Topic #29 Nyquist plots: Closed-loop response Reference textbook : - PowerPoint PPT Presentation

ME 779 Control Systems Topic #29 Nyquist plots: Closed-loop response Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Nyquist plots: Closed-loop response CLOSED LOOP FREQUENCY RESPONSE   C j ( ) G j ( )      ( ) 1 ( ) ( ) R j G j H j 2

Nyquist plots: Closed-loop response Peak Magnitude  C j ( )  20log dB M  r R j ( ) 3 dB is considered good 3

Nyquist plots: Closed-loop response Constant M-circles for unity feedback systems  G ( j )   M ( j )   1 G ( j ) (    ) G j x jy  2 2 x y   ( ) M j   2 2 (1 ) x y 4

Nyquist plots: Closed-loop response Constant M-circles for unity feedback systems     2 2 2 2 2 2 ( 1 ) M x M y x y      2 2 2 2 2 2 x ( 1 M ) ( 1 M ) y 2 M x M 2 2 M M    2 2 2 x y x   2 2 1 M 1 M 5

Nyquist plots: Closed-loop response Constant M-circles for unity feedback systems 2   2   2 M M        2 x y       2 2   2 1 M 1 M   2 M   Adding    M 2   1 The above equation represents a family   2 M   , 0   of circles with its center at and  2   1 M . M  radius 2 1 M . 6

Nyquist plots: Closed-loop response Constant M-circles for unity feedback systems M>1 M<1 Family of M-circles corresponding to the close loop magnitudes (M) of a unit feedback system 7

Nyquist plots: Closed-loop response Constant N-circles   G ( j )     M    1 G ( j ) y y      1 1 tan tan  x 1 x 8

Nyquist plots: Closed-loop response Constant N-circles A B   y y       1 1 N tan tan tan    tan(  )=N 1 x x  tan A tan B   tan( ) A B  1 tan tan A B y y   x 1 x  N    y y     1     1 x x 9

Nyquist plots: Closed-loop response Constant N-circles y y      2 2 x x y 0 N   2 2 N x x y 1 1  Add on both sides 2 4 4 N y 1 1 1 1        2 2 x x y 2 2 4 4 4 4 N N N 2 2 2       1 1 1 1            x y       2 2 N 4 2 N 10

Nyquist plots: Closed-loop response Constant N-circles 2 2 2       1 1 1 1            x y       2 2 N 4 2 N The above equation represents a family   1 1    , of circles with its center at   2 2 N 2   1 1    and radius   4 2 N 11

Nyquist plots: Closed-loop response Constant N-circles 12

Nyquist plots: Closed-loop response Example Determine the closed-loop magnitude ratio and bandwidth of the feedback system whose forward transfer function is 10  G s ( )   given by and H(s)=1, by (1) direct ( 2)( 4) s s s computation and (2) using M and N circles 10  ( ) G s   ( 2)( 4) s s s 13

Nyquist plots: Closed-loop response Example 14

Nyquist plots: Closed-loop response Example 15

Nyquist plots: Closed-loop response Closed-loop magnitude and phase Closed- Closed- Closed-loop loop Closed-loop loop Freq., Magnitude Phase Freq., Magnitude Phase rad/s ratio angle, deg rad/s ratio angle, deg 0.1 1.0 355 1.2 1.3 280 0.5 1.1 335 1.3 1.2 269 0.8 1.2 316 1.4 1.2 258 0.9 1.2 308 1.5 1.1 248 1.0 1.2 300 1.6 1.0 238 1.1 1.3 290 1.7 0.9 230 1.8 0.8 222 1.9 0.7 216 2.0 0.6 210 16

Nyquist plots: Closed-loop response Example 17

Nyquist plots: Closed-loop response Example 18

Nyquist plots: Closed-loop response Conclusion 19