CDS 101, Lecture 9.125 Nov 02R. M. Murray, Caltech1CDS 101: Lecture 9.1PID and Root LocusRichard M. Murray25 November 2002Goals:y Define PID controllers and describe how to use themy Introduce the root locus technique and describe how to use it to choose loop gainy Show some of the limitations of feedback due to RHP poles and zerosReading: y Astrom, Sec 6.1-6.4, 6.6y Optional: PPH, Sec 13y Advanced: Lewis, Chapter 12 + Sec 13.125 Nov 02 R. M. Murray, Caltech CDS 2Review from Last Week Lecture 8.1: Frequency Domain DesignLoop Shaping for Stability and Performancey Steady state error, bandwidth, trackingMain ideasy Performance specifications give bounds on loop transfer functiony Use controller to shape responsey Gain/phase relationships constrain design approachy Standard compensators: proportional, lead, PI-100-5005010-1100101102-300-200-1000100()Ls()Ps()Cs()Cs()Ls()PsCDS 101, Lecture 9.125 Nov 02R. M. Murray, Caltech225 Nov 02 R. M. Murray, Caltech CDS 3Overview: PID controlIntuitiony Proportional term: provides inputs that correct for “current” errorsy Integral term: insures that steady state error goes to zero (if not, control gets bigger)y 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=+ +∫25 Nov 02 R. M. Murray, Caltech CDS 4Frequency domain compensation with PIDTransfer function for PID controllery Roughly equivalent to a PI controller with lead compensationy Idea: gives high gain at low frequency plus phase lead at high frequencyy Place below 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 9.125 Nov 02R. M. Murray, Caltech325 Nov 02 R. M. Murray, Caltech CDS 5Tools for Designing PID controllersZeigler-Nichols tuningyDesign PID gains based on step responsey Works OK for many plants (a bit underdamped)y Good way to get a first cut controllery Frequency domain version also existsCaution: PID amplifies high frequency noiseySol’n: pole at high frequencyCaution: Integrator windupyProlonged error causes large integrated errory Effect: get large undershoot (to reset integrator)y Sol’n: move pole at zero to very small valuey Fancier sol’n: anti-windup compensationC(s)++-dryeuP(s)1() (1 )DICs K TsTs=++1.2 /Ka=2*ITL=/2DTL=Point of maximumslopeLaStep responseBode DiagramsFrequency (rad/sec)Magnitude (dB)02040608010-410-210010210425 Nov 02 R. M. Murray, Caltech CDS 6Example: PID cruise controlZiegler-Nichols design for cruise controlleryPlot step response, extract L and a, compute gainsy Result: sluggish ⇒ increase loop gain Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-100-5005010010-310-210-1100101-200-1000100Time (sec.)AmplitudeStep Response0 10 20 30 40 50 6000.511.51/()/mrPssbm s a=⋅++0 10 20 30 40 50-0.100.10.20.30.40.52.490.039La==stepslope1.2 /Ka=2*ITL=/2DTL=1() (1 )DICs K TsTs=++()Ls()Ps()Cs()Ls()Cs()PsCDS 101, Lecture 9.125 Nov 02R. M. Murray, Caltech425 Nov 02 R. M. Murray, Caltech CDS 7Pole Zero Diagrams and Root Locus PlotsPole zero diagram verifies stabilityyRoots of 1 + PC give closed loop polesy Can trace the poles as a parameter is changed:Root locus = locus of roots as parameter value is changedyCan plot pole location for any single parameter; just repeatedly solve for rootsy Common choice in control is to vary the loop gain (K)C(s)+-ryeuP(s)-7 -6 -5 -4 -3 -2 -1 0 1 2 3-8-6-4-202468Real AxisImag Axis1() (1 )DICs K TsTs=++ααOriginal polelocation (α= 0)Pole goesto ∞Pole goesunstable for some αPole goesto terminalvaluePoles mergeand split25 Nov 02 R. M. Murray, Caltech CDS 8Root Locus for Loop GainLoop gain as root locus parameteryCommon choice for control designy Special properties for loop gainà Roots go from poles of PC to zeros of PCà Excess poles go to infinityà Can compute asymptotes, break points, etcy Very useful tool for control designy MATLAB: rlocusAdditional commentsyAlthough loop gain is the most common parameter, don’t forget that you can plot roots versus any parametery Need to link root location to performance…-7 -6 -5 -4 -3 -2 -1 0 1 2 3-8-6-4-202468Real AxisImag AxisC(s)+-ryeuP(s)αOpen loop polelocation (α= 0)Closed pole goes to openloop zerosAsymptotes for excess poles at (360˚/(P-Z))Real axis to theleft of odd # of real poles & zeros is on root locusCDS 101, Lecture 9.125 Nov 02R. M. Murray, Caltech525 Nov 02 R. M. Murray, Caltech CDS 9Second Order System ResponseSecond order system responseySpring mass dynamics, written in canonical formy Performance specificationsGuidelines for pole placementyDamping ratio gives Re/Im ratioy Setting time determined by –Re(λ)-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-10-8-6-4-20246810Imag AxisReal AxisRoot Locus Editor (C)2222()2( )( )nnnn nd ndHsss s js jωωςωω ςωω ςωω==+ + ++ +−1.8 /3.9 /rnsnTTωςω≈≈21dnωως=−2/1SS0pMeeπςς−−≈=44%16%4%Mp-3.9-1.7-1Slope0.250.50.707ζTs < xMp< yDesired regionfor closed loop poles25 Nov 02 R. M. Murray, Caltech CDS 10Effect of pole location on performanceIdea: look at “dominant poles”yPoles nearest the imaginary axis (nearest to instability)y Analyze using analogy to second order systemPZmap complements information on Bode/Nyquist plotsySimilar to gain and phase calculationsy Shows performance in terms of the closed loop polesy Particularly useful for choosing system gainy Also useful for deciding where to put controller poles and zeros (with practice)Time (se c.)AmplitudeStep Response0 1 2 3 4 5 6 7 8 9 1000.20.40.60.811.21.4From: U(1)To: Y(1)Real Axi sImag A xisPole-zero map-5 -4 -3 -2 -1 0 1-1.5-1-0.500.511.5 Time (se c.)AmplitudeStep Response0 1 2 3 4 5 6 7 8 9 1000.20.40.60.811.21.4From: U(1)To: Y(1)Real Axi sImag A xisPole-zero map-5 -4 -3 -2 -1 0 1-1.5-1-0.500.511.5 Time (se c.)AmplitudeStep Response0 1 2 3
View Full Document
Unlocking...