L7-1_loopanal

L7-1_loopanal - CDS 101/110a: Lecture 7-1 Loop Analysis of...

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1 CDS 101/110a: Lecture 7-1 Loop Analysis of Feedback Systems Douglas MacMynowski 8 November 2010 Goals: Show how to compute closed loop stability from open loop properties Describe the Nyquist stability criterion for stability of feedback systems Define gain and phase margin and determine it from Nyquist and Bode plots Reading: Åström and Murray, Feedback Systems, Ch 9.1-9.4 Advanced : Lewis, Chapters 7 Review From Last Week 101 e Frequency (rad/sec) Phase (deg) 100 101 -200 -150 -100 -50 0 10-2 10-1 100 Magnitude F d D.MacMynowski, Caltech CDS CDS 101/110, 8 Nov 10 2 -c -k C ( s ) P ( s ) + + y eu -1 r
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2 Closed Loop Stability Q: how do open loop dynamics affect the closed loop stability? Given open loop transfer function C ( s ) P ( s ) determine when system is stable C ( s ) P ( s ) + + d y eu -1 r Brute force answer: compute poles of closed loop transfer function Alternative: look for conditions on PC that lead to instability Example: if PC s 1forsome s • Poles of H yr = zeros of 1 + PC • Easy to compute, but not so good for design P(s)C(s) itude (dB) -10 0 10 20 D. MacMynowski, Caltech CDS CDS 101/110, 8 Nov 10 3 Example: if PC ( s ) = - 1 for some i , then system is not asymptotically stable Condition on PC is much nicer because we can design PC ( s ) by choice of C ( s ) However, checking PC ( s ) = - 1 is not enough; need more sophisticated check Frequency (rad/sec) Phase (deg) Magni -60 -50 -40 -30 -20 0.1 1 10 -360 -270 -180 -90 0 Game Plan: Frequency Domain Design Goal: figure out how to design C ( s ) so that 1+ C ( s ) P ( s ) is stable and we get good performance • Poles of H yr = zeros of 1 + PC • Would also like to “shape” H yr to specify performance at different frequencies Low frequency range: (good tracking) Bandwidth: frequency at which closed loop response = 1/ 2 open loop gain 1 -150 -100 -50 0 50 100 Magnitude (dB) 0 Bode Diagram D. MacMynowski, Caltech CDS CDS 101/110, 8 Nov 10 4 Idea: use C ( s ) to shape PC (under certain constraints) Need tools to analyze stability
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This note was uploaded on 01/04/2012 for the course CDS 101 taught by Professor Murray,r during the Fall '08 term at Caltech.

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L7-1_loopanal - CDS 101/110a: Lecture 7-1 Loop Analysis of...

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