CDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech1CDS 101: Lecture 7.1Loop Analysis of Feedback SystemsRichard M. Murray11 November 2002Goals: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 Astrom, Section 3.5y Optional: Packard, Poola and Horowitz, Chapter 30-31y Advanced: Lewis, Chapter 711 Nov 02 R. M. Murray, Caltech CDS 2Week 6: Frequency Response & Transfer Functions(0) 0xAx BuyCxDux=+=+=1() ( )HsCsIABD−=−+sin( )uA tω=()ss()sin ( )yHjAtHjωωω=⋅+ ∠21 22 111212yu yu yunnHHHdd==1skxyΣf-1sb-xx1mC(s) P(s)++-dryeuReview from Last WeekCDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech211 Nov 02 R. M. Murray, Caltech CDS 3Additional info from last week’s lectureDecibel (dB)y Logarithmic scale used to define ratio between amplitudes (gain)y 20 decibel (dB) = factor of 10 in gain: plot 20*log10(gain)History and backgroundy 1 bel = logarithm of the ratio of power. Decibel = 1/10 bel.y Since power goes as the square of the amplitude, we use 20*log(gain) for ratio of amplitudes (power goes as the square of the amplitude in electrical circuits)P(s)C(s)Frequency (rad/sec)Magnitude (dB)-60-50-40-30-20-10010200.1 1 1020 dB = factor of 10010110−310−310−11011 Nov 02 R. M. Murray, Caltech CDS 4Closed 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 stabley Useful for design since we specify C(s)Brute 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 10-360-270-180-900CDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech311 Nov 02 R. M. Murray, Caltech CDS 5Game Plan: Frequency Domain DesignGoal: figure out how to design C(s) so that 1+C(s)P(s) is stable + good performancey Low frequency range:(good tracking)y Bandwidth: as high as possible, but stay stabley 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)PC1PC 11PCPC≈+1PC ⇒11 Nov 02 R. M. Murray, Caltech CDS 6Nyquist CriterionCan determine 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 frequen-cy response along the Ny-quist “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=2CDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech411 Nov 02 R. M. Murray, Caltech CDS 7Simple 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 the loopy Use when phase is monotonicy General case requires NyquistC(s)++-dryeuP(s)bode(sys)nyquist(sys)Real AxisImaginary AxisNyquist Diagrams-1.5 -1 -0.5 0 0.5 1 1.5 2-3-2-10123From: U(1)To: Y(1)Can generate Nyquist plot from the Bode plot + reflection around real axisω=0ω=∞Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-40-30-20-10010From: U(1)10-1100101-200-150-100-500To: Y(1)ω=-∞11 Nov 02 R. M. Murray, Caltech CDS 8Example: Proportional + Integral* cruise control systemC(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=⋅++RemarksyN = 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 0-1.5-1-0.500.511.5CDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech511 Nov 02 R. M. Murray, Caltech CDS 9More 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)20.9 1()0.5 1sLssss+=⋅−++111 ( 0.35)( 0.07 1.2 )( 0.07 1.2 )sLss js j+=++ ++ +−N = -1, P = 1 ⇒ Z = N+P = 0 (stable)9unstable pole11 Nov 02 R. M. Murray, Caltech CDS 10Comments 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 proofsNyquist plots for systems with poles on the jωaxisCautions with using MATLABy MATLAB doesn’t generate portion of plot corresponding to 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-CDS 101, Lecture 7.111/11/2002R. M. Murray, Caltech611 Nov 02 R. M. Murray, Caltech CDS 11Robust 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
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