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L8-2_limits

# L8-2_limits - CDS 101/110a Lecture 8-2 Limits on...

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1 CDS 101/110a: Lecture 8-2 Limits on Performance D. MacMynowski 17 November 2010 Goals: Describe limits of performance on feedback systems Introduce Bode’s integral formula and the “waterbed” effect Show some of the limitations of feedback due to RHP poles and zeros Reading: Åström and Murray, Feedback Systems, Ch 11 Algebraic Constraints on Performance C ( s ) + + - d r y eu P ( s ) + n Sensitivity function Complementary sensitivity ft i Goal: keep S & T small S small low tracking error T small good noise rejection (and robustness [CDS 110b]) Problem: S + T = 1 Can’t make both S & T small at the same frequency function Magnitude (dB) D. MacMynowski, Caltech CDS CDS 101/110, 17 Nov 10 2 Solution: keep S small at low frequency and T small at high frequency Loop gain interpretation: keep L large at low frequency, and small at high frequency Transition between large gain and small gain complicated by stability (phase margin)

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2 Sensitivity • From Rowley & Battin, Fundamentals & Applications of Modern Flow Control , Ch 5 • Example plotted is: • Plot is typical; as magnitude decreases, phase increases D. MacMynowski, Caltech CDS CDS 101/110, 17 Nov 10 3 Bode’s Integral Formula and the Waterbed Effect Bode’s integral formula for S = 1/(1+PC) = 1/(1+L): Let p k be the unstable poles of L ( s ) and a ssume relative degree of L ( s ) 2 Theorem: the area under the sensitivity function is a conserved quantity: Waterbed effect:
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L8-2_limits - CDS 101/110a Lecture 8-2 Limits on...

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