6 003 Signals and Systems Lecture 13 6 003 Signals and Systems March 18 2010 Feedback and Control Feedback simple elegant and robust framework for control CT Feedback and Control X E C controller Y plant S sensor We started with robotic driving di desiredFront do distanceFront March 18 2010 Feedback and Control Feedback and Control Using feedback to enhance performance Reducing sensitivity to unwanted parameter variation Example power amplifier Examples power amplifier 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 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 and Control Feedback can be used to compensate for parameter variation Feedback reduces the change in gain due to change in F0 power amplifier MP3 player X F0 K 8 F0 12 MP3 player X Y F0 100 Y 8 F0 12 speaker 1 10 20 Gain to Speaker KF0 H s 1 KF0 If K is made large so that KF0 1 then 1 H s F0 no feedback 100F0 feedback 0 1 100F 10 10 8 F0 12 0 independent of K or F0 F0 0 1 10 20 6 003 Signals and Systems Lecture 13 Check Yourself Crossover Distortion Feedback can compensate for parameter variation even when the variation occurs rapidly power amplifier MP3 player March 18 2010 X Example using transistors to amplify power Y F0 K 50V 8 F0 12 speaker MP3 player Feedback greatly reduces sensitivity to variations in K or F0 speaker KF0 1 lim H s 1 KF0 K 50V What about variations in Aren t those important Crossover Distortion Crossover Distortion This circuit introduces crossover distortion Crossover distortion can have dramatic effects For the upper transistor to conduct Vi Vo VT For the lower transistor to conduct Vi Vo VT Example crossover distortion when the input is Vi t B sin 0 t Vo t 50V Vo 50V Vi Vi VT Vo Vo t Vi VT 50V 50V Crossover Distortion Crossover Distortion Feedback can reduce the effects of crossover distortion As K increases feedback reduces crossover distortion 50V 50V MP3 player Vo t K 4 K Vi speaker Vo K 50V 50V 2 t 6 003 Signals and Systems Lecture 13 Crossover Distortion Feedback and Control 50V Demo original no feedback K 2 K 4 K 8 K 16 original Vi March 18 2010 Using feedback to enhance performance Examples Vo K 50V Vo t 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 t J S Bach Sonata No 1 in G minor Mvmt IV Presto Nathan Milstein violin Control of Unstable Systems Control of Unstable Systems Feedback is useful for controlling unstable systems Magnetic levitation is unstable Example Magnetic levitation i t io i t io fm t y 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 Modeling Magnetic Levitation The magnet generates a force that depends on the distance y t The net force accelerates the mass i t io i t io fm t fm t y t y t Mg Mg fm t fm t M g f t M a M y t i t i0 y t Mg y t 3 magnet f t 1 M A A y t 6 003 Signals and Systems Lecture 13 March 18 2010 Modeling Magnetic Levitation Levitation with a Spring Over small distances magnetic force grows linearly with distance Relation between force and distance for a spring is opposite in sign F K x t y t M y t f t i t i0 x t y t y t Mg y t magnet f t f t 1 M A A y t Mg K y t Modeling Magnetic Levitation Block Diagrams Over small distances magnetic force nearly proportional to distance Block diagrams for magnetic levitation and spring mass are similar f t Spring and mass F K x t y t M y t i t i0 K Mg x t K M y t y t A y t A y t Magnetic levitation f t Ky t f t y t K F Ky t M y t 1 M A y t A x t 0 Check Yourself A y t K M fm t y t y t y t A y t A Mg y t Magnetic levitation F Ky t M y t x t 0 y t i t io Spring and mass F K x t y t M y t A Magnetic Levitation is Unstable How do the poles of these two systems differ x t y t K M K M y t A y t A y t 4 magnet f t 1 M A A y t 6 003 Signals and Systems Lecture 13 March 18 2010 Magnetic Levitation Stabilizing Magnetic Levitation We can stabilize this system by adding an additional feedback loop to control i t Stabilize magnetic levitation by controlling the magnet current i t io f t i t 1 1i0 i t i0 fm t i t 0 9i0 Mg y t Mg y t i t y t f t magnet 1 M A y t A Stabilizing Magnetic Levitation Magnetic Levitation Stabilize magnetic levitation by controlling the magnet current Increasing K2 moves poles toward the origin and then onto j axis x t i t io fm t K K2 M y t y t A y t A s plane y t Mg fi t K2 1 M A y t A K fo t But the poles are still marginally stable Magnetic Levitation Inverted Pendulum Adding a zero makes the poles stable for sufficiently large K2 As a final example of stabilizing an unstable system consider an inverted pendulum x t K K2 M s z0 y t A y t A y t m t d2 x t dt2 mg s plane t l mg l x t lab frame inertial cart frame non inertial d2 x t d2 t mg l sin t m l cos t ml2 z z z dt2 dt2 z z I Try it Demo designed by Prof James Roberge 5 force distance force distance 6 003 Signals and Systems Lecture 13 Check Yourself Inverted Pendulum March 18 2010 Inverted Pendulum This unstable system can be stablized with feedback Where are the poles of this system t m mg d2 x t dt2 t x t ml2 mg t l d2 x t dt2 t l mg x t l d2 x t d2 t mgl sin t m l cos t dt2 dt2 mg l Try it Demo originally designed by Marcel Gaudreau Feedback and Control Using feedback to enhance performance Examples m improve performance of an op amp circuit control position …
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