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Control System Design - PID ControlBo Bernhardsson and Karl Johan ÅströmDepartment of Automatic Control LTH,Lund UniversityBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Control System Design - PID Control1Introduction2The Basic Controller3Performance and Robustness4Tuning Rules5Relay Auto-tuning6Limitations of PID Control7SummaryTheme: The most common controller.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

IntroductionPID control is widely used in all areas where control is applied(solves ( 90% of all control problems)A PID controller is more than meets the eyeThe tuning adventure (Tore KJ)Telemetric, Eurotherm 1979Adaptive control and auto-tuningSTU, patents, NAF (Sune Larsson) SDM20Satt Control, Alfa Laval Automation, ABBFisher Control, Emerson 1979–Research and the PID books 1988, 1995, 2006, ?Interactive Learning Modules Guzman, Dormidohttp://aer.ual.es/ilm/Revival of PID Control - publications, conferencesTechnology transitionsPneumatic, mechanical,electric, electronic, computerModeling: the FOTD model P(s) 1 KsT e sLTo PID or not to PID - that is the questionBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Predictions about PID Control1982: The ASEA Novatune Team 1982 (Novatune is a usefulgeneral digital control law with adaptation):PID Control will soon be obsolete1989: Conference on Model Predictive Control:Using a PI controller is like driving a car only looking at the rearview mirror: It will soon be replaced by Model Predictive Control.2002: Desborough and Miller (Honeywell):Based on a survey of over 11 000 controllers in the refining,chemicals and pulp and paper industries, 98% of regulatorycontrollers utilise PID feedbackSimilar studies in Japan and GermanyPID is here to stay!Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Typical ScenariosProcess controlStandard distributed control system for 500-10000 loopsOne control room, commissioning, tuning, operations, upgradinghandeled by operators and instrument engineersLoops are tuned and retuned at installation and during operationAutomatic tuningEquipment manufacturersAutomotive systems: emissions, cruise control, antiskid, .Motor drives, robots and motion controlDedicated equipment for air conditioningControllers may be tuned based on models or by bumptests andempirical rulesInstallation tuning and upgrading very different for differentapplicationsTasks: regulation, command signal followingBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Entech Experience & Protuner ExperiencesBill Bialkowsk Entech - Canadian consulting company for pulp andpaper industry Average paper mill has 3000-5000 loops, 97% use PIthe remaining 3% are PID, MPC, adaptive etc.50% works well, 25% ineffective, 25% dysfunctionalMajor reasons why they don’t work wellPoor system design 20%Problems with valve, positioners, actuators 30%Bad tuning 30%Process Performance is not as good as you think. D. Ender, ControlEngineering 1993.More than 30% of installed controllers operate in manualMore than 30% of the loops increase short term variabilityAbout 25% of the loops use default settingsAbout 30% of the loops have equipment problemsBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

PID versus More Advanced ControllersPresentErrorPastFuturet Tdtu(t) kp e kiZ0te(τ )dτ kdde,dtTimeTd kd / kpPID predicts by linear extrapolation, Td prediction horizonAdvanced controllers predict using a mathematical modelBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

The Amazing Property of Integral ActionConsider a PI controlleru ke kiZ0te(τ )dτAssume that all signals converge to constant values e(t) e0 , u(t) u0Rtand that 0 ( e(τ ) e0 )dτ converges, then e0 must be zero.Proof: Assume e0 , 0, thenu(t) ke0 kiZ0te(τ )dτ ke0 kiZ0t e(τ ) e0 dτ ki e0 tThe left hand side converges to a constant and the left hand side does notconverge to a constant unless e0 0, futhermoreu( ) kiZ0 e(τ ) e0 dτA controller with integral action will always give the correct steady stateprovided that a steady state exists. It adapts to changing disturbances.Integral action is sometimes even called adaptive.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Interactive Learning ModulesA series of interactive learning tools for PID control has beendeveloped by Tore and KJ in collaboration with Control Groups inSpain (Jose-Luis Guzman Almeria, Sebastian Dormido Madrid), YvesPiquet (creator of Sysquake, a highly interactive version of Matlab).Executable modules for PC, Mac and Linux are available for freedownload fromhttp: www. http://aer.ual.esPID Basics, PID Loop Shaping and PID WindupILM DemoBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Control System Design - PID Control1Introduction2The Controller3Performance and Robustness4Tuning Rules5Relay Auto-tuning6Limitations of PID Control7SummaryTheme: The most common controller.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Static Characteristicsuu maxSlope Kubu mineProportional bandP: Controlleru K e ub ,Bo Bernhardsson and Karl Johan ÅströmK gain,ub bias or resetControl System Design - PID Control

A PID AlgorithmA PID controller is much more thanZu(t) kp e(t) kiWe have to considerFilter for measurementnoiseSet point weigthingActuator limitations:0te(τ )dτ kdde(t)dtIntegrator WindupMode switchesBumpless parameter changesComputer implementationRate limitationsDealing with these issues is a good introduction to practical aspects ofany control algorithm.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Integral Action or ResetIt was noticed early that proportional control gives steady state error. Abias term ub called reset was introduced to eliminate steady stateerrors.u kp e ubBias was adjusted manually and then replaced by the following way toadjust bias automatically. (Filter out low frequency component of u andadd it by positive feedback.)euΣKI11 sTi 1 .A simple calculation gives U (s) k 1 sTiVoilá a PI controller!Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Derivative ActionA derivative is the limity(t) y(t T )dy(,dtTsY (s) (1 e sTY (s)TApproximate the time delay by a low pass filtere sT (1,1 sTsY (s) (Block diagrame1 1 sY (s)1 Y (s) T1 sT1 sTukpΣ 11 sTdIs this how the body does it?Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Parallel and Series Form PIDParallel or non-interactive form: C f b(s) kp 1 1 sTdsTi kp(1 sTi s2 Ti Td )sTiwith independent gain parametrizationC f b(s) kp kd s2 kp s kiki kd s ssSeries form or interactive form: C̃ f b(s) k̃p 1 1sT̃i(1 sT̃d ) k̃p 1 s(T̃i T̃d ) s2 T̃i T̃dsT̃iRelations between coefficientskp k̃pT̃i T̃d,T̃iTi T̃i T̃d ,Td T̃i T̃dT̃i T̃dParallel form is more general. Equivalence only if Ti 4Td .Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

FilteringFilter only derivative part (absolute essential) C f b(s) k 1 sTd kd ski1 kp sTi1 sT fs1 sT fFilter the measured signal (several advantages)Better noise attenuation and robustness due to high frequencyroll-offProcess dynamics can be augmented by filter and design can bemade for an ideal PIDC f b(s) C f b(s) 1 sTi s2 Ti Tdkd s2 kp s ki kis(1 sT f )s(1 sT f )1 sTi s2 Ti Tdkd s2 kp s ki kis(1 sT f s2 T f2 /2)s(1 sT f s2 T f2 /2)Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

2DOF in PID ControllersA 2DOF structure makes set-point response independent ofdisturbance response. Set-point weighting “Poor man’s” 2DOF, allowsa moderate adjustment of set point response through parameters band c. Comment on practical controllers. U (s) kp bR(s) Y (s) ki( R(s) Y (s)) kd s cR(s) Y (s)srki /sreΣkpki /sΣuΣP(s)ykpkd suΣkd sController 1b 1 1P(s)Bo Bernhardsson and Karl Johan ÅströmControllerb c 0Control System Design - PID Control 1y

Avoiding Windup ykd sActuatore r ykiνkpΣu1syP(s)Σ Σ esktA local feedback loop keeps integrator output close to the actuatorlimits. The gain kt or the time constant Tt 1/ kt determines howquickly the integrator is reset. Intuitive Explanation - Cherchez l’erreur!Useful to replace kt by a general transfer function.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Dow Chemical Version of Anti-windupMany process industries (also in Sweden) had their own controldepartments and they developed their own systems based on standardcomputers. Dow, Monsanto and Billerud were good examples. dydtkdeekpΣkiΣ1sIvsat wΣsat ΣǫktThe integrator is reset based on its output and not based on thenominal control signal as in previous scheme.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Controlu

The Proportional BandThe proportional band is the range of the error signal where thecontroller (actuator) does not saturate.u K (bysp y) I K Tddy.dtSolving for the predicted process outputyp y Tddy,dtgives the proportional band ( yl , yh ) (also PB 100/K) asyl bysp I umaxKyh bysp I umin,Kwhere umin , umax are the values of the control signal for which theactuator saturates.Anti-windup changes the proportional band.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Anti-windup and Proportional BandTt 0.1Tt 0.311y00y0510Tt 1.01502015Tt 1.41015201015201yy0005101520Bo Bernhardsson and Karl Johan Åström05Control System Design - PID Control

Anti-windup in Series ImplementationeuΣKI11 sTieΣKuI11 sTiThese schemes are natural for pneumatic controllersHave been used by Foxboro (Invensys) for a long timeTracking time constant Tt TiBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Manual and Automatic ControlMost controllers have several modesManual/automaticIn manual control the controllers output is adjusted manually byan operator often by increase/decrease buttonsMode switching is an important issueSwitching transients should be avoidedEasy to do if the same integrator is used for manual andautomatic controlBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

PID Controller with Tracking ModeyspKbySPyMVwTRDsK Td1 sTd / N 1eyspP1sKTiIv1Ttw –No tracking if w v!Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID ControlPIDv

Anti-windup for Controller with Tracking Mode yK Td seKK /TiΣ1/sΣvActuatormodel ΣuActuator es1/TtActuator modelSPMVTRPIDvuActuatorNotice that there is no tracking effect if u v!The tracking input can be used in many other waysBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Control System Design - PID Control1Introduction2The Controller3Performance and Robustness4Tuning Rules5Relay Auto-tuning6Limitations of PID Control7SummaryTheme: The most common controller.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

RequirementsDisturbancesEffect of feedback on disturbancesAttenuate effects of load disturbancesModerate measurement noise injectionRobustnessReduce effects of process variationsReduce effects of modeling errorsCommand signal responseFollow command signalsArchitectures with two degrees of freedom (2DOF)Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Tune for Load DisturbancesG. Shinskey Intech Letters 1993: “The user should not test the loopusing set-point changes if the set point is to remain constant most ofthe time. To tune for fast recovery from load changes, a loaddisturbance should be simulated by stepping the controller output inmanual, and then transferring to auto. For lag-dominant processes, thetwo responses are markedly different.”For typical process control problemsTune kp, ki , and kd for load disturbances, filtering formeasurement noise and β , and γ for set-points u(t) kp β r(t) y(t) kiZ0t drr(τ ) y(τ ) dτ kd γThe literature is often very misleading!Motion control is differentBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Controldt dy f dt

PerformanceDisturbance reduction by feedbackYcl SYol 1Yol1 PCLoad disturbance attenuation (typically low frequencies)G yd sP( ,1 PCki Gud PC1 PCMeasurement noise injection (typically high frequencies)G xn PC,1 PC Gun Command signal followingG xr kiC( C G f ( kp kd s)1 PCsG f (γ kd s2 β kp s ki )PG f (γ kd s2 β kp s ki ),G urs PG f ( kd s2 kp s ki )s PG f ( kd s2 kp s ki )Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Criteria IE and IAETraditionally the criteriaZ IE IT AE Z0 0e(t)dt,I AE t p e(t)pdt, ZQE 0Z0p e(t)pdt, 2I E2 Z0 e2 (t)dt2( e (t) ρ u (t))dtwhere e is the error for a unit step in the set point or the loaddisturbance have often been used to evaluate PID controllersNotice that for a step u0 in the load disturbance we haveZ u( ) ki0e(t)dtFor a unit step disturbance we have u( ) 1 and henceI E 1/ ki . If the responses are well damped we have I E ( I AEand integral gain is then a measure of load disturbance attenuation.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Load Disturbance AttenuationP 2(s 1) 4 PI: kp 0.5, ki 0.250p G xd (ω )p10 110 210 210 1100ω10Approximations for low (red dashed) and high frequencies (bluedashed)P1s(( ,1 PCCkiBo Bernhardsson and Karl Johan ÅströmP(P1 PCControl System Design - PID Control110

Measurement Noise InjectionP (s 1) 4 PID: kp 1, ki 0.2 , kd 1, Td 1 T f 0.21p Gun (ω )p10010 210 110010ω110First order filter (dashed), second order filter (full)skd s2 kp s ki 2s(1 sT f (sT f ) /2) s K ki Peaks of Gun at ω ms and at ω ( 2/T f Gun CS Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control210

RobustnessGain and phase margins m and ϕ mMaximum sensitivities Ms maxω p S(iω )p, Mt maxω pT (iω )pH 11 PC 1 C 1 P 1 PC CPC 1 PCP 1 PC PC 1 PC Dimensions! For SISO systems the H norm of G s isγ 2 max(1 p Pp2 )(1 p Cp2 )p1 PCp2With scaling of process and controllerγ max 1 p PCp1PC max p1 PCp1 PC1 PCBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

CirclesMs Mt 2Ms Mt 1.4replacementsContourCenterRadiusMs 11/ MsMtMs , MtMs Mt MM2 2tMt 1Ms (2Mt 1) Mt 1 2Ms ( Mt 1) 2M 2 2M 12M ( M 1)Bo Bernhardsson and Karl Johan ÅströmMtMt2 1Ms Mt 12Ms ( Mt 1)2M 12M ( M 1)Control System Design - PID Control

Stability Region for P (s 1) 4403530ki2520151052000152104568k0kdExplains why derivative action is difficultDon’t fall off the edge!Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Robustness Region for P (s 1) 4 & Ms e with stability regionBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control3.5

Projections on the kp ki planekd 01kd 110.80.80.80.60.60.60.40.40.40.20.20.20 0.500.511.5kd 310 0.5100.511.5kd 3.10 0.510.80.80.80.60.60.60.40.40.40.20.20.20 0.500.511.50 0.50Bo Bernhardsson and Karl Johan Åström0.5kd 2111.50 0.500.511.5kd 3.300.5Control System Design - PID Control11.5

Edges Correspond to Cusps in the Nyquist PlotIm G l (iω ) 1Re G l (iω )Nyquist curve of the loop transfer function for PID control of theprocess P(s) 1/(s 1)4 , with a controller having parameterskp 0.925, ki 0.9, and kd 2.86.Cusps are avoided in this example by minimizing IAE instead (dashedcurve) kp 1.33, ki 0.63, and kd 1.78Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Time ResponsesStep in load disturbanceStep in set 501.51uu10.50.50010203040500Process P(s) 1/(s 1)4 , with controller having parameterskp 0.925, ki 0.9, and kd 2.86 (max ki solid lines IAE 3.0)and kp 1.33, ki 0.63, and kd 1.78 (min IAE 2.2 dashedlines). Damping ratios of zeros ζ 0.16 and 0.37.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Tuning based on OptimizationA reasonable formulation of the design problem is to optimizeperformance subject to constraints on robustness and noise injection.Performance criteria IE or IAE for load disturbance attenuationSmall difference between IE and IAE for PILarger differences for PI because of derivative cliffNecessary to use an edge constraintRobustness Ms and MtNoise injection max p Gun (iω )p or pp Gun pp2Maximize performance with noise attenuation and robustness asconstraints (Shinskey: Minimize effect of load disturbances)Minimize noise injection with performance and robustness asconstraints (Horowitz: minimize cost of control)Many efficient algorithms availableKey issues: How to find the modelBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Control System Design - PID Control1Introduction2The Controller3Performance and Robustness4Tuning Rules5Relay Auto-tuning6Limitations of PID Control7SummaryTheme: The most common controller.Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Tuning RulesWhen do you need rules?Why not model by physics or experiments and design acontroller?Typical processes - essentially monotone - modeled by FOTDZiegler-Nichols Tuning 1942 (for historical reasons)Lambda tuning - Common in pulp and paper industrySIMC - Skogestad: Probably the best simple PID tuning rules inthe worldOptimization, criteria and constraintsAMIGO - Minimize IE, maiximze Integral gain subject torobustness constraint and edge constraint for PIDMIAEO - Minimize IAE subject to robustness constraint (for localreasons and insight)How to get the models?Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Ziegler-Nichols Tuning - CommissioningProcess control scenario: You have a controller with adjustableparameters and a process. How do you find suitable values of thecontroller parameters? Ziegler-Nichols idea was to tune controllerbased on simple experiments on the processThe step response method - open loop experimentMake an open loop step response (bump test)Pick out features of the step response and determine parametersfrom a tableThe frequency response method - closed loopConnect the controller change controller parameters, observeprocess behavior and adjust parmetersThe rules were developed by picking out typical process models,tuning controller by hand or simulation (MITs differential analyzer andpneumatic), and correlating controller parameters to process featuresBo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Assessment of Ziegler-Nichols MethodsGreat simple idea: base tuning on simple process experiments,Published in 1942 in Trans. ASME 64 (1942) 759–768.Tremendously influential for establishing process controlSlight modifications used extensively by controller manufacturersand process engineersThe Million question: What structure (series or parallel) did theyuse?BUT poor executionUses too little process information: only 2 parametersStep response method: a, LFrequency response method: Tu , K uBasic design principle quarter amplitude damping is not robust,gives closed loop systems with too high sensitivity ( Ms 3) andtoo poor damping (ζ ( 0.2)Bo Bernhardsson and Karl Johan ÅströmControl System Design - PID Control

Lambda TuningProcess model and desired command responseP(s) K p sLe .1 sTG yysp The controller becomesC(s) P 1 (s)1e sL .1 sTclG yysp (s)1 sT ,1 G yysp (s)K p(1 sTcl e sL )Cancellation of the process pole s 1/T !! Approximations of e sLgive PI and PID controllers, for example e sL ( 1 s

Predictions about PID Control 1982: The ASEA Novatune Team 1982 (Novatune is a useful general digital control law with adaptation): PID Control will soon be obsolete 1989: Conference on Model Predictive Control: Using a PI controller is like driving a car only looking at the rear view mirror: It will soon be replaced by Model Predictive Control.