CDS 101, Lecture 8.115 November 2004R. M. Murray, Caltech1CDS 101: Lecture 8.1Frequency Domain Design using PIDRichard M. Murray15 November 2004Goals:y Describe the use of frequency domain performance specificationsy Show how to use “loop shaping” using PID to achieve a performance specificationReading: y Åström and Murray, Analysis and Design of Feedback Systems, 7.6ff and Ch 815 Nov 04 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 8.115 November 2004R. M. Murray, Caltech215 Nov 04 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=+Frequency (rad/sec)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/ )dBX2BWSSy 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=+ ≈15 Nov 04 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 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 8.115 November 2004R. M. Murray, Caltech315 Nov 04 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()Cs15 Nov 04 R. M. Murray, Caltech CDS 6Gain/phase relationshipsGain and phase for transfer function w/ real coeffs 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−0s1s2sCDS 101, Lecture 8.115 November 2004R. M. Murray, Caltech415 Nov 04 R. M. Murray, Caltech CDS 7Overview: PID controlIntuitiony Proportional term: provides inputs that correct for “current” errorsy Integral term: insures steady state error goes to zeroy Derivative term: provides “anticipation” of upcoming changesA bit of history on “three term control”y First appeared in 1922 paper by Minorsky: “Directional stability of automatically steered bodies” under the name “three term control”y Also realized that “small deviations” (linearization) could be used to understand the (nonlinear) system dynamics under controlUtility of PIDy PID control is most common feedback structure in engineering systemsy For many systems, only need PI or PD (special case)y Many tools for tuning PID loops and designing gains (see reading)PID+-ryeuP(s)pIDuKeKeKe=+ +∫&15 Nov 04 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 8.115 November 2004R. M. Murray, Caltech515 Nov 04 R. M. Murray, Caltech CDS 9-100-5005010010-210-1100101102-300-200-1000()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: infinite SS gain + phase lagy Nyquist: no easy interpretationy Note: this example is unstable+-ryeuP(s)IpKKs+0pK >0IK >/zIpKKω=()Cs()Cs()Ls()Ls15 Nov 04 R. M. Murray, Caltech CDS 10Proportional + Integral + Derivative (PID)Transfer function for PID controllery Idea: gives high gain at low frequency plus phase lead at high frequencyy Place ω1and ω2below desired crossover freqC(s)++-dryeuP(s)1()pIDCs K K Kss=+⋅+Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams0102030405010-310-210-1100101102103-100-50050100pIDuKeKeKe=+ +∫&1()ue p I DHs K K Kss=+⋅+11ITω=21DTω=1(1 )DIkTsTs=++(1/)(1/)DIDIkT s T s TTs++=CDS 101, Lecture 8.115 November 2004R. M. Murray, Caltech615 Nov 04 R. M. Murray, Caltech CDS 11Example: Cruise Control using PID - Specification1/()/mrPssbms a=⋅++Performance 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 uncertaintyObservationsy Purely proportional gain won’t work: to get gain above desired level will not leave adequate phase marginey Need to increase the phase from ~0.5 to 2 rad/sec and increase gain as wellFrequency (rad/sec)Phase (deg); Magnitude (dB)-150-100-5005010-210-1100101102-200-150-100-50015 Nov 04 R. M. Murray, Caltech CDS 12Frequency (rad/sec)Phase (deg); Magnitude
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