S u ym cs e r we have the following closed loop system

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Unformatted text preview: nsfer function. Noninvertible part of Ta .s/ consists of I I I I plant zero at infinity (pole excess of P .s/ is 1) plant nonminimum-phase zero at s D 40 plant delay e 0:1s controller nonminimum-phase zeros at s D 5:1 j6:8 . s C 40/.s 2 10:2s C 72:25/e Tref .s/ D 40 72:25 . s C 1/4 0:1s Let's then choose ; where is some time constant to choose (note that Tref .0/ D 1). 2DOF design: simple split (Cb .s/ D 1) (contd) 1 For D 12 (chosen to remain with the same settling time) and K.s/ D we have: Bode diagrams of T .s/ and Ta .s/K.s/ 0 -5 -10 Magnitude (dB) Tref .s/ Ta .s/ Output responses to step reference (T .s/ and Ta .s/K.s/) 1 0.8 0.6 Amplitude 1 2 3 -15 -20 -25 0.4 0.2 -30 -35 -40 0 10 0 10 10 Frequency (rad/sec) 10 -0.2 0 0.5 1 Time (sec) 1.5 2 Better, but not that great transients either. . . The cause of oscillations is I dominating lightly damped zeros of Tref .s/ at s D 5:1 j6:8 (these oscillations do not show up at the Bode magnitude diagram). 2DOF design: can different split of C.s/ help? d y P .s/ ym u Cf .s/ Cb .s/ F .s/ r n In this case, nonminimum-phase zeros of Ta .s/ D I I I P .s/Cf .s/ 1CP .s/C.s/ are nonminimum-phase zeros of P .s/, nonminimum-phase zeros of Cf .s/, unstable poles of Cb .s/. nonminimum-phase zeros of Cb .s/ are not those of Ta .s/ Important: I Consequently, it makes sense to I put all nonminimum-phase zeros of C.s/ into Cb .s/, so that the won't show up in Ta .s/. 2DOF design: split with Cb .s/ D In this case 2500 s 2 10:2sC72:25 72:25 s 2 C70sC2500 . s C 40/e 0:1s Ta .s/ 2 Tinv;2 .s/; s C 11:37s C 2985 where Tinv;2 .s/ is some (stably) invertible transfer function. Let's then choose . s C 40/e 0:1s ; Tref .s/ D 40. s C 1/2 where is some time constant to choose (note that Tref .0/ D 1 again). 2DOF design: split with Cb .s/ D 2500 s 2 10:2sC72:25 72:25 s 2 C70sC2500 (contd) Tref .s/ Ta .s/ For D 1 (chosen to remain with the same settling time) and K.s/ D 7 we have: Bode diagrams of T .s/ and Ta .s/K.s/ 0 -5 -10 Magnitude (dB) Output responses to step reference (T .s/ and Ta .s/K.s/) 1 0.8 0.6 Amplitude 1 2 3 -15 -20 -25 0.4 0.2 -30 -35 -40 0 10 0 10 10 Frequency (rad/sec) 10 -0.2 0 0.5 1 Time (sec) 1.5 2 Much better transients. . ....
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This note was uploaded on 06/30/2011 for the course MECHANICAL 034406 taught by Professor Leonidmirkin during the Spring '10 term at Technion.

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