6 003 Signals and Systems CT Feedback and Control October 27 2009 Mid term Examination 2 Tomorrow October 28 7 30 9 30pm Walker Memorial 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 21 11 inches front and back Designed as 1 hour exam two hours to complete Feedback and Control Feedback is pervasive in natural and artificial systems We have previously investigated feedback in DT systems Example robotic steering x n controller y n plant sensor y n n Feedback and Control Today investigate continuous time feedback and control 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 Op Amp Much of the utility of op amps depends on their having high gain Ideal op amp lim K V K V Vo K V V Op Amp K j K j log scale The gain of an op amp depends on frequency 105 104 103 102 log scale 1 10 102 103 104 105 106 10 102 103 104 105 106 0 2 log scale 1 Frequency dependence of LM741 op amp Check Yourself K j K j log scale What system function has this frequency response 105 104 103 102 0 2 log scale 1 1 K0 s 2 10 K0 s s 102 3 103 104 K0 s s 105 4 106 K0 s 5 none Check Yourself K j K j log scale What system function has this frequency response 105 104 103 102 0 2 log scale 1 1 1 K0 s 2 10 K0 s s 102 3 103 104 K0 s s 105 4 106 K0 s 5 none Op Amp K j K j log scale Low gain at high frequencies limits applications audio frequencies 105 104 103 102 log scale 1 10 102 103 104 105 106 10 102 103 104 105 106 0 2 log scale 1 Unacceptable frequency response for an audio amplifier Op Amp An ideal op amp has fast time response Vi Vo K Step response Vo t s t Vi t u t 1 A t t Check Yourself Determine the step response of an LM741 s t A t seconds 1 A 2 105 40 s 2 A 2 105 1 s 40 3 A 80 105 40 s 4 A 80 105 1 s 40 5 none of the above Check Yourself Determine the step response of an LM741 System function K0 K s s Impulse response h t K0 e t u t Step response Z t Z t t K0 e s t h d K0 e d K0 1 e t u t 0 0 Parameters A K0 2 105 1 1 s 40 Check Yourself Determine the step response of an LM741 s t 2 105 t seconds 1 40 1 A 2 105 40 s 2 A 2 105 1 s 40 3 A 80 105 40 s 4 A 80 105 1 s 40 5 none of the above Op Amp Performance parameters for real op amps fall short of the ideal K j log scale Frequency Response high gain but only at low frequencies 105 audio frequencies 104 103 102 log scale 1 10 102 103 104 105 106 Step Response slow by electronic standards 2 105 t seconds 1 40 Op Amp The op amp s poor performance parameters result from a dominant pole at a relatively low frequency s plane 40 40 rad s 40 rad s 6 4 Hz 2 rad cycle Op amps are designed to have a dominant pole easy to customize their behavior using feedback Improving Performance Using feedback to improve performance parameters circuit Vi LTI model K s Vo R1 Vi K s V0 K s Vo Vi 1 K s R2 V Vo R2 R1 R2 K0 s 0 1 K s K0 s K0 Vo Vo Check Yourself What is the most negative value of the closed loop pole that can be achieved with feedback s plane 1 1 2 1 K0 3 1 K0 4 5 none of the above Improving Performance Using feedback to improve performance parameters circuit Vi LTI model K s Vo R1 Vi K s V0 K s Vo Vi 1 K s R2 V Vo R2 R1 R2 K0 s 0 1 K s K0 s K0 Vo Vo Check Yourself What is the most negative value of the closed loop pole that can be achieved with feedback Open loop system function K0 s pole s Closed loop system function K0 s K0 pole s 1 K0 The feedback constant is 0 1 most negative value of the closed loop pole is s 1 K0 Check Yourself What is the most negative value of the closed loop pole that can be achieved with feedback 3 s plane 1 K0 1 1 2 1 K0 3 1 K0 4 5 none of the above Improving Performance K j K j log scale K j0 Feedback extends frequency response by a factor of 1 K0 K0 2 105 1 0 1 1 K0 0 01 0 001 log scale 1 10 102 103 0 104 105 106 1 K0 2 log scale 1 10 102 103 104 105 106 Improving Performance Feedback produces higher bandwidths by reducing the gain at low frequencies It trades gain for bandwidth 0 open loop H j log scale 105 104 103 102 10 4 10 2 10 1 1 0 1 log scale 1 102 104 106 108 Improving Performance Feedback makes the time response faster by a factor of 1 K0 K0 2 105 Step response s t K0 1 e 1 K0 t u t 1 K0 K0 1 K0 s t 1 0 t seconds 1 40 Improving Performance Feedback produces faster responses by reducing the final value of the step response It trades gain for speed Step response s t K0 1 e 1 K0 t u t 1 K0 s t 0 2 105 0 5 10 5 1 5 10 5 t seconds 1 40 The maximum rate of voltage change ds t is not increased dt t 0 Improving Performance Feedback improves performance parameters of op amp circuits can extend frequency response can increase speed Performance enhancements are achieved through a reduction of gain Motor Controller 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 v t robotic arm t v t We wish to build the robot arm with a DC motor v t DC motor t This problem is similar to the head turning servo in 6 01 Check Yourself What is the relation between v t and t for a DC motor v t 1 2 3 4 5 DC motor t v t cos t v t t v t cos t v t none of the …
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