Thanks For Stimulating Collaboration! Advances In PID .
Thanks for Stimulating Collaboration!Advances in PID ControlKarl Johan Åström Interactive learning modules PID control Feedforward control Tuning and adaptation Department of Automatic Control, Lund UniversityEvent based controlTore and I are very grateful for the excellent interactions.Reseach GoalsSome Recent PapersPrimary goals: New edition of Advanced PID Control Develop basis for a new generation of auto-tuners Understand trade-offs between performance (loaddisturbance attenuation, measurement noise injection) androbustness Understand when FOTD models are sufficient and whenbetter modeling is required - Can design be based onFOTD models Hast, Åström, Bernhardsson, Boyd PID Design by ConvexConcave Optimization. ECC 2013 Garpinger, Åström, Hägglund Performance and robustnesstrade-offs in PID Control. Journal of Process Control,24:5(2014) 568-577. Romero Segovia, Hägglund Åström Measurement noisefiltering for PID controllers JPC 24(2014) 299-313 Romero Segovia, Hägglund Åström Measurement noisefiltering for common PID tuning rules CEP 32(2014) 43-63 Berner, Åström , Hägglund Towards a New Generation ofRelay Autotuners IFAC World Gongress 2014 Garpinger, Hägglund Modeling for Optimal PID Design.IFAC World Congress 2014 Boyd, Hast, Åström. MIMO PID Tuning via Iterated LMIRestriction. Submitted Automatica 2014Secondary goals: Understand current tuning rules: Lambda, SIMC, AMIGO Design rules for noise filtering Suitable optimization methodsOutlineDesign Criteria A trade-off between conflicting requirementsLoad disturbance attenuationRobustness to process uncertaintyMeasurement noiseSetpoint response1. Introduction 2. Performance and Robustness3. Performance and Measurement Noise4. OptimizationSet-point response can be treated separately (2 DOFsetpoint weighting)Performance:ZIE 5. Next Generation Auto-tuners6. Summary e(t)dt 1/ ki ,I AE 1,1 P(iω ) C(iω )Mt max0Z 0p e(t)pdtRobustness:Ms maxωωP(iω ) C(iω )1 P(iω ) C(iω )Measurement noise: Noise gain, SDU, filteringLevel Curves for Performance and Robustness PI Control – Lag-Dominated Dynamics Performance (IAE 1/ ki blue) and robustness ( Ms , Mt red)IE level curves are horizontal lines 1P(s) 1/((s 1)(0.1s 1)(0.01s 1)(0.001s 1)), τ 0.067Unconstrained optimal controller poor robustness Ms 8!ZN step and ZN frequency .61.2 1.400Approximately: ki gives performance and kp sets robustness12.51.8204060kp80100120
PI Control – Balanced Dynamics PI Control – Delay-Dominated Dynamics4P(s) 1/(s 1) , τ 0.33Unconstrained optimal controller has robustness Ms 2.8ZN step and ZN frequency P1 (s) e s /(1 0.05s)2 , τ 0.92Unconstrained optimal controller has robustness Ms 2!IE and IAE minimization equivalent for small Ms , p33.54Tuning – Lag-Dominated 60.7kp0.80.911.1Tuning – Balanced DynamicsLagdominantBalanced201.90.4S S 1830.35160.314ki8ki1.60.151.30.121.81.7λ00λSS ASλS SS λ1.71.51.42 0.21.5S S 4 1.8S S A6 bda tuning has very low gainsS and S give similar tuningLambda tuning gives constant integral time Ti kp/ ki0.20.40.6kp0.811.2 Tuning methods S , A and λ gives similar results All controllers have constant integral time Ti kp/ kiTuning Delay-Dominated DynamicsOutlineDelay dominated1λλλ0.90.820.71. Introduction1.90.61.81.7kiS0.50.40.30.2SS 1.21.6A2. Performance and RobustnessS 3. Performance and Measurement Noise1.4S S 4. Optimization5. Next Generation Auto-tuners1.50.11.36. SummaryZN00 0.050.10.150.20.25kp0.30.350.4Lambda tuning too high integral gainObvious why Skogestad modified his methodAll controllers have constant integral time Ti kp/ kiMeasurement NoisendCPI DuΣApproximation of GunPxΣWe have S (y Gun (s) G ControllerProcessController transfer functionki1CPI D (s) kp kd s,Gf s1 sT f s2 T f2 /2ki kp s kd s2C SC (1 PC(s K ki )(1 sT f (sT f )2 /2)For low frequencies (small s) the numerator of Gun isdominated by the integral gain ki and we haveCPI D (s) (C CPI D G fki,sG f (s) ( 1,S(s) (s.s K kiHenceTransfer function from noise to control signal Gun (s) sfor s small, and S ( 1 large ss K kiC SC1 PCPI DGun (s) ( Gun For controllers with integral action we have Gun (0) 1/ Kwhere P(0) K .2ki kp s kd s2(s K ki )(1 sT f (sT f )2 /2)
Bode Plots of Noise Transfer Function GunLag dominatedDelay dominatedBalanced1110Bode Plots Controller Transfer Function CLag -210010 10 Stochastic Modeling2σ u2 ( πNoise gainσuσ yfknw 010210-21001021010Finding a Suitable Filter Time ConstantnduCPI DxΣTfyΣP G fProcessControllerπσ 2y f Φ0,010-210Gain crossover frequency Frequency ω f 2/T fki kp s kd s2(s K ki )(1 sT f (sT f )2 /2)!410 k2p 2ki kdk2ki 2 d3KTfTf20010Measurement noise stationary with spectral density Φ(ω )Z σ u2 p Gun (iω )p2 Φ(ω )dω Z σ 2y f p G f (iω )p2 Φ(ω )dωWhite noise101010-2Validity of approximation (error in mid frequency range Mspeak)Differences PI/PID lag dominated/delay dominatedGun (s) ( 21021010 010210101010010-110-210210PIPI1010Delay dominatedBalanced110Gf Φ0 vuu ki T fk2 k2p 2ki kd 2 d2 .(tKTf 11 sT f s2 T f2 /2CPI D (s) kp ki kd s,sC CPI D G fDevelop sound design procedure for PI and PID control ofa given processApply procedure to a representative test batchAnalyse results to find insights and understandingExplore and try to find simple design rulesFinding a Suitable Filter Time ConstantPI Control Lag-dominant DynamicsP1 (s) An iterative design procedure1(s 1)(0.1s 1)(0.01s 1)(0.001s 1)FOTD parameters: K 1, T 1.04, L 0.08, and τ 0.071. Design controler for nominal process P0 e.g. by minimizingIAE subject to robustness constraints, G f 1.0.62. Compute ω c for PG f3. Choose T f α /ω c , α 0.01, 0.02, 0.05, 0.01, 0.15, 0.24. Repeat from 2 with until convergence0.20.500.4-0.20.3-0.40.2-0.60.15. Make trade-off plots (load disturbance 0Can be applied to any design procedure, particularly simple fordesign methods based on the FOTD model.010010-110-510-210-210PID Control Lag-dominant 10PI red x PID blue 410210SDU3410
PI Control - FOTD 032120AMIGO104SIMC5300.20.4τ0.60.801Effect of T f on FOTD ParametersT f /(α L0 )54Lambda LLambdaT2T f /(α L0 )583AMIGOLambda L4SIMCT f /(α L0 )10Lambda LT f /(α L0 )5PID Control - FOTD 20.4τ0.60.801Effect of T on FOTD Parametersnd2ΣxPΣy1.5L a /T fCPI Du10.5 G 0Process1Ta /T fController0.50With filtering the effective process dynamics changes fromP to PG f How to determine the FOTD parameters? The step response method2( L a Ta )/T f 10L L0 (1 0.65 τ 8 )T f ,T T0 1.1 τ 8 T f .Outline0.10.20.30.40.50.6τ0.70.80.9Convex OptimizationThe basic convex optimization:minimizexsubject to1. Introduction2. Performance and Robustnessf0 ( x )f i ( x) 0, i 1, . . . , mh j ( x) 0, j 1, . . . , n,f i ( x), convex functions, h( x) affine functions of x.3. Performance and Measurement Noise4. Optimizationf ( x)405. Next Generation Auto-tuners6. Summary If a local minimum exists itis a global minimum Efficient and fast numericalalgorithms Good software tools CVX200 2Convex-Concave ProcedureG l PG f kp The approximated problem 2 ki kd ssis linear in the parameters kp, ki , kd .The robustness constraint thatG l (iω ) is outside the circlef i ( x) i ( x k) i ( x k )T ( x x k ) 0, i 1, . . . , mr p G l cp 0does not give a convex problem.Convex-concave optimization can beapplied since G l is linear in theparameters. For each frequency theconstraint to be outside the circle isreplaced by being outside a halfplane (the dashed line)Composition always possible if Hessian of f ( x) ( x) isboundedConverges to a local minimum or saddle-point.Sacrifices global optimality but gains convexity and hencespeed.Feasible starting point is needed.40ℑ L(iω )f0 ( x) 0 ( x k) 0 ( x k )T ( x x k )is convex and solved to generate a new solution point x k 1 .Iterate until convergence. 1Loop transfer functionf ( x) ( x) ( f ( x) ( x k ) ( x k)T ( x x k )subject to0xConvex-Concave Optimization for PID ControlReplace concave part by linearization around current solutionpoint x kminimize 1 1 2 2 1ℜ L(iω )0
Heat RodsC(s) kp 0eΣki kd ssC(s)1su0P(s)ΣΣ11.54sIE 0.086, IAE 0.10CPI (s) 2.94 y 1Optimization problemmax.System output, y(t)10.100ki ( L k 1) 1/1.4 ℜ( L 1) 0p L k 1p ( L k cT ) rT ℜ( L cT ) 0p L k cT ps.t.IE 0.021,0.2Convex approximationmaximize kisubject to p S(iω )p 1.4pT(iω )p 1.448.25 0.46ssIAE 0.031CPID (s) 7.40 012ℑ L(iω ) P(s) e Nyquist Plot and Load Step Response3 1Control signal, u(t) 20.50 3 3 0.5 1 1.501C(s)1suΣ0P(s)Σ0.1ymax.kis.t.1/1.4 ℜ ( L k 1) p L k 1pmax.kis.t.1/1.4 ℜ ( L k 1) p L k 1pC(s) 3.31 6.62suSolved using CVX in MATLAB. Converges within twelve iterations(4 s).y(t)1Σ 101020300.5 3 2 101 0.5 1ℜ L(iω ) 1.501020Process model OptimizationSmax 1.4,C CPI D G f Optimization: Minimize q( P(0) K I ) 1 q subject toqT q Tmax ,0Control signalyController transfer function1CPI D (s) K p K i K d s,s 3.34sThe Wood-Berry Distillation ColumnProcess11 sT f s2 T f2 /23.20s0 3 G Controller10.1 0.1 2nP00.2u(t) Σ 1ℜ L(iω )IE 0.31, IAE 0.57Nyquist plotℑ L(iω )A grid of 1000 frequencies between10 2 and 102 rad/s.x 2System output ( L 1) 0 d 3C(s) 3.61 6.26sIE 0.15, IAE 0.74Multivariable PID Controllersq Sq Smax , 3300maximize kisubject to p S(iω )p 1.4κ 1/1.4Gf 20Nyquist Plot and Load Step Responsesx T Qx Ak x bk 0Optimization problem10The oscillatory behavior related to cusp in Nyquist curve( L 1) 0Convex approximation(s 1)3kiC(s) kp kd ss0t Adding a Curvature Constraint1 1 2Convex approximationmaximize kisubject to p S(iω )p 1.4CPI D00 0.1P(s) Nyquist plot1 1Optimization problem13Step responsey(t)eΣ2ℑ L(iω )100Nyquist Plot and Step ResponsesIntuitively it may seem like optimization of IE or IAE will give thesame result provided the system is well damped,(s 1)3kiC(s) kp kd ss 1ℜ L(iω )IE or IAEP(s) 2 q Q CSq Qmax512.8e s 16.7s 1P(s) 6.6e 7s10.9s 1Tmax 1.4, 18.9e 3s21.0s 1 19.4e 3s14.2 1Qmax 3/σ min ( P(0)) 0.738.Derivative action time constant: τ 0.3Sampled with N 300 logarithmicallyspaced frequency samples in the interval 10 3 , 103Initialization: K P 0,K I ǫ P(0)† ,K D 0,ǫ 0.01.30
Wood and his ColumnGeneral and Diagonal PID ControllersOptimal PID controller (converged in 7 iterations)q( P(0) K I ) 1 q 2.25. 0.1750 0.04700.0913 0.0345KP , KI , 0.0751 0.07090.0402 0.0328 0.1601 0.0051KD ,0.0201 0.1768Diagonal PID controller (converged in 8 iterations)q( P(0) K I ) 1 q 13.36, 0.021000.15350,, KI KP 0 0.01360 0.0692 0.17140KD ,0 0.1725General PID ControllerStep ResponsesT110q S(iω )q10-111y1 (t)y1 (t)10T1200-210-310-210-110010ω1102100.53100qT (iω )q100.5-110-210T21-110010ω110210T2231011y2 (t)-210y2 (t)-3100q Q (iω 0t60Extensions800Exchanging objectives and constraints Frequency dependen bounds1. Introduction Other closed loop transfer functions2. Performance and Robustness Low frequency disturbance attenuationsS(s) P(s) ( s( P(0) K I ) 1 P(0)3. Performance and Measurement NoiseUnstable plants Robustness to plant variations More general controllers604. OptimizationHigh frequency roll-off 40tOutline 205. Next Generation Auto-tuners6. SummaryReflectionsKey IssuesPutting it all together Relay feedbackGood excitation - the secret to good modeling A long term plan - include what we have learned Auto-tuners for building simulationAuto-tuners for controllers Avoid waiting for steady state!How short can it be? Design of identification experiment Short experimentation time The classic, integrators, filters, asymmetry, adaptivehysteresisInput signal and excitation essential!!The beauty of the relay-autotunerHow to design the second phase? Chirp pulse? Computational issues? Implementation: Coding, box, DCS, web, cloud Behavior in closed loop the primary goal!Fitting error, cross-validation, AIC, Vinnicombe Criteria How to assess a model? 6Short experimentsGood robust tuning rules with design parameterImplementation issues Matlab, Python, FMIBoxes, PLCs, DCS systemsSimple version: PI controlComplex version: Selection of PI or PID and bettermodelingStand alone box: Matlab, Python, FMISoftwaremWeb, cloud80
Modeling IssuesModeling for PI & PID ControlAMIGO Tuning - complete testbatch6ki [ PI D ]/ ki [ PI ] vs τ5210y4321101002468t1012010 True time delay is a fundamental limitation FOTD lumps true time delay and high order dynamics Does not matter for τ 0.40.10.2circles: P(s) 0.30.40.5Ke sL ,1 sT0.60.70.80.91Ke sL(1 sT )2squares: P(s) FOTD OK for τ 0.4 better model required for smaller τ ! Better models than FOTD are required for PID control andτ 0.4 0Derivative action small improvement for τ 0.8 ModelsThree and Four Parameter ModelsTwo parameter models2P(s) K exp(-L) /(1 sT)0.25b,P(s) s a0.2 sL0.15P(s) K e0.1y0.050-0.05-0.1Three parameter modelsP(s) be sL ,s aP(s) Ke sL1 sT0.5100.51tP(s) b exp(-sL)/(s 2 a1*s 100.511.521.50.511.50110210100110The key to successful system identificationSymmetric relay dominant at one frequency The asymmetric relay has the dominant frequency at theperiod Tp, a low frequency component and some highfrequencies Highly desirable to have excitation at other frequenciesModifications of the relay 21010P(s) (b1*s b2)/(s2 a1*s a2)P(s) b exp(-sL)/(s 2 a1*s a2)0 010-1 -11010 -2 210 -101010110IntegratorFiltersChange hysteresis: Ulf HolmbergMix of integrator and relay: WallerFlat spectrum: KristianAsymmetric relayChirp signals -270-210210-1100-270-2102 -90-180101.5t 1.52P(s) K exp(-L) /(1 sT)1011ub1 s b2e sLs2 a1 s a20-20.50.50101001.50.5Three and Four Parameter Models100.05y0.05ybP(s) 2e sLs a1 s a21P(s) -220-0.5Five parameter model101.51P(s) (b1*s b2)/(s2 a1*s a2)b1 s b2P(s) 2,s a1 s a21020.5Four parameter models101.5ub,s2 a1 s a2KP(s) e sL(1 sT )2P(s) 01.5210The Chirp SignalAsymmetric Relay and ChirpAsymmetrical relay experiment combined chirp signalexperimentDouble experiment time. Constant amplitude,L 0.01, w 15 (1 0.5 t), tmax 2.7,0.15 ω L 0.35 u(t) (a b t) sin (c d t)t Frequency varies between a and c d tmax amplitude betweena b tmax42b exp(-sL)/(s a1 s 678900.511.522.53-270-210-11001011010Parameters: a1 10.366 0.033,a2 9.574 0.028,b 9.566 0.027,L 0.0109 0.0002Notice both high and low frequency excitation7210
Effect of Proper ExcitationNyquist Plots0.3Only relayRelay and chirp0.20.1P(s) b exp(-sL)/(s 2 a1*s a2)b exp(-sL)/(s 2 a1 s .800.20.40.60.81reOutlineSummary Trade-off plots give a lot of insight 1. IntroductionRational ways of designing filters 2. Performance and Robustness 3. Performance and Measurement Noise 4. Optimization 6. SummaryAutomatic tuning How to package the results P(iω ), Pnmp(iω )Lag dominated0-90I PSfrPI-270-110I-90P PID-180Balanced0010DPI1102-270-110010D1101P PID-270-1100D1011021010111010Mc10PI-180210I-90P PID-180Delay dominated00010010-11010-1-110ω10 c0ω c ( P(0) ki ,11010Mc -1-110ω10 c011010-110ω10 c01eγ ( arg P(iω c ) π ϕ m ) ,p P(iω c )p110γ (1KJÅ European Journal of Control 6:1 (2000) 2-208PID Design ToolInteractive Learning ModulesConvex optimization Assessment PlotsSimple rules related to FOTD or Ti , TdThe equation for noise gainFeedforward (not covered in the talk)Computations 5. Next Generation Auto-tunersEffect of parameters ki , kp, τAssessment of tuning rules (close to green line)Better excitation: asymmetric relay and chirpSimple tunersElaborate tuners with extensive computations - cloud?Assessment plots
Controller Process C PID S SP u y d n Controller transfer function G f! 1 1 # sT f # s2 T 2 f *2 C PID dse ! kp# ki s # kd s, C ! C PID G f Transfer function from noise to control signal G undse ! C 1 # PC! SC For controllers with integral action we have G un d0e ! 1*K where P d0e ! K . Approximation of G un We have S (s s # Kk i for s small .
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
advances in agronomy adv anat em advances in anatomy embryology and cell biology adv anat pa advances in anatomic pathology . advances in organometallic chemistry adv parasit advances in parasitology adv physics advances in physics adv physl e advances in physiology education adv poly t advances in polymer technology
10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan
service i Norge och Finland drivs inom ramen för ett enskilt företag (NRK. 1 och Yleisradio), fin ns det i Sverige tre: Ett för tv (Sveriges Television , SVT ), ett för radio (Sveriges Radio , SR ) och ett för utbildnings program (Sveriges Utbildningsradio, UR, vilket till följd av sin begränsade storlek inte återfinns bland de 25 största
Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
LÄS NOGGRANT FÖLJANDE VILLKOR FÖR APPLE DEVELOPER PROGRAM LICENCE . Apple Developer Program License Agreement Syfte Du vill använda Apple-mjukvara (enligt definitionen nedan) för att utveckla en eller flera Applikationer (enligt definitionen nedan) för Apple-märkta produkter. . Applikationer som utvecklas för iOS-produkter, Apple .
och krav. Maskinerna skriver ut upp till fyra tum breda etiketter med direkt termoteknik och termotransferteknik och är lämpliga för en lång rad användningsområden på vertikala marknader. TD-seriens professionella etikettskrivare för . skrivbordet. Brothers nya avancerade 4-tums etikettskrivare för skrivbordet är effektiva och enkla att
Den kanadensiska språkvetaren Jim Cummins har visat i sin forskning från år 1979 att det kan ta 1 till 3 år för att lära sig ett vardagsspråk och mellan 5 till 7 år för att behärska ett akademiskt språk.4 Han införde två begrepp för att beskriva elevernas språkliga kompetens: BI
