ees42042 fundamental of control systems bode...
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EES42042 Fundamental of Control Systems
Bode PlotsBode Plots
DR. Ir. Wahidin Wahab M.Sc.Ir. Aries Subiantoro M.Sc.
2
Bode PlotsPlot of db Gain and phase vs frequencyIt is assumed you know how to construct Bode PlotsMATLAB program bode.m available for fast Bode plottinguseful for determining Gain and Phase margins
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.1The HP 35670A Dynamic Signal Analyzer obtainsfrequency responsedata from a physicalsystem. Thedisplayed data can be used to analyze, design, or determine a mathematical modelfor the system.
Courtesy of Hewlett-Packard.
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.2Sinusoidal frequencyresponse:a. system;b. transfer function;c. input and output
waveforms
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.3System withsinusoidal input
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.4Frequency responseplots for
G(s) =1/(s + 2):
separate magnitudeand phase
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.5Frequency response plots for G(s) = 1/(s + 2) : polar plot
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.6Bode plots of
G(s)=(s + a):
a. magnitude plot;b. phase plot.
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 10.1Asymptotic and actual normalized and scaledfrequency response data for
G(s) = (s + a)
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.7Asymptotic and actual normalized and scaled magnitude response of
G(s) = (s + a)
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.8Asymptotic and actual normalized and scaled phase response of (s + a)
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.9Normalized and scaledBode plots fora. G(s) = s;b. G(s) = 1/s;c. G(s) = (s + a);d. G(s) = 1/(s + a)
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Gain Margin
Factor by which gain has to be increased to encircle (-1,0) point in polar plot
( ){ }
( )( )[ ]110
1
1
1
log20Margin Gain dbIn
1margin Gain
t.f.loopopen 180arg
such that frequency crossover phase Define
ω
ω
ωω
jG
jG
G(s)jG
−=
=
=°−=
14
Phase MarginThe amount of lag which when applied to the open loop t.f.will cause the polar plot encircle (-1,0) point
( )( )[ ]2
2
2
arg180Margin Phase0dbor 1
such that frequency crossover gain Define
ωω
ω
jGjG
+°=
=
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.54Effect of delay upon frequencyresponse
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.10 Closed-loop unity feedback system
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.11Bode log-magnitudeplot for Example 10.2:a. components;b. composite
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 10.12Bode phaseplot for Example 10.2:a. components;b. composite
19
Example
( )( )51)(
t.f.loopopen
++=
sssKsG
+- G(s)
R(s) C(s)
20
Example
Positive Gain margin of 21 degrees there system is stable
Now try increasing gain from 10 to 100
21
Example Magnitude response of open loop t.f.
10-1
100
101
102
-150
-100
-50
0
50dB Magnitude Response
DB
Gai
n
Angular Frequency - rad/sec
Gain crossover
frequency
22
Example Phase Response of open loop t.f.
10-1
100
101
102
-300
-250
-200
-150
-100
-50Phase Response
Ang
le -
degr
ees
Angular Frequency - rad/sec
Phase Crossover
frequency
-180o
23
Example Magnitude response of open loop t.f.
10-1
100
101
102-150
-100
-50
0
50dB Magnitude Response
DB
Gai
n
Angular Frequency - rad/sec
10-1
100
101
102
-300
-250
-200
-150
-100
-50Phase Response
Ang
le -
degr
ees
Angular Frequency - rad/sec
Phase margin
Gain Margin
24
ExampleIn this instance gain margin is +8db and
the phase margin is +210
Therefore system is stableNow try gain K=100
25
Example
10-1
100
101
102
-100
-50
0
50dB Magnitude Response
DB
Gai
n
Angular Frequency - rad/sec
10-1
100
101
102
-300
-250
-200
-150
-100
-50Phase Response
Ang
le -
degr
ees
Angular Frequency - rad/sec
Negative phase margin
Negative gain margin
26
ExampleNegative gain and phase margins mean system is unstable for gain K=100actual values are– gain margin = -12dB– phase margin = -30o
27Notes on Gain and Phase Margins
Measure of nearness of polar plot to (-1,0) pointNeither ON THEIR OWN give sufficient description of system stability– both must be used together
28Notes on Gain and Phase Margins
For minimum phase systems both margins should be positive– non-minimum phase occurs when poles of
OLTF exist in RHP – see Ogata pp. 486-487
29Notes on Gain and Phase Margins
Satisfactory values of gain and phase margin– phase margin should be in the range 30o-60o
– gain margin should be >6dBthese values lead to satisfactory damping ratios in the closed loop systemBode plot sketches should be enough to give you an idea of potential problems
ClosedClosed--Loop Transient Loop Transient
2121
ζζ −=pM
221 ζωω −= np
( ) 24421 242BW +−+−= ζζζωω n