Control Systems - chibum.files.wordpress.comBode Diagram Bode diagram: a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency

Control Systems

Frequency Response Method

Bode Diagram

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Control SystemsChibum Lee -Seoultech

Lecture Outline

The concept of the frequency response

Bode diagram

Basic factors of bode diagram

2


Concept of Frequency Response

Frequency response: the steady state response of

the system to a sinusoidal input

• In frequency response method, we vary the frequency of

the input signal over a certain range and study resulting response

Basic principle

For linear system, sine wave input sine wave output

with same frequency (in steady state)

(Note: magnitude and phase may be different)

3



Fkyybym

y = displacement from spring equilibrium

)(1

)(

)(2

sGkbsmssF

sY

4



Proof)

)())((

)()(

21 npspsps

spsG

)sin()( tXtx

js

b

js

a

psps

jsjs

X

pspsps

spsXsGsY

n

n

n

1

1

21 ))(()())((

)()()()(

n

j

tp

j

tjtj jebeaety1

)(

5



substitute 𝑠 = 𝑗𝜔

( ) ( ) / ( )G s Y s X s

)sin()(

ShiftPhase

AmplitudeOutput

tYty

|)(| jGXY )()](Re[

)](Im[tan 1

jG

jG

jG

)sin()( tXtxFrequency

AmplitudeInput

XjGYjXjGjY )()()()(

))sin()(cos()(

))sin()(cos()(

)(

)(

)(

tjtXXejX

tjtXeXejX

XesX

tj

ttj

st

6



For sinusoidal inputs

• Magnitude: amplitude ratio of the output sinusoid to

the input sinusoid

• Phase: phase shift of the output sinusoid with respect to

the input sinusoid

A positive phase angle of 𝐺 𝑗𝜔 is called phase lead,

negative is phase lag

)(

)()(

jX

jYjG

)(

)()(

jX

jYjG

7


Usefulness of Frequency Response

The transfer function can be determined experimentally

from input and output signals (without detailed

modeling)

• Ex:

101

102

103

104

-150

-100

-50

0

Ma

gn

itu

de

(d

B)

HallSensor Response Fit

exp

fitted

101

102

103

104

-300

-200

-100

0

Frequency (Hz)

Ph

ase

(d

eg

ree

s)

exp

fitted

8


Bode Diagram

Bode diagram: a graph of the transfer function of a

linear, time-invariant system versus frequency, plotted

with a log-frequency axis, to show the system's

frequency response.

Consists of 2 plots:

• Magnitude of 𝐺 𝑗𝜔

(generally X: logscale Y: dB scale)

• Phase of 𝐺 𝑗𝜔

(generally X: logscale Y: linear scale)

101

102

103

104

-150

-100

-50

0

Ma

gn

itu

de

(d

B)

HallSensor Response Fit

exp

fitted

101

102

103

104

-300

-200

-100

0

Frequency (Hz)

Ph

ase

(d

eg

ree

s)

exp

fitted

9


Properties of the Bode Diagram

Bode diagram

• Generally, magnitude is plotted in decibels

• Why log scale in magnitude?

Use of log in magnitude convert ‘multiplication’ into ‘sum’

system responses:

• What about Phase?

Naturally satisfies

• By knowing the bode diagram of each block, we can build more

complex system easily by addition

1 2 1 2G G G G

)(log20)( 10 jGjGdB

2101102110 log20log20log20 GGGG

10


Basic Factors: Gain

Pure gain K

• Constant magnitude

for all frequencies

• Phase angle is

0 if K>0

-180 deg. if K<0

( )G s K

Magnitude

Phase

dB

degre

es

𝜔

𝜔00

-1800

20log10|K|

K>0

K<0

KX(s) Y(s)

11


Basic Factors: Integrator

Integrator

• The magnitude is 0 dB at 𝜔= 1,

drops 20 dB for every decade

in 𝜔

• Phase is constant -90 deg.

1/sX(s) Y(s)

j

jGs

sG1

)(1

)(

10

1

1010

log20

)(log20)(log20

jG

900

/1tan

1 1

j

12


Basic Factors: Derivative

Derivative

• The magnitude is 0 dB at 𝜔 = 1,

rises 20 dB for every decade

in 𝜔

• Phase is constant 90 degrees

sX(s) Y(s) jjGssG )()(

1010 log20)(log20 jG

900

tan 1

j

13


Basic Factors: Multiple Integrator

Multiple integrator

• Magnitude

• Phase

1/s nX(s) Y(s)ns

sG1

)(

decn dB

nj n

/20

log20)(

1log20 1010

90)(

1n

j n

14


Basic Factors: Multiple Derivative

Multiple derivative

• Magnitude

• Phase

( ) nG s s s nX(s) Y(s)

decaden dB

nj n

/20

log20)(log20 1010

90)( nj n

15


Lecture Outline

Bode diagram of 1st order system

Bode diagram of 2nd order system

Bode diagram of general system

Specification from Bode Diagram

16


1st Order Systems

1st order system: mixture of gain and integrator

• Magnitude

For

For

1/(1+sT)X(s) Y(s)

22

101010 1log201log201

1log20 TTj

Tj

111 22 TT

dBTj

0)1(log201

1log20 1010

TTT

2211

TT 10

22

10 log201log20

TjTjG

sTsG

1

1)(

1

1)(

17


1st Order Systems

1st order system: mixture of gain and integrator

• Phase

For

For

01

TT

TjTjT

11

0)0(tan|)(tan 11

TT

90)(tan|)(tan 11

TT

)tan()1(11

1TTj

Tj

1/(1+sT)X(s) Y(s)Tj

TjGsT

sG

1

1)(

1

1)(

18


1st Order Systems

Ex.

• Corner frequency

T = 1/10 1/T = 10

• DC gain = 2 = 6 dB

low frequency asymptote

20/(s+10)X(s) Y(s)

10

20)(

ssG

Bode Diagram

Frequency (rad/s)

-40

-30

-20

-10

0

10

Magnitude (

dB

)

0.1 1 10 100 1000-90

-75

-60

-45

-30

-15

0

Phase (

deg)

19


Bode Diagram

Frequency (rad/s)

-40

-30

-20

-10

0

10

Magnitude (

dB

)

0.1 1 10 100 1000-180

-165

-150

-135

-120

-105

-90

Phase (

deg)

1st Order Systems

For unstable system:

• The phase is ‘flipped’ about -90 degree lines

• Ex.

20/(s-10)X(s) Y(s)

)1/(1)( sTsG

20


1st Order Systems

1st order system: mixture of gain and differentiator

1+sTX(s) Y(s)

21


2nd Order Systems

2nd order system

rewritten

For 1 system: 2 real roots 2 first order systems

• easy to plot

Mostly 0 < < 1

1 2

1 1( )G s K

s p s p

22

2

2)(

nn

n

sssG

12

1)(

2

2

nn

sssG

22


2nd Order Systems

Magnitude

• For

• For

• For (intermediate frequency) influence of

22

2

2

10210 21log20

21

1log20)(

nn

nn

jjjG

dB 0)1(log20)( 10 jG

n

jG

10log40)(

1211

22

2

2

nnn

22

2

222

2

2

211

nnnnn

1n

23


2nd Order Systems

Phase

• For

• For

• For (intermediate frequency)

2

1

2

2

1

2

tan211

21

1

n

n

nn

nn

jjj

0)( jG

180)( jG

n 90)( jG

1

0tan

1

2

tan1 1

2

1

n

n

n

0tan

1

2

tan1 1

2

1

n

n

n

24


2nd Order Systems

2nd order system response depends on

25


-80

-60

-40

-20

0

20

Magnitude (

dB

)

10-2

10-1

100

101

102

-360

-315

-270

-225

-180

Phase (

deg)

Bode Diagram

Frequency (rad/s)

2nd Order Systems

For unstable 2nd order system (negative

• Magnitude same

• Phase flipped

about -90 degree line

22

2

2

10210 21log20

21

1log20)(

nn

nn

jjjG

2

1

2

1

2

tan

21

1

n

n

nn

jj

26


General Systems

To draw a Bode diagram for a cascaded control system

• Open-loop transfer function G1(s)G2(s) into

separate integrators / 1st order / 2nd order systems

• Draw asymptotes for individual parts

• Summate the asymptotes to get one asymptote

• Draw transfer function along with new asymptote

X(s) Y(s)G1(s) G2(s)

27


General Systems

For the general loop-gain system

• The Log-Magnitude (dB)

• Phases

X(s) Y(s)G1(s) G2(s)

)1()1)(1()(

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

21

21

p

N

mba

TjTjTjj

TjTjTjKjGjG

p

N

mba

TjTjTjjN

TjTjTjKjGjG

1log201log201log20log20

1log201log201log20log20)()(log20

1021011010

101010102110

)1()1()1()(

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

21

21

p

mba

TjTjTjjN

TjTjTjKjGjG

28


Example

Ex.

G1(s)= G2(s)= G3(s)=

X(s) Y(s)

10

100

s

s s

G1(s)G2(s)G3(s)

s

1

1

10s

100

1

s

29


Bode Diagram

Frequency (rad/s)

-70

-60

-50

-40

-30

-20

-10

0

10

Magnitude (

dB

)

0.1 1 10 100 1000-90

-75

-60

-45

-30

-15

0

Phase (

deg)

Example

X(s) Y(s)

10

100

s

s s

30


Bode Diagram

Frequency (rad/s)

0.1 1 10 100 1000-80

-60

-40

-20

0

20

40M

agnitude (

dB

)

Specification from Bode Diagram

Frequency domain terms

31

DC gain

𝜔𝑏: cutoff frequency

or bandwidth𝜔𝑟: resonance

frequency

DC gain-3dB

Resonance gain

Cut-off rate


Outline

Gain change on Bode diagram

Addition of poles on Bode diagram

Addition of zeros on Bode diagram

Non-minimum phase system

Examples of bode diagram

32


-100

-80

-60

-40

-20

0

Magnitude (

dB

)

0.1 1 10 100 1000-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/s)

0:1s+ 10

1s+ 10

10s+ 10

Gain change

Multiplication by a gain shifts the Bode magnitude plot

up or down, depending on whether K>1.

Phase doesn’t change

10

10

10

1

10

1.0

s

s

s

33


Addition of Poles to a Bode Diagram

Multiplication of 𝐺 𝑠 by a stable 1st order system

will ‘bend’ the magnitude plot at the corner frequency.

High frequency magnitude will drop -20 db/decade

faster.

Phase will shift down by 90 degrees after the corner

frequency.

Similarly for an n-th order system.

34


-100

-80

-60

-40

-20

0

Magnitude (

dB

)

0.1 1 10 100 1000-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/s)

1s+ 1

100(s+ 1)(s+ 100)

Addition of Poles to a Bode Diagram

100

100

1

1

ss

-20 db/dec

-40 db/dec

90 degphase drop

1

1s vs.

35


Addition of Zeros to a Bode Diagram

A very similar procedure will occur for 1st order zeros

High frequency magnitude will rise +20 db/decade

faster.

Phase will shift up by 90 degrees after the corner

frequency.

Similarly for an n-th order system.

Second order systems follow the same procedure, except

it’s +40 db/dec and +180 deg

36


-80

-60

-40

-20

0

Magnitude (

dB

)

0.1 1 10 100 1000-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/s)

1s+ 1

(s+ 100)

(s+ 1)100

Addition of Poles or Zeros to a Bode Diagram

1 100

1 100

s

s

-20 db/dec

0 db/dec

90 degphase jump

1

1s vs.

37


Non-Minimum Phase Systems


• If systems have zeros in the RHP, they did not follow previous

phase rules. non-minimum phase

• For minimum phase system, the transfer function can be

determined from the magnitude curve alone, but

for non-minimum phase system, it can’t be.

Non-minimum phase systems are often difficult to control.

38


0

5

10

15

20

Magnitude (

dB

)

0.1 1 10 100 1000-90

0

90

180

Phase (

deg)

Bode Diagram

Frequency (rad/s)

s+ 10s+ 1

s! 10s+ 1


Minimum Phase system vs. Non-Minimum Phase system

Re-10

Im

-1Re

10

Im

-1

1

10)(1

s

ssG

1

10)(2

s

ssG

Samemagnitude

Different phase

39


Example

Ex.)2)(2(

)3(10)(

2

ssss

ssG

40


Example

Ex.)

50

1

50

6.01)(5.01(

)1.01(5)(

2

2ssss

ssG

41


Example

)50

1

50

6.01)(5.01(

)1.01(5)(

2

2ssss

ssG

42


Example

)50

1

50

6.01)(5.01(

)1.01(5)(

2

2ssss

ssG

43

Control SystemsChibum Lee -Seoultech 44




Example

Ex.)5)(2(

)1(10)(

ss

ssG

47

Control Systems - chibum.files.wordpress.comBode Diagram Bode diagram: a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency

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