C d 4 10 40 12 50 1 10 10 0 10 frequency radsec 1 10

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Unformatted text preview: nce frequencies. This means that design should not hinge upon increasing gain in the whole region, but rather can I target narrow frequency bands, around resonances. This gives rise to I multiple crossover regions high-gain and low-gain regions can interlace. and implies that I This property is of great importance as it enables us to I exploit phase lag (even nonminimum-phase or delay) elements in feedback loops, thus going beyond Bode's gain-phase relation bounds. Outline "Flexible" loops Pendulum on cart DC motor with flexible transmission 2-degrees-of-freedom controller configuration 2DOF example: DC motor with flexible transmission Experimental setup DC motor Load DC motor is connected with load via flexible transmission. We want: I I complete steady-state rejection of step disturbances, dampened output response. The plant Plant transfer function (obtained experimentally) is 5. s C 40/.s C 40/2 P .s/ D 2 e .s C s C 156:25/.s 2 C 3:172s C 1936/ 0:1s ; with the following Bode plot and step response: Bode diagram of P .s/ 2.5 20 10 Magnitude (dB) 0 -10 -20 Step response of P .s/ 2 1.5 Amplitude -30 -40 0 1 -360 Phase (deg) 0.5 -720 -1080 0 -1440 10 0 10 1 10 Frequency (rad/sec) 2 10 3 -0.5 0 1 2 3 4 5 Time (sec) 6 7 8 9 10 Loop shaping Delay and NMP zero add considerable phase lag: Nichols chart of P .s/ 25 20 15 Open-Loop Gain (dB) 10 5 0 -5 -10 -15 -720 -630 -540 -450 -360 -270 Open-Loop Phase (deg) -180 -90 0 Thus, we have I no hope to squeeze both resonances before the first critical point and have to deal with multiple crossover regions and add phase lag. Loop shaping: adding integrator First, to satisfy step disturbance rejection requirement, let's add I controller: Nichols chart of P .s/ 1 s 40 30 20 10 Open-Loop Gain (dB) 0 -10 -20 -30 -40 -50 -60 -900 System: Pint Gain (dB): -20.4 Phase (deg): -564 Frequency (rad/sec): 44 System: Pint...
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