# lect03 - Control Theory(035188 lecture no 3 Leonid Mirkin...

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Control Theory (035188) lecture no. 3 Leonid Mirkin Faculty of Mechanical Engineering Technion — IIT Outline “Flexible” loops Pendulum on cart DC motor with flexible transmission 2-degrees-of-freedom controller configuration 2DOF example: DC motor with flexible transmission

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What we understand by “flexible” loops I Loops with one or several resonant frequencies 1 : -100 -80 -60 -40 -20 0 20 Magnitude (dB) 10 -1 10 0 10 1 10 2 -180 -135 -90 -45 0 Phase (deg) Frequency (rad/sec) 1 Typical example is flexible mechanical structures. What is special about flexible systems Resonances in frequency domain give rise to I slowly decaying oscillations, dominating time response: 0 10 20 30 40 50 60 70 80 90 100 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Impulse Response Time (sec) Amplitude Ability to cope with oscillations (dampen them) is main leitmotiv in control of flexible systems, frequently more important than high/low gain tradeoff.
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 Here pendulum is mounted on a cart, which is controller by a DC motor. A local servo loop is already closed around the motor, yet it does not dampen oscillations of the pendulum. So, our goal here is to I close the second loop, dampening pendulum oscillations.

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Plant After closing the motor position loop, plant becomes of the form: P.s/ D NUL 42s 2 .s C 18/.s 2 C 0:02s C 23/ : -60 -40 -20 0 20 40 60 Magnitude (dB) 10 -1 10 0 10 1 10 2 10 3 -270 -225 -180 -135 -90 -45 0 Phase (deg) Frequency (rad/sec) Bode diagram of P.s/ 0 5 10 15 20 25 30 35 40 -10 -5 0 5 10 Time (sec) Amplitude Response of P.s/ to a short pulse These oscillations should be dampened by feedback . Loop shaping: before we start Some observations are in order: 1. Static gain P.s/ j s D 0 and velocity gain 1 s P.s/ j s D 0 are both zero. 2. Gain at sub-resonance frequencies ( ! < 2 ) is pretty low. Thus, plant filters out low-frequency disturbances well (no help required 2 ). Also, 3. gain at over-resonance frequencies ( ! > 50 ) is low. Thus, plant filters out high-frequency noise also well (no help required). Therefore, I we only need to interfere around the resonance (i.e., in 2 < ! < 50 ). 2 In fact we cannot do much. For example, static gain can only be increased by canceling the poles at the origin, which is obviously illegal.
Loop shaping: proportional gain Comparison of these two plots 0 45 90 135 180 225 270 315 360 -60 -40 -20 0 20 40 60 Open-Loop Phase (deg) Open-Loop Gain (dB) Nichols chart of L.s/ D P.s/ -180 -135 -90 -45 0 45 90 135 180 -60 -40 -20 0 20 40 60 Open-Loop Phase (deg) Open-Loop Gain (dB) Nichols chart of L.s/ D NUL P.s/ shows clearly that negative gain ( positive feedback ) is preferable. Note that I loop gain has two crossover frequencies in this case, which means that we have two regions of interest for loop phase ( arg L. j !/ ). Loop shaping: keeping far from the critical point If we keep the plant’s first crossover ( ± 2:64 ) and want phase margin of 50 ı , -180 -135 -90 -45 0 45 90 135 180 -60 -40 -20 0 20 40 60 Open-Loop Phase (deg) Open-Loop Gain (dB) Nichols chart of L.s/ D NUL P.s/ Open-Loop Phase (deg) Open-Loop Gain (dB) -180 -135 -90 -45 0 45 90 135 180 -60 -40 -20 0 20 40 60 System: L Phase Margin (deg): 123 Delay Margin (sec): 0.15

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