6 003 Signals and Systems CT Feedback and Control March 16 2010 Feedback and Control Feedback simple elegant and robust framework for control X E C Y controller plant S sensor Last time robotic driving di desiredFront do distanceFront Feedback and Control This week using feedback to enhance performance Examples increasing speed and bandwidth controlling position instead of speed reducing sensitivity to parameter variation reducing distortion stabilizing unstable systems magnetic levitation inverted pendulum Op amps An ideal op amp has many desireable characteristics V K Vo K V V V high speed large bandwidth high input impedance low output impedance It is difficult to build a circuit with all of these features 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 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 s t A t seconds K j K j log scale Determine for the unit step response s t of an LM741 105 104 103 102 0 2 1 1 40 s 40 s 2 2 1 s 3 40 10 4 2 40 s 0 none of the above 102 103 log scale 104 1 5 2 40 s 105 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 s t A t seconds K j K j log scale Determine for the unit step response s t of an LM741 105 104 103 102 0 2 1 1 40 s 40 s 2 2 1 s 3 40 10 4 2 40 s 0 none of the above 102 103 log scale 104 1 5 2 40 s 105 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 We can use feedback to improve performance of op amps circuit Vi 6 003 model K s Vo Vi K s R1 V0 R2 K s Vo Vi 1 K s V Vo R2 R1 R2 Vo Vo Dominant Pole Op amps are designed to have a dominant pole at low frequencies simplifies the application of feedback s plane 40 40 rad s 40 rad s 6 4 Hz 2 rad cycle Improving Performance Using feedback to improve performance parameters circuit Vi 6 003 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 6 003 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 H j H j log scale H 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 above t Check Yourself What is the relation between v t and t for a DC motor To first order the rotational speed t of a DC motor is proportional to the input voltage v t v t t DC motor v t t t t First order model integrator V A 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 above t Motor Controller Use proportional feedback to …
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