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 open-loop transfer function for K=1 and determine gain margin, phase margin and absolute stability? K  ( ) ( ) G s H s    4 3 2 5 8 6 s s s s 2

Bode plots of higher order systems 1  G ( s ) H ( s )    2 ( 2 2 )( 3 ) s s s s 1    ( ) ( ) G j H j          2 j ( j ) 2( j ) 2 (( j ) 3) 3

Bode plots of higher order systems 1 3    ( ) ( ) G j H j           2 (2 ) 2 ) 2 ( 1) j j j 3 4

Bode plots of higher order systems     2 Comparing the second order term ( ) 2( ) 2 j j      with a standard second order term: 2 2 2 j n n 1     2 n 2 For the first order integral factor,  c =3 rad/s For ζ > 0.5, the response at resonance is less than the response at frequencies less than the resonant frequencies 5

Bode plots of higher order systems Computation of Bode magnitude using asymptotic properties of the integral second-order term 1 1     2 n      2 2 (1 ) 2 r j r   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 -40 -40 -52 -52 -32 -32 -72 -52 -12 0 6

Bode plots of higher order systems 1 Computation of Bode magnitude using j   asymptotic properties of the integral first-order 1 term 3 x1 x3 x2 x1 x3 x1 x1 x10 Frequency 3 30 30 14 30 10 10 100 , rad/s Magnitude 0 -20 -20 -14 -20 -10 -10 -30 , dB 7

Bode Magnitude Frequency, rad/s  n  c 0.01 0.1 0.14 0.3 1 3 10 14 30 100 2 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 20log1/ 3 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -6   2 20log n 1 40 20 17 10 0 -3 -10 -20 -23 -30 -40 20 log  j 0 0 0 0 0 -3 -12 -32 -40 -52 -72 20log 1      2 (1 r ) j (2 r ) 1 0 0 0 0 0 -1 -3 -10 -14 -20 -30 20 log   j 1 3 Bode magnitude, 24 4 1 -6 -16 -23 -41 -78 -93 -118 -158 dB 8

Bode plots of higher order systems Bode Magnitude Phase crossover Frequency 1.09 rad/s 9

Bode phase Frequency, rad/s  n  c 0.01 0.1 0.14 0.3 1 3 10 14 30 100 2 0 0 0 0 0 0 0 0 0 0 0 1  3 270 270 270 270 270 270 270 270 270 270 270 1  , degrees  j 360 360 360 343 297 270 221 192 180 180 180 1       2 (1 r ) j (2 r ) , degrees 1 , degrees 360 360 360 360 336 330 315 291 285 270 270    j 1 3 Bode phase, 270 270 250 253 183 150 86 33 15 0 0 degrees 10

Bode plots of higher order systems Bode phase Gain crossover Frequency 0.16 rad/s 11

Bode plots of higher order systems   0.01 Nichols plot   10 12

Bode plots of higher order systems Draw the Bode magnitude and phase plot of the following open-loop transfer function for K=1 and determine gain margin, phase margin and absolute stability?     2 K s 10 s 100  G s H s ( ) ( )     4 3 2 20 100 500 1500 s s s s 13

Bode plots of higher order systems   2 s 10 s 100     G ( s ) H ( s )       2 15 . 15 4 . 18 0 . 68 23 . 64 ) s s s s Natural frequencies, damping factors and decade ranges of the second-order factors Second -order Resonant Decade range of Damping term frequency, frequencies, rad/s ratio rad/s   10 1 to 100 0.5 2 10 100 s s   4.86 0.486 to 48.6 0.07 2 0 . 68 23 . 64 s s 14

Bode plots of higher order systems Corner frequencies and decade ranges of the integral first-order terms First- Corner Decade order frequency, range of terms rad/s frequencies, rad/s (s/4.2+1) 4.2 0.42 to 42 (s/15.2+1) 15.2 0.15 to 152 15

Bode plots of higher order systems Computation of Bode magnitude using asymptotic properties of the derivative second-order term    10 rad/s =0.5      2 (1 ) 2 r j r   n 1 x10 x2 1 1 x3 x10 x1 x1 x4 Frequency, 10 100 100 49 49 152 152 15.2 10 42 rad/s Magnitude, 0 40 40 28 28 48 48 8 0 24 dB 16

Bode plots of higher order systems 1 Computation of Bode magnitude using asymptotic properties of the integral second-order term      2 (1 ) 2 r j r      4.86 rad/s =0.07 n x1 x10 x1 x2 x1 x3 x1 x2 Frequency, rad/s 4.9 49 49 100 49 152 4.9 10 Magnitude, dB 0 -40 -40 -52 -40 -60 0 -12 x1 x4 x10 x1 Frequency, rad/s 10 42 152 15.2 -12 -36 -60 -20 Magnitude, dB 17

Bode plots of higher order systems Computation of Bode magnitude using 1 j  asymptotic properties of the integral first-      1 order term (corner frequency, 15.2 rad/s)   15.2 1 x10 x3 1 1 x2 Frequency, 15.2 152 152 49 49 100 rad/s Magnitude 0 -20 -20 -10 -10 -16 , dB 1 x3 1 x3 Frequency, rad/s 15.2 42 15.2 49 Magnitude, dB 0 -10 0 -10 18

Bode plots of higher order systems Bode Magnitude (dB) Frequency rad/s 0.1 0.15 0.42 0.49 1 4.2 4.9 10 15.2 42 49 100 -20log15.2 -24 -24 -24 -24 -24 -24 -24 -24 -24 -24 -24 -24 -20log4.2 -12 -12 -12 -12 -12 -12 -12 -12 -12 -12 -12 -12 20log100 40 40 40 40 40 40 40 40 40 40 40 40 -20log4.86 2 -27 -27 -27 -27 -27 -27 -27 -27 -27 -27 -27 -27   2 0 0 0 0 0 0 0 0 20log 1 r 8 2 4 2 8 40 jr - 0 0 0 0 0 11 16 -12 -20 -36 -40 -52   2 20log 1 r j 0.14 r - 0 0 0 0 0 -3 -4 -8 -12 -20 -21 -28   20 log 1 / 4 . 2 j - 0 0 0 0 0 0 0 -2 -3 -10 -10 -16   20 log 1 / 15 . 2 j -23 -23 -23 -23 -23 -15 -11 -45 -50 -65 -66 -79 19

Bode plots of higher order systems Bode Phase (degrees Frequency rad/s 0.1 0.15 0.42 0.49 1 4.2 4.9 10 15.2 42 49 100 0 0 0 0 6 27 33 90 131 166 168 174      2 100 j 10 - 360 360 360 360 360 334 270 185 183 181 181 180      2 23 . 64 j 0 . 68 - 360 360 354 353 347 315 311 293 285 276 270 270    1 / 4 . 2 j - 360 360 0 0 0 345 342 327 315 290 287 279    1 j / 15 . 2 360 360 354 353 353 301 236 175 194 193 186 183 20

Bode plots of higher order systems Additional Phase (degrees) Frequency rad/s 5 6 7 8 9 11 12      2 100 10 j 34 43 54 66 78 101 110      2 23 . 64 j 0 . 68 - 248 198 191 188 186 184 184    1 / 4 . 2 j - 310 305 301 298 295 291 289    - 1 / 15 . 2 j 342 338 335 332 329 324 322 214 164 161 164 168 180 185 21

Bode plots of higher order systems Bode magnitude 22

Bode plots of higher order systems Nyquist plot of the open-loop transfer function of (K=1) 23

Bode plots of higher order systems Bode phase 24

Bode plots of higher order systems Nichols chart 25

Bode plots of higher order systems Conclusion 26