No Slide Titlekarimpor.profcms.um.ac.ir/imagesm/354/stories/control... · 2015-07-10 · Lecture 11 Dr. Ali Karimpour Sep 2015 2 Lecture 11 Frequency domain analysis Topics to be

Lecture 11

Dr. Ali Karimpour Sep 2015

CONTROL

ENGINEERING

Ali Karimpour

Associate Professor

Ferdowsi University of Mashhad

Lecture 11


2

Lecture 11

Frequency domain analysis

Topics to be covered include:

Frequency domain specification.

Peak of resonance and resonance frequency.

Bandwidth.

Gain margin.

Phase margin.

Polar plot. Stability analysis with polar plot.

Nichols chart or gain phase plot. Stability analysis with gain phase plot.

Bode plot. Stability analysis with Bode plot.

Effect of adding poles and zeros on loop transfer function.



Bandwidth.

Gain margin.

Phase margin.

Lecture 11


3

One degree-of-freedom configuration

)()()(1

)()()()(

)()(1

1)(

)()(1

)()()( sn

sKsG

sKsGsdsG

sKsGsr

sKsG

sKsGsy d

T(s) T(s)S(s)

S(s) is : Sensitivity Function

T(s) is : Complementary Sensitivity Function Why?

Lecture 11


4

Frequency (rad/s)

T

S

L

Frequency domain specification

PM

P

Peak of resonance ( )PM

Resonance frequency ( )P

Lecture 11


5


Closed-loop bandwidth ( )B

B

-3

Open - loop bandwidth ( )O

O

Gain crossover frequency ( )c

c

Lecture 11


6

Stability margins

Stability Is Not A Yes/No Proposition

Lecture 11


7

Stability margins

)(sG)(sC

)(sGc

+

-

)(sR

ω=ωc

ω=ω180

G (jω) Gc(jω)

Phase Margin

We define phase margin as the phase

(angle) that the frequency response

would have to change to move to the

-1 point.

c : Is the gain crossover frequency

Phase margin (PM) is the most widely

used measure of relative stability.

)()(180cccm

jGjGPM

Physical meaning:

Lecture 11


8

Example 1: Derive PM and gain crossover frequency of following

system.

)10)(5(

500

sss

)(sC+

-

)(sR

)10)(5(

500)(

ssssG

Stability margins

sec/5.6 radc

10PM......

PM and gain crossover frequency from Nyquist (polar plot)

PM and gain crossover frequency from Bode plot

PM and gain crossover frequency from Nichols (gain phase plot)

Lecture 11


9

Stability margins

Phase and Gain Margin

-1

Same Phase Margin

Gain Margin ??

Thus we need another measure

of relative stability.

Lecture 11


10

Stability margins

)(sG)(sC

)(sGc

+

-

)(sR

ω=ωc

ω=ω180

G (jω) Gc(jω)

Gain Margin

We define gain margin as the gain

that the frequency response would

have to increase to move to the -1

point.

180 : Is the phase crossover frequency

Gain margin is another widely used

measure of relative stability.

)()(/1 180180 jGjGGM c

)()(/1log20 180180 jGjGGM c

Physical meaning:

Lecture 11


11

Example 2: Derive GM and phase crossover frequency of following

system.

)10)(5(

500

sss

)(sC+

-

)(sR

)10)(5(

500)(

ssssG

Stability margins

sec/07.7180

rad 5.1GM......

GM and phase crossover from Nyquist (polar plot)

GM and phase crossover from Bode plot

GM and phase crossover from Nichols (gain phase plot)

dbGM 5.3

Lecture 11


12

Stability margins

Phase Margin and Gain Margin

-1

Same Phase Margin

Thus we need another measure

of relative stability.

Same Gain Margin

Lecture 11


13

G (jω) Gc(jω)

Sensitivity peak

)()(1

1)(

jGjGjS

c

)()(1

1)(max

scs

sjGjG

jSM

1

sM)(sG)(sC

)(sGc

+

-

)(sR

The sensitivity peak, Ms is defined as follows:

)()(1 00 jGjG c

0

)( 0

1 jS

Lecture 11


14

Example 3: Derive the sensitivity peak of following system.

)10)(5(

500

sss

)(sC+

-

)(sR

)10)(5(

500)(

ssssG

Stability margins

sec/?? rads ??

sM......

Sensitivity peak from Nyquist (polar plot)

Sensitivity peak from Bode plot

Sensitivity peak from Nichols (gain phase plot)

Lecture 11


)(sC)(sL

+

-

)(sR

15

Frequency (rad/s)

T

S

L


1- Peak of resonance ( )PM

3- Resonance frequency ( )P 4- Closed-loop bandwidth ( )B

2- Open - loop bandwidth ( )O

6- Gain crossover frequency ( )c5- Sensitivity Peak ( )s

M

Lecture 11


16

ω=ωc

ω=ω180

L (jω)


)(sC)(sL

+

-

)(sR5- Sensitivity Peak ( )

sM

6- Gain crossover frequency ( )c

7- Phase crossover frequency ( )180

8- Phase Margin ( )PM

9- Gain Margin ( )GM

Lecture 11


17





Bandwidth.

Gain margin.

Phase margin.





Lecture 11


18

-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

Imagin

ary

Axis

Nyquist chart (polar plot))(sG

)(sC)(sGc

+

-

)(sR

1 10793.2

2 12337.1

3 13882.0

4 15154.0

5 16238.0

6 17127.0

7 17920.0

8 18716.0

9 19312.0

20 22902.0

?c 6.2 ?180 7 ?GM db????PM 45

1

2

)10)(5(

150)()(Let

ssssGsGc

)()( jGjGc

3

7 20

45

10793.2)101)(51(

150)1()1(

jjjjGjGc

Lecture 11


19

Frequency charts

Nyquist

(polar plot)

Bode plot

Nichols chart

Lecture 11


20

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

-30

-20

-10

0

10

20Nichols Chart

Open-Loop Phase (deg)

Open-L

oop G

ain

(dB

)

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

-30

-20

-10

0

10

20Nichols Chart


Open-L

oop G

ain

(dB

)

)(sG)(sC

)(sGc

+

-

)(sR

)10)(5(

150)()(Let

ssssGsGc

)()(log20 jGjGc

db33.9

db71.2

db71.1

db29.5

db42.8

db23.11

db80.13

db18.16

db39.18

1

3

68

)()( jGjGc

1 10793.2

2 12337.1

3 13882.0

4 15154.0

5 16238.0

6 17127.0

7 17920.0

8 18716.0

9 19312.0

20 22902.0 db77.35

20

2

4

5

?c 6.2 ?180 7 ?GM db8.13?PM 45

Nichols chart (gain phase plot)

Lecture 11


21

Bode plot )(sG)(sC

)(sGc

+

-

)(sR

)10)(5(

150)()(Let

ssssGsGc

)()( jGjGc )()(log20 jGjGc

1 10793.2 db33.9

2 12337.1 db71.2

3 13882.0 db71.1

4 15154.0 db29.5

5 16238.0 db42.8

6 17127.0 db23.11

7 17920.0 db80.13

8 18716.0 db18.16

9 19312.0 db39.18

20 22902.0 db77.35 ?c 5.2 ?180 7 ?GM db8.13?PM 48

-80

-60

-40

-20

0

20

Magnitu

de (

dB

)

100

101

102

-270

-225

-180

-135

-90P

hase (

deg)

Bode Diagram

Frequency (rad/sec)

-80

-60

-40

-20

0

20

Magnitu

de (

dB

)

100

101

102

-270

-225

-180

-135

-90P

hase (

deg)

Bode Diagram

Frequency (rad/sec)

1003020107654321

Lecture 11


22

University entrance exam 1393

. اشدفاز کدام یک از توابع زیر دارای کم ترین تغییرات فاز می بدیاگرام-4مثال

3(1

1)(4(

)1(

1)(

1(1

1)(2(

)1(

1)(

22

22

ssg

ssg

ssg

s

ssg

Lecture 11


University entrance exam 1393

قطبی یک سیستم مرتبه دوم نوعی دیاگرام-5مثال

کدام عبارت صحیح است؟. در دو حالت در زیر ترسیم شده است

.است( 2)سیستم فراجهشبیشتر از ( 1)سیستم فراجهش1).است( 1)سریع تر از سیستم ( 2)پاسخ سیستم 2).م استسیستمیراینشان دهنده فرکانس نوسانات موهومیفرکانس تالقی با محور 3). هر سه عبارت صحیح است4)

22

2

2nn

n

ss

23

Lecture 11


24

)(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

Imagin

ary

Axis

1

2

3

7 20

45

Closed loop values from Nyquist chart

2004.1)1( jT

?)2( jT

?pM

?)1( jT

?)3( jT

?)( jT

Lecture 11


25

M circles (constant magnitude of T)

Lecture 11


26

N circles (constant phase of T)

Lecture 11


27

Constant gain and phase loci in Nichols chart

M circles and N circles

on Nichols chart

GC

cG+

-

R

Lecture 11


28

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

Lecture 11


29





Bandwidth.

Gain margin.

Phase margin.





Lecture 11


30

Effect of adding poles on Bode plot.

-

c2er)(sG

s1

1

Adding poles

-

c2er)(sG

trSystem speedBW

/1

Lecture 11


31

Adding poles to open loop transfer functions

اضافه کردن قطب به تابع انتقال حلقه باز

-

c2er)(sG

s1

1223

2

22

2)21()(

)()(

nnn

n

ssssR

sCsM

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)

Am

plit

ude τ=0

τ=1.0

τ=2.0

τ=5.0P.O.

tr

System speed

More problem as

poles go to ??

BW

Lecture 11


32

Effect of adding zeros on Bode plot.

-

c2er)(sG s1

Adding zeros

-

c2er)(sG

/1

trSystem speedBW

Lecture 11


33

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

sss

s

sR

sCsM

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 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Step Response

Time (sec)

Am

plit

ude

τ=0

τ=0.5

τ=2.0

τ=0.2

τ=5.0

τ=10

Lecture 11


34

Example 6: Derive the Bode plot of following system.

(rad/sec)Frequency

10101010 3210

0

10

20

30

20

10

90

0

90

0

10

20

30

40

50

Magnitu

de (

dB

)

100

101

102

103

0

45

90P

hase (

deg)

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)

M

agn

itu

de

(db

)

1

1)(

s

sasG

)(log20 jG

1

1log201log20

jja

)( jG

)1

1()1(

jja

1aLet

a/1 /1

alog20

?

)(tan)(tan 11 am

221)tan(

a

am

222

222

)1(

)(2)1)(()tan(

a

aaaam

2222 2)1( aa

a

1

1

1sin

a

am

m

Lecture 11


35

Example 7: Derive the Bode plot of the following system.

(rad/sec)Frequency

10101010 3210

0

10

20

30

20

10

90

0

90

0

10

20

30

40

50

Magnitu

de (

dB

)

100

101

102

103

0

45

90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)

M

agn

itu

de

(db

)

1

1)(

s

sasG

)(log20 jG

1

1log201log20

jja

)( jG

)1

1()1(

jja

1aLet

/1 a/1

alog20

1

1sin

a

am

Lecture 11


150:

)25(

150:

2 ssb

ssaanswer

2- The polar plot of an open

loop system with negative

unit feedback is shown.

a) Find the open loop

b) transfer function.

c) Find the closed loop

d) transfer function.

36

Exercises

1- Derive the gain crossover frequency, phase

crossover frequency, GM and PM of following

system by use of Bode plot.

)10(

1

ss

)(sC

200+

-

)(sR

38,,5.12: 180 mc andGMAnswer

Lecture 11


37

Exercises

3- Bode plot of an open loop system with negative unit feedback is

shown.

a) Find the open loop transfer function.

b) Find the closed loop transfer function.

20020

200:

)20(

200:

2 ssb

ssaanswer

-80

-60

-40

-20

0

20

40

Magnitu

de (

dB

)

100

101

102

103

-180

-135

-90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Lecture 11


38

4- 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-L

oop G

ain

(dB

)Exercises

Lecture 11


39-360 -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-L

oop G

ain

(dB

)

1.0

5- The Nichols chart of a open loop system with negative unit feedback

is shown.

a) Find the error constants

b) 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

radBWradBWdbMc

radradPMdbGMb

kkkaanswer

closedlooploopopenp

c

avp

23

57

12

Exercises

Lecture 11


40

6- Draw Nichols chart of following system (Final exam).

Exercises

7- Draw gain-phase plot of a minimum phase type one system with no

zero and three poles and GM=2 db and PM=45° (Final exam).

8- Bode plot of a minimum phase system is: (Final exam).

a- Derive phase and gain crossover

Frequency, Gm and PM.

b- Determine the nonzero error constant.

c- If 0.01 sec delay added inside the

feedback loop, derive new Bode plot

in the same figure.

d- Derive phase and gain crossover

Frequency, Gm and PM of new system.

Lecture 11


41

9- Nichols chart of a system is given, determine

a- Gain and phase cross over

frequency.

b- GM and PM.

c- Open loop and closed loop BW.

d- Type of system.

e- Nonzero error constant.

f- …

g- …

Exercises

Lecture 11


42

Example 8: Derive the GM and PM of following system

by use of Bode plot.

)10)(5(

500

sss

)(sC+

-

)(sR

?c sec/5.6 rad

10PM

?180 sec/2.7 rad

dbGM 3

(rad/sec)Frequency

10101010 3210

40

20

0

20

80

60

90

180

270

0

10

20

30

40

50

Magnitu

de (

dB

)

100

101

102

103

0

45

90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)

M

agn

itu

de

(db

)

Documents

No Slide Titlekarimpor.profcms.um.ac.ir/imagesm/354/stories/control... · 2015-07-10 · Lecture 11 Dr. Ali Karimpour Sep 2015 2 Lecture 11 Frequency domain analysis Topics to be