MIT16_30F10_lec03

MIT16_30F10_lec03 - Topic #3 16.30/31 Feedback Control...

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Topic #3 16.30/31 Feedback Control Systems Frequency response methods Analysis Synthesis Performance Stability in the Frequency Domain Nyquist Stability Theorem
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Fall 2010 16.30/31 3–2 FR: Introduction Root locus methods have: Advantages: Good indicator of transient response; Explicitly shows location of all closed-loop poles; Trade-offs in the design are fairly clear. Disadvantages: Requires a transfer function model (poles and zeros); Difficult to infer all performance metrics; Hard to determine response to steady-state (sinusoids) Hard to infer stability margins Frequency response methods are a good complement to the root locus techniques: Can infer performance and stability from the same plot Can use measured data rather than a transfer function model Design process can be independent of the system order Time delays are handled correctly Graphical techniques (analysis and synthesis) are quite simple. September 15, 2010
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Fall 2010 16.30/31 3–3 Frequency Response Function Given a system with a transfer function G ( s ) , we call the G ( j ω ) , ω [0 , ) the frequency response function (FRF) G ( j ω ) = | G ( j ω ) | G ( j ω ) The FRF can be used to ±nd the steady-state response of a system to a sinusoidal input since, if e ( t ) y ( t ) G ( s ) and e ( t ) = sin 2 t , | G (2 j ) | = 0 . 3 , G (2 j ) = 80 , then the steady-state output is y ( t ) = 0 . 3 sin(2 t 80 ) The FRF clearly shows the magnitude (and phase) of the re- sponse of a system to sinusoidal input A variety of ways to display this: 1. Polar ( Nyquist ) plot Re vs. Im of G ( j ω ) in complex plane. Hard to visualize, not useful for synthesis, but gives de±nitive tests for stability and is the basis of the robustness analysis. 2. Nichols Plot | G ( j ω ) | vs. G ( j ω ) , which is very handy for sys- tems with lightly damped poles. 3. Bode Plot Log | G ( j ω ) | and G ( j ω ) vs. Log frequency. Simplest tool for visualization and synthesis Typically plot 20 log | G | which is given the symbol dB September 15, 2010
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Fall 2010 16.30/31 3–4 Use logarithmic since if log | G ( s ) | = ( s + 1)( s + 2) ( s + 3)( s + 4) = log | s + 1 | + log | s + 2 | − log | s + 3 | − log | s + 4 | and each of these factors can be calculated separately and then added to get the total FRF. Can also split the phase plot since ( s + 1)( s + 2) = ( s + 1) + ( s + 2) ( s + 3)( s + 4) ( s + 3) ( s + 4) The keypoint in the sketching of the plots is that good straightline approximations exist and can be used to obtain a good prediction of the system response.
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MIT16_30F10_lec03 - Topic #3 16.30/31 Feedback Control...

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