CDS 101: Lecture 8.2Tools for PID & Loop ShapingRichard M. Murray17 November 2004Goals:y Show how to use “loop shaping” to achieve a performance specificationy Introduce new tools for loop shaping design: Ziegler-Nichols, root locus, lead compensationy Work through some example control design problemsReading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 817 Nov 04 R. M. Murray, Caltech CDS 2Tools 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)02040608010-410-210010210417 Nov 04 R. M. Murray, Caltech CDS 3Example: 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/()/mrPssbmsa=⋅++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()Ps17 Nov 04 R. M. Murray, Caltech CDS 4Pole 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 split17 Nov 04 R. M. Murray, Caltech CDS 5One 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 sz szsp sp∠=∠ + + +∠ + −∠+ −−∠+LLφi= phase contribution from s0to -piψI= phase contribution from s0to -zi17 Nov 04 R. M. Murray, Caltech CDS 6Root 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+→+=17 Nov 04 R. M. Murray, Caltech CDS 7Second 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 poles17 Nov 04 R. M. Murray, Caltech CDS 8Effect of pole location on performanceIdea: look at “dominant poles”y Poles nearest the imaginary axis (nearest to instability)y Analyze using analogy to second order systemPZmap complements informa-tion on Bode/Nyquist plotsy Similar 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 [and SISOtool])Time (sec.)AmplitudeStep Response0 1 2 3 4 5 6 7 8 9 1000.20.40.60.811.21.4From: U(1)To: Y(1)Real AxisImag AxisPole-zero map-5 -4 -3 -2 -1 0 1-1.5-1-0.500.511.5 Time (sec.)AmplitudeStep Response0 1 2 3 4 5 6 7 8 9 1000.20.40.60.811.21.4From: U(1)To: Y(1)Real AxisImag AxisPole-zero map-5 -4 -3 -2 -1 0 1-1.5-1-0.500.511.5 Time (sec.)AmplitudeStep Response0 1 2 3 4 5 6 7 8 9 1000.511.522.533.5From: U(1)To: Y(1)Real AxisImag AxisPole-zero map-5 -4 -3 -2 -1 0 1-1.5-1-0.500.511.5 xxxxxxxxxxxxx17 Nov 04 R. M. Murray, Caltech CDS 9Example: PID cruise controlStart with PID control design: Modify gain to improve performancey Use MATLAB sisotooly Adjust loop gain (K) to reduce overshoot and decrease settling timeàζ≈ 1 ⇒ less than 5% overshootà Re(p) < -0.5 ⇒ Tsless than 2 sec1/()/mrPssbmsa=⋅++1() (1 )DICs K TsTs=++17 Nov 04 R. M. Murray, Caltech CDS 10Example: Pitch Control for Caltech Ducted FanSystem descriptiony Vector thrust engine attached to wingy Inputs: fan thrust, thrust angle (vectored)y Outputs: position and orientationy States: x, y, θ+ derivativesy Dynamics: flight aerodynamicsControl approachy Design “inner loop” control law to regulate pitch (θ) using thrust vectoringy Second “outer loop” controller regulates the position and altitude by commanding the pitch and thrusty Basically the same approach as aircraft control laws17 Nov 04 R. M. Murray, Caltech CDS 11Performance Specification and Design ApproachDesign approachy Open loop plant has poor phase marginy Add phase lead in 5-50 rad/sec rangey Increase the gain to achieve steady state and tracking performance specsy Avoid integrator to minimize phasePerformance Specificationy ≤ 1% steady state errorà Zero
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