6.003: Signals and Systems Lecture 13 March 18, 2010

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1 6.003: Signas and Systems CT Feedback: simpe, eegant, and robust framework for contro. E C controer pant S sensor We started with robotic driving. March 18, 2010 d i = desiredfront d o = distancefront Using feedback to enhance performance. Exampes: improve performance of an op amp circuit. contro position of a motor. reduce sensitivity to unwanted parameter variation. reduce distortions. stabiize unstabe systems magnetic evitation inverted penduum Reducing sensitivity to unwanted parameter variation. Exampe: power ampifier power ampifier 8 < < 12 speaker Changes in (due to changes in temperature, for exampe) ead to undesired changes in sound eve. Feedback can be used to compensate for parameter variation. power ampifier 8 < < 12 speaker Feedback reduces the change in gain due to change in < < 12 H(s) = 1 β If is made arge, so that β 1, then H(s) 1 β independent of or! β Gain to Speaker < < (no feedback) (feedback) 1

2 Check oursef power ampifier Feedback can compensate for parameter variation even when the variation occurs rapidy. 8 < < 12 speaker Exampe: using transistors to ampify power. 50V β Feedback greaty reduces sensitivity to variations in or. im H(s) = 1 1 β β 50V speaker What about variations in β? ren t those important? This circuit introduces crossover distortion. For the upper transistor to conduct, V i V o >V T. For the ower transistor to conduct, V i V o < V T. 50V V o Crossover distortion can have dramatic effects. Exampe: crossover distortion when the input is V i (t) = B sin(ω 0 t). 50V V o (t) V i V o t V i V o V T V i V T 50V 50V Feedback can reduce the effects of crossover distortion. 50V s increases, feedback reduces crossover distortion. 50V V o (t) = 4 speaker V i V o t 50V 50V 2

3 Demo 50V Using feedback to enhance performance. origina Exampes: no feedback V i V o = 2 improve performance of an op amp circuit. = 4 contro position of a motor. = 8 reduce sensitivity to unwanted parameter variation. 50V = 16 reduce distortions. origina stabiize unstabe systems V o (t) magnetic evitation inverted penduum t J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Mistein, vioin Contro of Unstabe Systems Feedback is usefu for controing unstabe systems. Contro of Unstabe Systems Magnetic evitation is unstabe. Exampe: Magnetic evitation. Equiibrium (y = 0): magnetic force is equa to the weight. Increase y increased force further increases y. Decrease y decreased force further decreases y. Positive feedback! The magnet generates a force that depends on the distance. The net force acceerates the mass. = =Ma = Mÿ(t) M 3

4 Over sma distances, magnetic force grows ineary with distance. Levitation with a Spring Reation between force and distance for a spring is opposite in sign. ( ) F = = Mÿ(t) M Over sma distances, magnetic force neary proportiona to distance. Bock Diagrams Bock diagrams for magnetic evitation and spring/mass are simiar. Spring and mass ( ) F = = Mÿ(t) ÿ(t) ẏ(t) M Magnetic evitation 1 M F = =Mÿ(t) ÿ(t) ẏ(t) =0 M Check oursef Magnetic Levitation is Unstabe How do the poes of these two systems differ? Spring and ( mass ) F = = Mÿ(t) M ÿ(t) ẏ(t) Magnetic evitation F = =Mÿ(t) M =0 M ÿ(t) ẏ(t) 4

5 Magnetic Levitation We can stabiize this system by adding an additiona feedback oop to contro i(t). Stabiizing Magnetic Levitation Stabiize magnetic evitation by controing the magnet current. i(t) =1.1i 0 i(t) =0.9i 0 i(t) α M Stabiizing Magnetic Levitation Stabiize magnetic evitation by controing the magnet current. Magnetic Levitation Increasing 2 moves poes toward the origin and then onto jω axis. ÿ(t) ẏ(t) 2 M s-pane f i (t) 2 1 M f o (t) But the poes are sti marginay stabe. Magnetic Levitation Inverted Penduum dding a zero makes the poes stabe for sufficienty arge 2. ÿ(t) ẏ(t) 2 M (s z 0 ) s-pane s a fina exampe of stabiizing an unstabe system, consider an inverted penduum. d 2 m ab frame (inertia) cart frame (non-inertia) Try it: Demo [designed by Prof. James Roberge]. 2 d2 d 2 }{{} m = sin m cos }{{}}{{}}{{ }}{{} I force distance distance force 5

6 6.003: Signas and Systems Lecture 13 March 18, 2010 Check oursef: Inverted Penduum Inverted Penduum This unstabe system can be stabized with feedback. Where are the poes of this system? m d2 d 2 m m 2 d2 = sin m d2 cos Try it. Demo. [originay designed by Marce Gaudreau] Using feedback to enhance performance. Exampes: improve performance of an op amp circuit. contro position of a motor. reduce sensitivity to unwanted parameter variation. reduce distortions. stabiize unstabe systems magnetic evitation inverted penduum 6

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6.003: Signals and Systems

6.003: Signals and Systems CT Feedback and Control October 20, 2011 1 Mid-term Examination #2 Wednesday, October 26, 7:30-9:30pm, No recitations on the day of the exam. Coverage: Lectures 1 12 Recitations