{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 2
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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}