Control SystemsChibum Lee -Seoultech
Lecture Outline
The concept of the frequency response
Bode diagram
Basic factors of bode diagram
2
Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
Concept of Frequency Response
Fkyybym
y = displacement from spring equilibrium
)(1
)(
)(2
sGkbsmssF
sY
4
Control SystemsChibum Lee -Seoultech
Concept of Frequency Response
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
Control SystemsChibum Lee -Seoultech
Concept of Frequency Response
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
Control SystemsChibum Lee -Seoultech
Concept of Frequency Response
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
Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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)
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
Lecture Outline
Bode diagram of 1st order system
Bode diagram of 2nd order system
Bode diagram of general system
Specification from Bode Diagram
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Control SystemsChibum Lee -Seoultech
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)(
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Control SystemsChibum Lee -Seoultech
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)(
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Control SystemsChibum Lee -Seoultech
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)
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
1st Order Systems
1st order system: mixture of gain and differentiator
1+sTX(s) Y(s)
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Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
2nd Order Systems
2nd order system response depends on
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Control SystemsChibum Lee -Seoultech
-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
Control SystemsChibum Lee -Seoultech
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)
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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
Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
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
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DC gain
𝜔𝑏: cutoff frequency
or bandwidth𝜔𝑟: resonance
frequency
DC gain-3dB
Resonance gain
Cut-off rate
Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
-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
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Control SystemsChibum Lee -Seoultech
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.
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Control SystemsChibum Lee -Seoultech
-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.
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Control SystemsChibum Lee -Seoultech
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
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Control SystemsChibum Lee -Seoultech
-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.
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Control SystemsChibum Lee -Seoultech
Non-Minimum Phase Systems
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
Control SystemsChibum Lee -Seoultech
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
Non-Minimum Phase Systems
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
Control SystemsChibum Lee -Seoultech
Example
Ex.)2)(2(
)3(10)(
2
ssss
ssG
40
Control SystemsChibum Lee -Seoultech
Example
Ex.)
50
1
50
6.01)(5.01(
)1.01(5)(
2
2ssss
ssG
41
Control SystemsChibum Lee -Seoultech
Example
)50
1
50
6.01)(5.01(
)1.01(5)(
2
2ssss
ssG
42
Control SystemsChibum Lee -Seoultech
Example
)50
1
50
6.01)(5.01(
)1.01(5)(
2
2ssss
ssG
43
Control SystemsChibum Lee -Seoultech 44
Control SystemsChibum Lee -Seoultech 45
Control SystemsChibum Lee -Seoultech 46
Control SystemsChibum Lee -Seoultech
Example
Ex.)5)(2(
)1(10)(
ss
ssG
47