MIT6_003S10_lec12

MIT6_003S10_lec12 - .003: Signals and Systems T Feedback...

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Unformatted text preview: .003: Signals and Systems T Feedback and Control arch 16, 2010 Feedback and Control Feedback: simple, elegant, and robust framework for control. + − X Y E S C controller plant sensor Last time: robotic driving. d i = desiredFront d o = 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 p-amps An “ideal” op-amp has many desireable characteristics. V o = K ( V + − V − ) V + V − K • high speed • large bandwidth • high input impedance • low output impedance • ... It is difficult to build a circuit with all of these features. p-Amp The gain of an op-amp depends on frequency. 10 5 10 4 10 3 10 2 ω [log scale] 1 10 10 2 10 3 10 4 10 5 10 6 ω [log scale] 1 10 10 2 10 3 10 4 10 5 10 6 | K ( jω ) | [log scale] − π 2 ∠ K ( jω ) | Frequency dependence of LM741 op-amp. p-Amp Low-gain at high frequencies limits applications. ∠ K ( jω ) | | K ( jω ) | [log scale] 10 5 10 4 10 3 10 2 audio frequencies ω [log scale] 1 10 10 2 10 3 10 4 10 5 10 6 − π 2 ω [log scale] 1 10 10 2 10 3 10 4 10 5 10 6 Unacceptable frequency response for an audio amplifier. p-Amp An ideal op-amp has fast time response. V i K V o Step response: V i ( t ) = u ( t ) V o ( t ) = s ( t ) 1 t A t heck Yourself Determine τ for the unit-step response s ( t ) of an LM741. τ A t [seconds] s ( t ) 10 2 10 3 10 4 10 5 | K ( jω ) | [log scale] − π 2 1 10 10 2 10 3 10 4 10 5 ω [log scale] ∠ K ( jω ) | 1. 40 s 2. 40 2 π s 3. 1 40 s 4. 2 π 40 s 5. 1 2 π × 40 s 0. none of the above heck Yourself Determine the step response of an LM741. System function: K ( s ) = αK s + α Impulse response: h ( t ) = αK e − αt u ( t ) Step response: s ( t ) = t h ( τ ) dτ = t αK e − ατ dτ = αK e − ατ t = K (1 − e − αt ) u ( t ) −∞ − α Parameters: A = K = 2 × 10 5 1 1 τ = α = 40 s heck...
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This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec12 - .003: Signals and Systems T Feedback...

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