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CALTECH CDS 101 - Loop Analysis of Feedback Systems

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CDS 101, Lecture 7.110 November 2003R. M. Murray, Caltech1CDS 101: Lecture 7.1Loop Analysis of Feedback SystemsRichard M. Murray10 November 2003Goals:y Show how to compute closed loop stability from open loop propertiesy Describe the Nyquist stability criterion for stability of feedback systemsy Define gain and phase margin and determine it from Nyquist and Bode plotsReading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 7y Advanced: Lewis, Chapter 711 Nov 02 R. M. Murray, Caltech CDS 23 Nov 03 RMM and HM, Caltech CDS 0Lecture 6.1: Transfer Functions(0) 0xAx BuyCxDux=+=+=&1() ( )HsCsIABD−=−+sin( )uA tω=()ss()sin ( )yHjAtHjωωω=⋅+ ∠21 22 111212yu yu yunnHHHdd==1skx&&yΣf-1sb-x&x1mC(s) P(s)++-dryeuFrequency (rad/sec)Phase (deg)100101-200-150-100-50010-210-1100101MagnitudeReview from Last WeekCDS 101, Lecture 7.110 November 2003R. M. Murray, Caltech211 Nov 02 R. M. Murray, Caltech CDS 3Closed Loop StabilityQ: how do open loop dynamics affect the closed loop stability?y Given open loop transfer function C(s)P(s) determine when system is stableBrute force answer: compute poles closed loop transfer functionAlternative: look for conditions on PC that lead to instabilityy Example: if PC(s) = -1 for some s = jω, then system is not asymptotically stabley Condition on PC is much nicer becausewe can design PC(s) by choice of C(s)y However, checking PC(s) = -1 is not enough; need more sophisticated checkC(s)++-dryeuP(s)1pcyrpcpcnnPCHPC d d n n==++• Poles of Hyr= zeros of 1 + PC • Easy to compute, but not so good for designP(s)C(s)Frequency (rad/sec)Phase (deg)Magnitude (dB)-60-50-40-30-20-10010200.1 1 10-360-270-180-90011 Nov 02 R. M. Murray, Caltech CDS 4Game Plan: Frequency Domain DesignGoal: figure out how to design C(s) so that 1+C(s)P(s) is stable and we get good performancey Low frequency range:(good tracking)y Bandwidth: frequency at which closed loop gain = ½⇒ open loop gain ≈ 1y Idea: use C(s) to shape PC(under certain constraints)y Need tools to analyze stability and performance for closed loop given PC1yrPCHPC=+• Poles of Hyr= zeros of 1 + PC • Would also like to “shape” Hyrto specifyperformance at differenct frequencies-150-100-50050100Magnitude (dB)10-410-310-210-1100101102103-270-225-180-135-90-450Phase (deg)Bode DiagramFrequency (rad/sec)PC11PCPC≈+1PC ⇒1PC CDS 101, Lecture 7.110 November 2003R. M. Murray, Caltech311 Nov 02 R. M. Murray, Caltech CDS 5Nyquist CriterionDetermine stability from (open) loop transfer function, L(s) = P(s)C(s).y Use “principle of the argument” from complex variable theory (see reading)Thm (Nyquist). Consider the Nyquist plot for loop transfer function L(s). LetP # RHP poles of L(s)N # clockwise encirclements of -1Z # RHP zeros of 1 + L(s)ThenZ = N + PC(s)++-dryeuP(s)• Nyquist “D” contour• Take limit as r → 0, R →∞•Trace from −∞ to +∞along imaginary axis•Trace frequency response for L(s) along the Nyquist “D” contour• Count net # of clockwise encirclements of the -1 pointRealImagrR-j∞+j∞Real AxisNyquist Diagrams-6 -4 -2 0 2 4 6 8 10 12-6-4-20246From: U(1)To: Y(1)ω=0ω=+j∞ω=-j∞jω< 0jω> 0+N=211 Nov 02 R. M. Murray, Caltech CDS 6Simple Interpretation of NyquistBasic idea: avoid positive feedbacky If L(s) has 180˚ phase (or greater) and gain greater than 1, then signals are amplified around loopy Use when phase is monotonicy General case requires NyquistC(s)++-dryeuP(s)bode(sys)nyquist(sys)Can generate Nyquist plot from Bode plot + reflection around real axisFrequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-40-30-20-10010From: U(1)10-1100101-200-150-100-500To: Y(1)Real AxisImaginary AxisNyquist Diagrams-1.5 -1 -0.5 0 0.5 1 1.5 2-3-2-10123From: U(1)To: Y(1)ω=0ω=∞ω=-∞CDS 101, Lecture 7.110 November 2003R. M. Murray, Caltech411 Nov 02 R. M. Murray, Caltech CDS 7Example: Proportional + Integral* speed controllerC(s)++-dryeuP(s)* slightly modified; more on the design of this compensator in next week’s lectureReal AxisImaginary AxisNyquist Diagrams-500 0 500 1000 1500 2000 2500-2000-1500-1000-5000500100015002000From: U(1)To: Y(1)1/()/mrPssbm s a=⋅++Remarksy N = 0, P = 0 ⇒ Z = 0 (stable)y Need to zoom in to make sure there are no net encirclementsy Note that we don’t have to compute closed loop response()0.01ipKCs Ks=++-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0 .5 0-1.5-1-0.500.511.511 Nov 02 R. M. Murray, Caltech CDS 8More complicated systemsWhat happens when open loop plant has RHP poles?y 1 + PC has singularities inside D countour ⇒ these must be taken into accountReal AxisImag AxisPole-zero map-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Real AxisImaginary AxisNyquist Diagrams-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-1.5-1-0.500.511.5From: U(1)To: Y(1)21()0. 115sLssss+=⋅−++111 ( 0.35)( 0.07 1.2 )( 0.07 1.2 )sLss js j+=++ ++ +−N = -1, P = 1 ⇒ Z = N+P = 0 (stable)9unstable poleCDS 101, Lecture 7.110 November 2003R. M. Murray, Caltech511 Nov 02 R. M. Murray, Caltech CDS 9Comments and cautionsWhy is the Nyquist plot useful?y Old answer: easy way to compute stability (before computers and MATLAB)y Real answer: gives insight into stability and robustness; very useful for reasoning about stabilityNyquist plots for systems with poles on the jωaxisCautions with using MATLABy MATLAB doesn’t generate portion of plot for poles on imaginary axisy These must be drawn in by hand (make sure to get the orientation right!)• chose contour to avoid poles on axis• need to carefully compute Nyquist plot at these points• evaluate H(ε+0j) todetermine direction21()(1)Hsss=+ω=-j∞ω=+j∞ω=0+ω=0-11 Nov 02 R. M. Murray, Caltech CDS 10Relative stability: gain and phase marginsNyquist plot tells us if closed loop is stable, but not how stableGain marginy How much we can modify the loop gain and still have the system be stabley Determined by the location where the loop transfer function crosses 180˚phasePhase marginy How much we can add “phase delay”and still have the system be stabley Determined by the phase at which the loop transfer function has unity gainBode plot interpretationy Look for gain = 1, 180˚ phase crossingsy MATLAB: margin(sys)Nyquist Diagram-1.5 -1 -0.5 0 0.5 1 1.5-3-2-101231GMPMFrequency (rad/sec)Phase (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)CDS 101, Lecture 7.110 November


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