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CALTECH CDS 101 - Lecture notes

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CDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech1CDS 101: Lecture 8.1Frequency Domain DesignRichard M. Murray19 November 2002Goals:y Describe the use of frequency domain performance specificationy Show how to use “loop shaping” to achieve a performance specificationy Work through a detailed example of a control design problemReading: y No new reading this weeky Advanced: Lewis, Chapter 1218 Nov 02 R. M. Murray, Caltech CDS 2Lecture 7.1: Loop Analysis of Feedback Systemsy Nyquist criteria for loop stabilityy Gain, phase margin for robustnessC(s)++-dryeuP(s)rR-j∞+j∞Nyquist Diagram-1.5 -1 -0.5 0 0.5 1 1.5-3-2-101231GMPMPhase (deg); Magnitude (dB)Bode Diagram-100-5005010-210-1100101-300-200-1000PMGMGm=7.005 dB (at 0.34641 rad/sec), Pm=18.754 deg. (at 0.26853 rad/sec)Thm (Nyquist). P # RHP poles of L(s)N # CW encirclementsZ # RHP zerosZ = N + PReview from Last WeekCDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech218 Nov 02 R. M. Murray, Caltech CDS 3Frequency Domain Performance SpecificationsSpecify bounds on the loop transfer function to guarantee desired performance1yrLHL=+11()112cLjLjω≈=++() () ()LsPsCs=C(s)++-dryeuP(s)11erHL=+Fre quency (rad/se c)Phase (deg); Magnitude (dB)Bode Diagrams-60-40-2002040Gm=6.498 dB (at 14.426 rad/sec), Pm=27.848 deg. (at 9.532 rad/sec)10-1100101102-300-200-1000100TrackingPMGMPC 20log(100/ )dBX2BWSSy Steady state error: ⇒ zero frequency (“DC”) gainy Bandwidth: assuming ~90˚phase margin⇒ sets crossover freqy Tracking: X% error up to frequency ωt⇒ determines gain bound (1 + PC > 100/X)()(0) 1/ 1 (0) 1/ (0)erHLL=+ ≈18 Nov 02 R. M. Murray, Caltech CDS 4Relative StabilityRelative stability: how stable is system to disturbances at certain frequencies?y System can be stable but still have very bad response at certain frequenciesy Typically occurs if system has low phase margin ⇒ get resonant peak in closed loop (Mr) + poor step responsey Solution: specify minimum phase margin. Typically 45˚ or morePhase (deg); Magnitude (dB)Bode Diagrams-60-40-200204010-1100101102-300-200-1000100Time (sec.)AmplitudeStep Response0 0.5 1 1.5 2 2.5 300.511.5From: U(1)To: Y(1)()LsPMGMFrequency (rad/sec)Magnitude (dB)-60-40-2002010-11001011021yrLHL=+MrCDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech318 Nov 02 R. M. Murray, Caltech CDS 5Overview of Loop ShapingPerformance specificationSteady state errorTracking errorBandwidthRelative stabilityApproach: “shape” loop transfer function using C(s)y P(s) + specifications giveny L(s) = P(s) C(s)à Use C(s) to choose desired shape for L(s)y Important: can’t set gain and phase independentlyFrequency (rad/sec)-100-5005010-1100101102-300-200-1000100()Ps()Cs()Ls()Ls()Ps()Cs18 Nov 02 R. M. Murray, Caltech CDS 6LTIViewMATLAB LTIViewy Allows simul-taneous view of up to six plotsy Can apply to any system in the MATLAB workspacey Useful for seeing how the various concepts relate to each othersCautiony Doesn’t allow much control over details of each plotCDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech418 Nov 02 R. M. Murray, Caltech CDS 7Gain/phase relationshipsGain and phase for transfer function with real coefficients are not independenty Given a given shape for the gain, there is a unique “minimum phase” transfer function that achieves that gain at the specified frequenciesy Basic idea: slope of the gain determines the phasey Implication: you have to tradeoff gain versus phase in control designFrequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-200-1000100200From: U(1)10-1100101102-200-1000100200To: Y(1)Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-100-50050From: U(1)10-1100101102-300-250-200-150-100-50To: Y(1)2s−211()(10)Hsss=⋅+31s∼-20 dB/dec-60 dB/dec-90˚ phase-270˚ phase1s∼1s−0s1s2s2s−1s−0s1s2s18 Nov 02 R. M. Murray, Caltech CDS 8Proportional FeedbackSimplest controller choice: u = Kpey Effect: lifts gain with no change in phasey Good for plants with low phase up to desired bandwidthy Bode: shift gain up by factor of Kpy Nyquist: scale Nyquist contour+-ryeuP(s)pK()Cs()CspK()Ps()Ps-150-100-5005010-1100101102-300-200-1000-30 -20 -10 0 10 20 30-60-40-200204060pK0pK >,()Ls()LsCDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech518 Nov 02 R. M. Murray, Caltech CDS 9-200-100010010-1100101102103-300-200-1000100()Ps()PsLead compensation-20 -10 0 10 20 30 40 50-60-40-200204060Use to increase phase in frequency bandy Effect: lifts phase by increasing gain at high frequencyy Very useful controller; increases PMy Bode: add phase between zero and poley Nyquist: increase phase margin+-ryeuP(s)saKsb++/KabKzaω=pbω=()Cs()Cs()Ls()Lsab<0K >18 Nov 02 R. M. Murray, Caltech CDS 10-100-5005010010-210-1100101102-300-200-1000()Ps()PsProportional + Integral Compensation-14 -12 -10 -8 -6 -4 -2 0 2 4-20-15-10-505101520Use to eliminate steady state errory Effect: lifts gain at low frequencyy Gives zero steady state errory Bode: shift gain up by factor of Kpy Nyquist: scale Nyquist contour+-ryeuP(s)IpKKs+0pK >0IK >/zIpKKω=()Cs()Cs()Ls()LsCDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech618 Nov 02 R. M. Murray, Caltech CDS 11Example: Pitch Control for Caltech Ducted FanSystem descriptiony Vector thrust engine attached to wingy Inputs: fan thrust, thrust angle (vectored)y Outputs: position and orientationy States: x, y, θ+ derivativesy Dynamics: flight aerodynamicsControl approachy Design “inner loop” control law to regulate pitch (θ) using thrust vectoringy Second “outer loop” controller regulates the position and altitude by commanding the pitch and thrusty Basically the same approach as aircraft control laws18 Nov 02 R. M. Murray, Caltech CDS 12Performance Specification and Design ApproachDesign approachy Open loop plant has poor phase marginy Add phase lead in 5-50 rad/sec rangey Increase the gain to achieve steady state and tracking performance specsy Avoid integrator to minimize phasePerformance Specificationy ≤ 1% steady state errorà Zero frequency gain > 100y ≤ 10% tracking error up to 10 rad/secà Gain > 10 from 0-10 rad/secy ≥ 45˚ phase marginà Gives good relative stabilityà Provides robustness to uncertaintyFrequency (rad/sec)Phase (deg); Magnitude (dB)-50050100101-200-150-100-5002()rPsJsdsmgl=++()saCs Ksb+=+2530015 300abK===⋅CDS 101, Lecture 8.111/18/2002R. M. Murray, Caltech718 Nov 02 R. M. Murray, Caltech CDS 13-150-100-50050100100101102103104-200-150-100-500()Ps()Ps0 0.1 0.2 0.3 0.4 0.500.511.5Control Design and AnalysisSelect parameters


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