MIT16_30F10_lec03

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

This preview shows pages 1–5. Sign up to view the full content.

Topic #3 16.30/31 Feedback Control Systems Frequency response methods Analysis Synthesis Performance Stability in the Frequency Domain Nyquist Stability Theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 17

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online