6 003 Signals and Systems CT Feedback and Control March 18 2010 Feedback and Control Feedback simple elegant and robust framework for control X E C Y controller plant S sensor We started with robotic driving di desiredFront do distanceFront Feedback and Control Using feedback to enhance performance Examples improve performance of an op amp circuit control position of a motor reduce sensitivity to unwanted parameter variation reduce distortions stabilize unstable systems magnetic levitation inverted pendulum Feedback and Control Reducing sensitivity to unwanted parameter variation Example power amplifier power amplifier MP3 player F0 speaker 8 F0 12 Changes in F0 due to changes in temperature for example lead to undesired changes in sound level Feedback and Control Feedback can be used to compensate for parameter variation power amplifier MP3 player X F0 K 8 F0 12 H s KF0 1 KF0 If K is made large so that KF0 1 then 1 H s independent of K or F0 Y speaker Feedback and Control Feedback reduces the change in gain due to change in F0 MP3 player X F0 100 Y 8 F0 12 1 10 Gain to Speaker 20 F0 no feedback 100F0 feedback 0 1 100F 10 10 8 F0 12 0 F0 0 10 20 Check Yourself power amplifier MP3 player X F0 K 8 F0 12 Y speaker Feedback greatly reduces sensitivity to variations in K or F0 lim H s K KF0 1 1 KF0 What about variations in Aren t those important Check Yourself What about variations in Aren t those important The value of is typically determined with resistors whose values are quite stable compared to semiconductor devices Crossover Distortion Feedback can compensate for parameter variation even when the variation occurs rapidly Example using transistors to amplify power 50V MP3 player speaker 50V Crossover Distortion This circuit introduces crossover distortion For the upper transistor to conduct Vi Vo VT For the lower transistor to conduct Vi Vo VT Vo 50V Vi Vo VT VT 50V Vi Crossover Distortion Crossover distortion can have dramatic effects Example crossover distortion when the input is Vi t B sin 0 t Vo t 50V Vi Vo 50V t Crossover Distortion Feedback can reduce the effects of crossover distortion 50V MP3 player K speaker 50V Crossover Distortion When K is small feedback has little effect on crossover distortion 50V Vi Vo t Vo K 50V K 1 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 2 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 4 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 10 t Crossover Distortion 50V Demo original no feedback K 2 K 4 K 8 K 16 original Vi Vo K 50V Vo t t J S Bach Sonata No 1 in G minor Mvmt IV Presto Nathan Milstein violin Feedback and Control Using feedback to enhance performance Examples improve performance of an op amp circuit control position of a motor reduce sensitivity to unwanted parameter variation reduce distortions stabilize unstable systems magnetic levitation inverted pendulum Control of Unstable Systems Feedback is useful for controlling unstable systems Example Magnetic levitation i t io y t Control of Unstable Systems Magnetic levitation is unstable i t io fm t y t Mg Equilibrium y 0 magnetic force fm t is equal to the weight M g Increase y increased force further increases y Decrease y decreased force further decreases y Positive feedback Modeling Magnetic Levitation The magnet generates a force that depends on the distance y t i t io fm t y t Mg fm t i t i0 Mg y t Modeling Magnetic Levitation The net force accelerates the mass i t io fm t y t Mg fm t M g f t M a M y t y t magnet f t 1 M A A y t Modeling Magnetic Levitation Over small distances magnetic force grows linearly with distance f t i t i0 Mg y t y t magnet f t 1 M A A y t Levitation with a Spring Relation between force and distance for a spring is opposite in sign F K x t y t M y t x t y t f t Mg K y t Modeling Magnetic Levitation Over small distances magnetic force nearly proportional to distance f t i t i0 K Mg y t f t Ky t y t f t K 1 M A A y t Block Diagrams Block diagrams for magnetic levitation and spring mass are similar Spring and mass F K x t y t M y t x t K M y t y t A y t A Magnetic levitation F Ky t M y t x t 0 K M y t A y t A y t Check Yourself How do the poles of these two systems differ Spring and mass F K x t y t M y t x t K M y t y t A y t A Magnetic levitation F Ky t M y t x t 0 K M y t A y t A y t Check Yourself How do the poles of the two systems differ Spring and mass F K x t y t M y t K Y M K X s2 M r s j s plane K M Magnetic levitation F Ky t M y t r K K s s M M 2 s plane Magnetic Levitation is Unstable i t io fm t y t Mg y t magnet f t 1 M A A y t Magnetic Levitation We can stabilize this system by adding an additional feedback loop to control i t f t i t 1 1i0 i t i0 i t 0 9i0 Mg y t Stabilizing Magnetic Levitation Stabilize magnetic levitation by controlling the magnet current i t io fm t y t Mg i t y t magnet f t 1 M A A y t Stabilizing Magnetic Levitation Stabilize magnetic levitation by controlling the magnet current i t io fm t y t Mg fi t fo t K2 1 M A K A y t Magnetic Levitation Increasing K2 moves poles toward the origin and then onto j axis x t K K2 M y t A s plane But the poles are still marginally stable y t A y t Magnetic Levitation Adding a zero makes the poles stable for sufficiently large K2 x t K K2 M s z0 y t A y t A s plane Try it Demo designed by Prof James Roberge y t Inverted Pendulum As a final example of stabilizing an unstable system consider an inverted pendulum m t d2 x t dt2 mg t l mg l x t lab frame inertial cart frame non inertial 2 d2 x t 2 d t ml mg l sin t m l cos t z dt2 z z dt2 z z I force distance force distance Check Yourself Inverted Pendulum Where are the poles …
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