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CALTECH CDS 101 - Limits of Performance

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CDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech1CDS 101: Lecture 9.1Limits of PerformanceRichard M. Murray22 November 2004Goals:y Describe limits of performance on feedback systemsy Introduce Bode’s integral formula and the “waterbed” effecty Show some of the limitations of feedback due to RHP poles and zerosReading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 922 Nov 04 R. M. Murray, Caltech CDS 2Lecture 8.1: Frequency Domain Design using PIDLoop Shaping for Stability & Performancey Steady state error, bandwidth, trackingMain ideasy Performance specs give bounds on loop transfer functiony Use controller to shape responsey Gain/phase relationships constrain design approachy Standard compensators: proportional, PI, PIDFrequency (rad/sec)Phase (deg); Magnitude (dB)-100-5005010015010-310-210-1100101-200-1000100()Cs()Cs()Ps()Ps()Ls()Ls1()ue p I DHs K K Kss=+⋅+Review from Last WeekCDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech222 Nov 04 R. M. Murray, Caltech CDS 3“Gang of Four”Four unique transfer functions define performancey Stability is always determined by 1/(1+PC)y Numerator determined by forward path between input and outputNoise and disturbancesy d = process disturbancesy n = sensor noisey Keep track of transfer functions between all possible inputs and outputsDesign represents a tradeoff between the quantitiesy Keep L=PC large for good performance (Her<< 1)y Keep L=PC small for good noise rejection (Hyn<< 1)C(s)++-dryeuP(s)+ n22 Nov 04 R. M. Murray, Caltech CDS 4Algebraic Constraints on PerformanceGoal: keep S & T smally S small ⇒ low tracking errory T small ⇒ good noise rejection (and robustness [CDS 110b])Problem: S + T = 1y Can’t make both S & T small at the same frequencyy Solution: keep S small at low frequency and T small at high frequencyy Loop again interpretation: keep L large at low frequency, and small at high frequencyy Transition between large gain and small gain complicated by stability (phase margin)C(s)++-dryeuP(s)+nSensitivityfunctionComplementarysensitivityfunctionMagnitude (dB)()Ls() 1Ls() 1Ls CDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech322 Nov 04 R. M. Murray, Caltech CDS 5Bode’s Integral Formula and the Waterbed EffectBode’s integral formula for S = 1/(1+PC) = 1/(1+L):y Let pkbe the unstable poles of L(s) and assume relative degree of L(s) ≥ 2y Theorem: the area under the sensitivity function is a conserved quantity:Waterbed effect:y Making sensitivity smaller over some frequency range requires increase in sensitivity someplace elsey Presence of RHP poles makes this effect worsey Actuator bandwidth further limits what you can doy Note: area formula is linear in ω; Bode plots are logarithmicFrequency (rad/sec)Magnitude (dB)Sensitivity Function-40-30-20-10010100101102103104Area below 0 dB + area above 0 dB = π∑Re pk= constant22 Nov 04 R. M. Murray, Caltech CDS 6Example: Magnetic LevitationSystem descriptiony Ball levitated by electromagnety Inputs: current thru electromagnety Outputs: position of ball (from IR sensor)y States: y Dynamics: F = ma, F = magnetic force generated by wire coily See MATLAB handout for detailsController circuity Active R/C filter networky Inputs: set point, disturbance, ball positiony States: currents and voltagesy Outputs: electromagnet currentIRreceivierIRtransmitterElectro-magnetBallCDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech422 Nov 04 R. M. Murray, Caltech CDS 7Equations of MotionProcess: actuation, sensing, dynamicsy u = current to electromagnety vir= voltage from IR sensorLinearization:y Poles at s = ± r ⇒ open loop unstableIRreceivierIRtransmitterElectro-magnetBallReal AxisImaginary AxisNyquist Diagram-4 -2 0 2 4 6 8-1.5-1-0.500.511.5Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagram-100-50050100101102103104-200-150-100-50050Note: RHP pole in L ⇒ need one net encirclement (CCW)22 Nov 04 R. M. Murray, Caltech CDS 8Control DesignNeed to create encirclementy Loop shaping is not useful herey Flip gain to bring Nyquist plot over -1 pointy Insert phase to create CCW encirclementCan accomplish using a lead compensatory Produce phase lead at crossovery Generates loop in Nyquist plotReal AxisImaginary AxisNyquist Diagram-4 -2 0 2 4 6 8-1.5-1-0.500.511.5Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagram-100-50050100101102103104-200-150-100-50050ω=0ω=∞ω=0CDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech522 Nov 04 R. M. Murray, Caltech CDS 9Performance LimitsNominal design gives low perfy Not enough gain at low frequencyy Try to adjust overall gain to improve low frequency responsey Works well at moderate gain, but notice waterbed effectBode integral limits improvementy Must increase sensitivity at some pointFrequency (rad/sec)Magnitude (dB)Sensitivity Function-40-30-20-10010100101102103104Time (sec.)AmplitudeStep Response0 0.04 0.08 0.12 0.1600.20.40.60.811.21.41.622 Nov 04 R. M. Murray, Caltech CDS 10Right Half Plane ZerosRight half plane zeros produce “non-minimum phase” behaviory Phase of frequency response has additional phase lag for given magnitudey Can cause output to move opposite from input for a short period of timeExample: vsFrequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-30-20-10010100101102-300-200-1000122()2nnsaHsssζωω+=++222()2nnsaHsssζωω−=++H1H1H2, H2Time (sec.)AmplitudeStep Response0 0.2 0.4 0.6 0.8 1 1.2-0.200.20.40.60.811.2H1H2CDS 101, Lecture 9.122 November 2004R. M. Murray, Caltech622 Nov 04 R. M. Murray, Caltech CDS 11Example: Lateral Control of the Ducted FanSource of non-minimum phase behaviory To move left, need to make θ> 0y To generate positive θ, need f1 > 0y Positive f1causes fan to move right initiallyy Fan starts to move left after short time (as fan rotates)θ),( yx1f2f1222()()()xfsmglHssJs ds mgl−=++y Poles: 0, 0, -σ± jωdy Zeros: mgl±Time (sec.)AmplitudeStep Response0 0.2 0.4 0.6 0.8 1-4-3.5-3-2.5-2-1.5-1-0.500.5Fan moves right andthen moves to the left22 Nov 04 R. M. Murray, Caltech CDS 12Stability in the Presence of ZerosLoop gain limitationsy Poles of closed loop = poles of 1 + L. Suppose C = k nc/dc, where kis the gain of the controllery For large k, closed loop poles approach open loop zerosy RHP zeros limit maximum gain ⇒ serious design constraint!Root locus interpretationy Plot location of eigenvalues as afunction of the loop gain ky Can show that closed loop poles gofrom open loop poles (k = 0) to


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