?

How to plot <T versus frequency?

How to derive φm and GM?

How to derive cross over frequencies?

How to derive open loop bandwidth?

How to derive closed loop bandwidth?

How to derive Mp?How to derive ωp?

How to derive type of system?

How to derive error coefficient?

GC

cG+-

R

10

Lecture 27

Ali Karimpour Apr 2009

Effect of adding poles on Bode plot.

-

c2er)(sG

s11

Adding poles

-

c2er)(sG

-80

-60

-40

-20

0

20

Mag

nitu

de (d

B)

100 101 102-270

-225

-180

-135

-90Ph

ase

(deg

)

Bode Diagram

Frequency (rad/sec)

tr System speed BW

/1

11

Lecture 27

Ali Karimpour Apr 2009

Effect of adding poles on Nyquist plot.

-

c2er)(sG

s11

Adding poles-1

G(jw)

-

c2er)(sG

12

Lecture 27

Ali Karimpour Apr 2009

Adding poles to open loop transfer functions

اضافه کردن قطب به تابع انتقال حلقه باز

-

c2er)(sG

s11

223

22

2 2)21()()()(

nnn

n

ssssRsCsM

5,2,1,05.01 n

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Step Response

Time (sec)

Ampl

itude τ=0

τ=1.0

τ=2.0τ=5.0 P.O.

tr

System speed

More problem as poles go to ??

BW

13

Lecture 27

Ali Karimpour Apr 2009

Effect of adding zeros on Bode plot.

-

c2er)(sG s1

Adding zeros

-

c2er)(sG

-80

-60

-40

-20

0

20

Mag

nitu

de (d

B)

100 101 102-270

-225

-180

-135

-90Ph

ase

(deg

)

Bode Diagram

Frequency (rad/sec)/1

tr System speed BW

14

Lecture 27

Ali Karimpour Apr 2009

Effect of adding zeros on Nyquist plot.

-

c2er)(sG s1

Adding zeros -1

G(jw)

-

c2er)(sG

15

Lecture 27

Ali Karimpour Apr 2009

Adding zeros to open loop transfer functions

اضافه کردن صفر به تابع انتقال حلقه باز

-

c2er)(sG s1

6)62(3)1(6

)()()( 23

22

ssss

sRsCsM

10,5,2,5.0,2.0,0

P.O. tr

System speed

BW

Note: For τ<0 system is unstable. Why?

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Step Response

Time (sec)

Ampl

itude

τ=0

τ=0.5τ=2.0

τ=0.2

τ=5.0τ=10

16

Lecture 27

Ali Karimpour Apr 2009

Example 1: Derive the Bode plot of following system.

(rad/sec)Frequency10101010 3210

0

10

20

30

20

10

90

0

90

0

10

20

30

40

50

Mag

nitu

de (d

B)

100 101 102 1030

45

90Ph

ase

(deg

)

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

M

agni

tude

(db)

11)(

ssasG

)(log20 jG

11log201log20

jja

)( jG

)1

1()1(

j

ja

1aLet

a/1 /1

alog20

?

)(tan)(tan 11 am

221)tan(

aa

m

222

222

)1()(2)1)(()tan(

aaaaam

2222 2)1( aa a

1

11sin

aa

m

m

17

Lecture 27

Ali Karimpour Apr 2009

Example 1: Derive the Bode plot of the following system.

(rad/sec)Frequency10101010 3210

0

10

20

30

20

10

90

0

90

0

10

20

30

40

50

Mag

nitu

de (d

B)

100 101 102 1030

45

90

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

M

agni

tude

(db)

11)(

ssasG

)(log20 jG

11log201log20

jja

)( jG

)1

1()1(

j

ja

1aLet

/1 a/1

alog20

18

Lecture 27

Ali Karimpour Apr 2009

Exercises تمرینها1 The Nichols chart of an open loop system with negative unit feedback is shown. a) Find the GM and PM.b) Find MP.

dbMbPMdbGMaanswer p 8.1:45,14:

-360 -315 -270 -225 -180 -135 -90 -45 0-120

-100

-80

-60

-40

-20

0

20

40

6 dB 3 dB

1 dB 0.5 dB

0.25 dB 0 dB

-1 dB

-3 dB -6 dB

-12 dB

-20 dB

-40 dB

-60 dB

-80 dB

-100 dB

-120 dB

Nichols Chart

Open-Loop Phase (deg)

Ope

n-Lo

op G

ain

(dB)

19

Lecture 27

Ali Karimpour Apr 2009-360 -315 -270 -225 -180 -135 -90 -45 0

-120

-100

-80

-60

-40

-20

0

20

40

6 dB 3 dB

1 dB 0.5 dB

0.25 dB 0 dB

-1 dB

-3 dB -6 dB

-12 dB

-20 dB

-40 dB

-60 dB

-80 dB

-100 dB

-120 dB

Nichols Chart

Open-Loop Phase (deg)

Ope

n-Lo

op G

ain

(dB

)

1.0

Exercises تمرینها2 The Nichols chart of a open loop system with negative unit feedback is shown. a) Find the error constantsb) Find the GM and PM and gain crossover frequency and phase crossover frequency.c) Find MP , open loop bandwidth and closed loop bandwidth.

sec/3.6sec,/7.4,3.5:sec/7sec,/75.3,32,10:

0,5,:

180

radBWradBWdbMcradradPMdbGMb

kkkaanswer

closedlooploopopenp

c

avp

23

57

12