ees42042 fundamental of control systems bode...

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)

13

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