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MIT 6 003 - Lecture Notes

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6 003 Signals and Systems Lecture 13 6 003 Signals and Systems October 27 2009 Mid term Examination 2 Tomorrow October 28 7 30 9 30pm Walker Memorial CT Feedback and Control No recitations tomorrow Coverage cumulative with more emphasis on recent material lectures 1 12 homeworks 1 7 Homework 7 includes practice problems for mid term 2 It will not collected or graded Solutions are posted Closed book 2 pages of notes 8 12 11 inches front and back Designed as 1 hour exam two hours to complete October 27 2009 Feedback and Control Feedback and Control Feedback is pervasive in natural and artificial systems Today investigate continuous time feedback and control We have previously investigated feedback in DT systems Examples Example robotic steering x n controller plant y n sensor y n 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 n Op Amp Op Amp Much of the utility of op amps depends on their having high gain The gain of an op amp depends on frequency V K V K j log scale lim K Vo K V V K j Ideal op amp 105 104 103 102 0 1 10 102 103 104 105 106 1 10 102 103 104 105 106 2 Frequency dependence of LM741 op amp 1 log scale log scale 6 003 Signals and Systems Lecture 13 Check Yourself October 27 2009 Op Amp Low gain at high frequencies limits applications K j log scale 105 104 103 102 0 2 1 1 K j K j K j log scale What system function has this frequency response K0 s 2 10 K0 s s 102 3 103 104 K0 s s 105 4 106 log scale K0 s audio frequencies 105 104 103 102 0 1 10 102 103 104 105 106 1 10 102 103 104 105 106 2 log scale log scale Unacceptable frequency response for an audio amplifier 5 none Op Amp Check Yourself An ideal op amp has fast time response Determine the step response of an LM741 Vi Vo K s t A Step response 1 t seconds Vo t s t Vi t u t 1 A 2 105 A 2 A 40 s 1 s 40 3 A 80 105 t t 2 105 4 A 80 105 40 s 1 s 40 5 none of the above Op Amp Improving Performance The op amp s poor performance parameters result from a dominant pole at a relatively low frequency Using feedback to improve performance parameters circuit s plane Vi LTI model K s Vo R1 40 Vi K s V0 Vo K s Vi 1 K s R2 40 rad s 40 rad s 6 4 Hz 2 rad cycle V Vo Op amps are designed to have a dominant pole easy to customize their behavior using feedback 2 R2 R1 R2 Vo K0 s 0 1 K s K0 s K0 Vo 6 003 Signals and Systems Lecture 13 Check Yourself October 27 2009 Improving Performance Feedback extends frequency response by a factor of 1 K0 K0 2 105 K j log scale K j0 What is the most negative value of the closed loop pole that can be achieved with feedback s plane 0 1 1 1 2 1 K0 3 1 K0 4 5 none of the above 1 K0 0 01 0 001 0 K j 1 1 10 102 103 104 105 106 log scale 1 K0 2 1 10 102 103 104 105 106 log scale Improving Performance Improving Performance Feedback produces higher bandwidths by reducing the gain at low frequencies It trades gain for bandwidth Feedback makes the time response faster by a factor of 1 K0 K0 2 105 H j log scale 105 104 103 102 10 1 0 open loop Step response 10 4 s t 10 2 K0 1 e 1 K0 t u t 1 K0 K0 1 K0 s t 1 0 1 1 40 0 1 t seconds log scale 1 102 104 106 108 Improving Performance Improving Performance Feedback produces faster responses by reducing the final value of the step response It trades gain for speed Feedback improves performance parameters of op amp circuits Step response can extend frequency response can increase speed Performance enhancements are achieved through a reduction of gain s t K0 1 e 1 K0 t u t 1 K0 s t 0 2 105 1 40 The maximum rate of voltage change 0 5 10 5 1 5 10 5 t seconds ds t is not increased dt t 0 3 6 003 Signals and Systems Lecture 13 Motor Controller October 27 2009 Check Yourself We wish to build a robot arm actually its elbow The input should be voltage v t and the output should be the elbow angle t robotic arm v t What is the relation between v t and t for a DC motor t v t We wish to build the robot arm with a DC motor DC motor v t DC motor v t 1 2 3 4 5 t t t v t cos t v t t v t cos t v t none of the above This problem is similar to the head turning servo in 6 01 Motor Controller Motor Controller Use proportional feedback to control the angle of the motor s shaft The closed loop system has a single pole at s v t amplifier DC motor A V s t s plane feedback potentiometer 1s A V 1 A s 1 1s As increases the closed loop pole becomes increasingly negative Motor Controller Motor Controller Find the impulse and step response The speed of a DC motor does not change instantly if the voltage is stepped There is lag due to rotational inertia The system function is V s The impulse response is First order model integrator V h t e t u t and the step response is therefore 1 s t 1 e t u t 1 A Step response of the models v t t t 1 Second order model integrator with lag pA A 2 V 1 pA t t The response is faster for larger values of Try it Demo 4 1 t tu t 2 t t p 1 e pt u t t 6 003 Signals and Systems Lecture 13 October 27 2009 Motor Controller Motor Controller Analyze second order model For second order model increasing causes the poles at 0 and p to approach each other collide at s p 2 then split into two poles with imaginary parts amplifier v t DC motor pA2 1 pA t s plane feedback potentiometer p pA2 pA2 p 1 pA 2 2 pA V 1 pA pA2 s ps p 1 1 pA r p p 2 s p 2 2 Increasing the gain does not increase speed of convergence Motor Controller Motor Controller Step response Step response s t s t 1 1 t t Motor Controller Feedback and Control Summary Step response CT feedback is …


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MIT 6 003 - Lecture Notes

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