CDS 101, Lecture 9.226 November 2003R. M. Murray, Caltech1CDS 101: Lecture 9.2PID and Root LocusRichard M. Murray26 November 2003Goals:y Define PID controllers and describe how to use themy Describe root locus diagram and show how to use it to choose loop gainReading: y Astrom, Sec 6.1-6.4, 6.6y Optional: PPH, Sec 13y Advanced: Lewis, Chapter 12 + Sec 13.126 Nov 03 R. M. Murray, Caltech CDS 2Overview: 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=+ +∫&CDS 101, Lecture 9.226 November 2003R. M. Murray, Caltech226 Nov 03 R. M. Murray, Caltech CDS 3Frequency 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++=26 Nov 03 R. M. Murray, Caltech CDS 4Tools for Designing PID controllersZeigler-Nichols tuningy Design PID gains based on step responsey Works OK for many plants (but underdamped)y Good way to get a first cut controllery Frequency domain version also existsCaution: PID amplifies high frequency noisey Sol’n: pole at high frequencyCaution: Integrator windupy Prolonged error causes large integrated errory Effect: 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)02040608010-410-2100102104CDS 101, Lecture 9.226 November 2003R. M. Murray, Caltech326 Nov 03 R. M. Murray, Caltech CDS 5PID vs lead/lag compensationPID ControlPros: easy to design, implementCons: low freq lag, high freq gainLead/Lag CompensationPros: low freq phase, high freq rolloffCons: more complicated (slightly)1()pIDCs K K Kss=+⋅+lagleadlag lead()sasaCs Ksbsb++=⋅++Magnitude (dB)10-310-210-11001011021001020304050-90-4504590Phase (deg)Bode DiagramFrequency (rad/sec)11ITω=21DTω=Magnitude (dB)10-310-210-110010110210301020304050-90-4504590Phase (deg)Bode DiagramFrequency (rad/sec)31lagbω=2lagaω=3 leadaω=4leadbω=26 Nov 03 R. M. Murray, Caltech CDS 6Example: PID cruise controlZiegler-Nichols design for cruise controllery Plot 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/()/mrPssbms 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 9.226 November 2003R. M. Murray, Caltech426 Nov 03 R. M. Murray, Caltech CDS 7Pole Zero Diagrams and Root Locus PlotsPole zero diagram verifies stabilityy Roots 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 changedy Can plot pole location versus any 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 split26 Nov 03 R. M. Murray, Caltech CDS 8One Parameter Root LocusBasic idea: convert to “standard problem”:y Look at location of roots as αis varied over positive real numbersy If “phase” of a(s)/b(s) = 180°, we can always choose a real αto solve eqny Can compute the phase from the pole/zero diagramTrace out positions in plane where phase = 180°y At each of these points, there exists gain αto satisfy a(s) + αb(s) = 0y All such points are on root locus() () 0as bsα+=1212() ( )( ) ( )()() ( )( ) ( )mnas szsz szGs kbs spsp sp+++==++ +LL001 001 0() ( ) ( )() ()mnGs s z s zspsp∠=∠ + + +∠ + −∠+ −−∠+LLφi= phase contribution from s0to -piψI= phase contribution from s0to -ziCDS 101, Lecture 9.226 November 2003R. M. Murray, Caltech526 Nov 03 R. M. Murray, Caltech CDS 9Root Locus for Loop GainLoop gain as root locus parametery Common 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 commentsy Although 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 locus()1()()0()nsds nsdsαα+→+=26 Nov 03 R. M. Murray, Caltech CDS 10Second Order System ResponseSecond order system responsey Spring mass dynamics, written in canonical formy Performance specificationsGuidelines for pole placementy Damping 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 polesCDS 101, Lecture 9.226 November 2003R. M. Murray, Caltech626 Nov 03 R. M. Murray, Caltech CDS 11Effect of pole
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