Topic #34 Bode plots of higher order systems Reference textbook : - - PowerPoint PPT Presentation
ME 779 Control Systems Topic #34 Bode plots of higher order systems Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1 Bode plots of higher order systems Draw the Bode magnitude and phase plot of the following
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Reference textbook:
Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012
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Draw the Bode magnitude and phase plot of the following
margin, phase margin and absolute stability?
4 3 2
3
) 3 )( 2 2 ( 1 ) ( ) (
2
s s s s s H s G
2
1 ( ) ( ) ( ) 2( ) 2 (( ) 3) G j H j j j j j
4
2
1 3 ( ) ( ) (2 ) 2 ) 2 ( 1) 3 G j H j j j j
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For ζ > 0.5, the response at resonance is less than the response at frequencies less than the resonant frequencies
2 2
2
n n
j
2
n
1 2
Comparing the second order term with a standard second order term: For the first order integral factor, c=3 rad/s
2
( ) 2( ) 2 j j
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x1 x10 x1 x2 x3 x1 x1 x10 x3
x1
Frequency, rad/s 1.4 14 14 30 30 10 10 100 30 3 Magnitude, dB
Computation of Bode magnitude using asymptotic properties of the integral second-order term
2
1 (1 ) 2 r j r
2
n
1 2
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x1 x3 x2 x1 x3 x1 x1 x10 Frequency , rad/s 3 30 30 14 30 10 10 100 Magnitude , dB 0 -20
Computation of Bode magnitude using asymptotic properties of the integral first-order term
1 1 3 j
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Frequency, rad/s n c 0.01 0.1 0.14 0.3 1 3 10 14 30 100
20log1/ 3
2
20log
n
j 1 log 20
40 20 17 10 0
2
20log 1 (1 ) (2 ) r j r
1 3 1 log 20 j
Bode magnitude, dB 24 4 1
2
Bode Magnitude
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Bode Magnitude
Phase crossover Frequency 1.09 rad/s
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Frequency, rad/s n c 0.01 0.1 0.14 0.3 1 3 10 14 30 100
3 1
j 1
, degrees 270 270 270 270 270 270 270 270 270 270 270
2
1 (1 ) (2 ) r j r
, degrees 360 360 360 343 297 270 221 192 180 180 180
1 3 1 j
, degrees 360 360 360 360 336 330 315 291 285 270 270 Bode phase, degrees 270 270 250 253 183 150 86 33 15
2
Bode phase
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Bode phase Gain crossover Frequency 0.16 rad/s
12
0.01
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Nichols plot
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2 4 3 2
10 100 ( ) ( ) 20 100 500 1500 K s s G s H s s s s s
Draw the Bode magnitude and phase plot of the following
margin, phase margin and absolute stability?
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Natural frequencies, damping factors and decade ranges of the second-order factors Second -order term Resonant frequency, rad/s Decade range of frequencies, rad/s Damping ratio
100 10
2
s s
10 1 to 100 0.5
64 . 23 68 .
2
s s
4.86 0.486 to 48.6 0.07
) 64 . 23 68 . 18 . 4 15 . 15 100 10 ) ( ) (
2 2
s s s s s s s H s G
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Corner frequencies and decade ranges of the integral first-order terms First-
terms Corner frequency, rad/s Decade range of frequencies, rad/s (s/4.2+1) 4.2 0.42 to 42 (s/15.2+1) 15.2 0.15 to 152
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1 x10 x2 1 1 x3 x10 x1 x1 x4 Frequency, rad/s 10 100 100 49 49 152 152 15.2 10 42 Magnitude, dB 0 40 40 28 28 48 48 8 0 24
Computation of Bode magnitude using asymptotic properties of the derivative second-order term
2
(1 ) 2 r j r
10 rad/s =0.5
n
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x1 x10 x1 x2 x1 x3 x1 x2 Frequency, rad/s 4.9 49 49 100 49 152 4.9 10 Magnitude, dB
x1 x4 x10 x1
Frequency, rad/s 10 42
152 15.2
Magnitude, dB
Computation of Bode magnitude using asymptotic properties of the integral second-order term
2
1 (1 ) 2 r j r 4.86 rad/s =0.07
n
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1 x10 x3 1 1 x2 Frequency, rad/s 15.2 152 152 49 49 100 Magnitude , dB
1 x3 1 x3
Frequency, rad/s 15.2 42
15.2 49
Magnitude, dB
Computation of Bode magnitude using asymptotic properties of the integral first-
1 1 15.2 j
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Frequency rad/s
0.1 0.15 0.42 0.49 1 4.2 4.9 10 15.2 42 49 100
20log100
40 40 40 40 40 40 40 40 40 40 40 40
2
20log 1 r jr
8 24 28 40
20log 1 0.14 r j r
11 16 -12 -20 -36 -40 -52
. 4 / 1 log 20 j
. 15 / 1 log 20 j
Bode Magnitude (dB)
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Frequency rad/s 0.1 0.15 0.42 0.49 1 4.2 4.9 10 15.2 42 49 100
10 100
2j
6 27 33 90 131 166 168 174
68 . 64 . 23
2j
360 360 360 360 360 334 270 185 183 181 181 180
. 4 / 1 j
360 360 354 353 347 315 311 293 285 276 270 270
. 15 / 1 j
360 360 345 342 327 315 290 287 279 360 360 354 353 353 301 236 175 194 193 186 183
Bode Phase (degrees
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Frequency rad/s 5 6 7 8 9 11 12
10 100
2
j
34 43 54 66 78 101 110
68 . 64 . 23
2
j
248 198 191 188 186 184 184
. 4 / 1 j 310 305 301 298 295 291 289
. 15 / 1 j
342 338 335 332 329 324 322 214 164 161 164 168 180 185
Additional Phase (degrees)
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Bode magnitude
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Nyquist plot of the open-loop transfer function of (K=1)
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Bode phase
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Nichols chart
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