Introduction PID Versus Advanced Control

Lecture 9 - PID ControlIntroduction Why PID?K. J. Åström– The most common controller– Widely used in all applications of control (90% of allcontrol problems)1. Introduction2. Derivative Filter3. Set Point Weighting An essential element of more sophisticated controllers4. Integrator Windup Are there any research issues?5. Computer Implementation Nonlinear features– Saturation and windup– Dead zones6. Tuning7. SummaryTheme: The most common controller. A glimpse of implementation.– What can be done by PID?– What cannot be done by PID?IntroductionPID versus Advanced Control Feedback is a very powerful concept with many usefulproperties–––––Reduction of effects of disturbancesCreate robust linear relationsFollow command with High FidelityRobust to process variationsBut risk for instability Advances in control theory have given a good insight intothe design problem PID a simple powerful form of feedback Apply advances in control to PID control Advanced control - other prediction methods Connect with the classic tradition of Ziegler and Nichols What are the benefits?c K. J. Åström, October 2002&1

The Amazing Property of Integral ActionConsider a PI controlleru ke kiZt0e(τ )dτAssume that there is an equilibrium with constant e(t) e0and constant u(t) u0 . The error e0 then must be zero. Proof:Assume e0 0, thenZ tZ tu ke0 kie(τ )dτ ke0 kie0 dτ ke0 ki e0 t00The right hand side is different from zero. Hence a contradictionunless e0 0.A controller with integral action will always give the correctsteady state provided that a steady state exists.A PID AlgorithmIn spite of the widespread use of PID it is only given moderateattention in education. Much information among the manufacturers. PID control is much more thanZ tde(t)u(t) ke(t) kie(τ )dτ kddt0We have to consider Derivative filter Computer implementation Set point (reference)weigthing Mode switches Bumpless parameter changes Integrator WindupDealing with these issues is a good introduction to practicalimplementation of any control algorithm.PID ControlDifferentiating Noisy SignalsBrusfri signal1. Introduction2. Derivative Filter3. Set Point Weighting4. Integrator Windup5. Computer ImplementationConsider the signaly(t) sin t an sin ω tIt has the derivatived y ( t) cos t anω cos ω tdtBrusfri derivata22 20 2510150Brusig signal sb2101510152 205Derivatan av sb 251015056. Tuning7. SummaryThe curves are generated with ω 100, a n 0.01.One percent error in the original signal gives 100% error inderivative!c K. J. Åström, October 2002&2

Approximate Differentiation - High FrequencyRoll-offReplace sT bySimulation of Approximate DerivativeBode DiagramsBrusfri signalsT1 sT / N20y(t) sin t an sin ω t0What does it mean?we have ForPSfragsmall sreplacementsGd (s) sT .Gain For large s we havePhaseGd (s) NFrequency.ωPhase (deg); Magnitude (dB)Gd (s) 20 40100806040200 110Brusfri derivata2400110210Approximate derivativesG d ( s) 1 s/510 20 2510150Brusig signal sb101510152 205Derivatan av sb2 2510150Brusig signal sb5Fuskderivatan av sb22 20Frequency (rad/sec)2 251015051015The system Gd (s) has the output T d y/dt for low frequency signals. The gain of Gd is not greater than N .A nice illustration of use of Bode Plots!Different ParameterizationsPID ControlParallel form: 1kG ( s) k 1 sTd (1 sTi s2 Ti Td )sTisTiSeries form: 1k̃(1 sT̃d) 1 s(T̃i T̃d ) s2 T̃i T̃dG̃ (s) k̃ 1 sT̃isT̃iRelations between coefficientsk k̃T̃i T̃dT̃i,Ti T̃i T̃d,Td T̃i T̃d1. Introduction2. Derivative Filter3. Set Point Weighting4. Integrator Windup5. Computer Implementation6. Tuning7. SummaryT̃i T̃dParallel form is more general. Equivalence possible only ifTi 4Td . Essential for tuning to know which form is used.c K. J. Åström, October 2002&3

Sfrag replacementsSet Point (Reference) ResponseSet Point (Reference) WeightingSet point weighting allows a moderate adjustment. A 2DOFstructure makes set-point response independent of disturbanceresponse.FΣeCIΣu10.50ndrA simple way to obtain some DOF benefitsPxΣy20v00b 1b 0.5b 010b 1b 0.5b 010CR2030402030403040nl 0.5 1 101020 Notice different signal paths y u and r u. Not a complete1sTdY (s)U (s) k bR(s) Y (s) ( R(s) Y (s)) 2DOF but often a good way to separate disturbance rejection fromTi1PSfrag sTd/ N replacementsresponse to reference signals.PID ControlEffects of Saturationr1. IntroductionControlleruProcessy2. Derivative Filter3. Set Point Weighting4. Integrator Windup5. Computer Implementation6. Tuning7. Summary Practically all systems have saturations in actuators The feedback loop is broken when saturation occurs Unstable modes in process and controller will grow An integrator is an unstable and it will wind up Windup protection is required in all controllers with integralaction Instabilities are essential difficulites!c K. J. Åström, October 2002&4

Integrator Windupy210One Way to Avoid Windup–yr0204060KT d se r y0.1 0.1204060KTi80–1sΣI2204060800Influence of the reset time constantTt.ry0.510200u0.0500.101020300010Tt 0.1Tt 102030Tt 3Tt 2 0.1I 0.4 0.8Tt 0.1, Tt 1300.15 0.05Tt 3Tt 2r10 Effect of Time Constant TtControl with Anti-Windup1Σes1Tt 20ActuatorΣKu0Actuatormodel8001020301020Rules of thumb Tt 0.5Ti for PI control or Tt Simulation made with PI control with Ti 1.c K. J. Åström, October 2002& 30Ti Td for PID.5

PID ControlComputer ImplementationPractically all control systems are today implemented usingcomputers. We will briefly discuss some aspects of this.1. Introduction2. Derivative FilterAD and DA converters are needed to connect sensors and actuators to the computer. A clock is also needed to synchronizethe operations. We will discuss3. Set Point Weighting4. Integrator Windup5. Computer Implementation Sampling and aliasing6. Tuning A basic algorithm7. Summary Converting differential equations to difference equations Wordlength issues Bumpless parameter changesSampling, Aliasing and Antialiasing FiltersA Basic AlgorithmThe following operations are executed by the computer.11. Wait for clock interrupt2. Convert setpoint r and process output y to numbers03. Compute control signal u 1012345 Samples of signals of different frequencies may be identical Nyquist frequency (Sampling frequency)/2 To represent a continuous signal uniquely from its samples thecontinuous signal cannot have frequencies above the Nyqyistfrequency which which is half the Nyquist frequency4. Convert control signal to analog value5. Update variables in control algorithm6. Go to step 1Desirable to make time between 1 and 4 as short as possible.Defer as much as possible of the computations to step 5. Antialiasing filters that reduce the frequency content above theNyquist frequency is essential.c K. J. Åström, October 2002&6

A Practical PID ControllerThe PID AlgorithmThe basic equationThe PID controller is described by: u(t) k br(t) y(t) kiDerivative filterTd dy fN dtZt0 yf yFeedback Gc (s) k kis d y f ( t)r(τ ) y(τ ) dτ kd( ),dtU (s) P(s) I (s) D (s) P(s) k bR(s) Y (s)1( R(s) Y (s))sTisTdY ( s)D (s) k1 sTd/ NI (s) k kd 1 ssTfFeedforward G f f (s) bk kisSet point weighting bSometimes also high frequency roll-off k1U (s) bR(s) Y (s) R(s) Y (s) sTd Y (s)(1 sT f )2sTiComputers can only add and multiply, it cannot integrate ortake derivatives. To obtain a programmable algorithm we mustapproximate. There are many ways to do this.Introduce the times tk when the clock ticks, assume thattk tk 1 h, ,where h is the sampling period.Integral PartThe Proportional Partp(tk) k (br(tk) y(tk))No approximation required!ki( t ) TiDifferentiateZte(τ )dτdik e(t)dtTiApproximate the derivative by a forward differencei(tk 1) i(tk )ke(tk) hTiThis equation can be written asi(tk 1) i(tk ) c K. J. Åström, October 2002&khe(tk )Ti7

Derivative PartD ( s) kHenceIn time domainDerivative Part ContinuedsTdY (s)1 sTd/ Nd( t k ) (1 sTd/ N ) D (s) ksTd Y (s)Td dddy kTdN dtdtApproximate derivative by backward differenced( t ) d( t k ) Td d(tk) d(tk 1)y(tk) y(tk 1) kTdNhhHence Td TdkTdd(tk 1) 1 d( t k ) y(tk) y(tk 1)NhNhhor TdkTd Nd(tk 1) d( t k ) y(tk) y(tk 1)Td NhTd NhNotice that the algorithm works well even if Td is small, this isnot the case if forward approximations are used.The Discrete PID AlgorithmAdd Protection Against WindupSummarizing we findp(tk) k (br(tk ) y(tk)) Tdd(tk) d(tk 1) kN y(tk) y(tk 1)Td Nhv p(tk) i(tk) d(tk)p(tk) k (br(tk ) y(tk))e( t k ) r( t k ) y ( t k ) Tdd(tk) d(tk 1) kN y(tk) y(tk 1)Td Nhu(tk) p(tk) i(tk) d(tk)i(tk 1) i(tk) khe( t k )TiTd d(tk) d(tk 1)y(tk) y(tk 1) kTdNhhu(tk) sat(v)e( t k ) r( t k ) y ( t k )i(tk 1) i(tk) khkhe( t k ) u vTiTr Useful to precompute parameters Make sure updating is done safely Organize the code rightc K. J. Åström, October 2002&8

Wordlength IssuesConsider updating of the integral parti(tk 1 ) i(tk ) khe(tk)TiBumpless Parameter ChangesA PID controller is often switched between three modes: off,manual and automatic control. It is important that there are noswitching transients.It is also important that parameter changes do not generatetransients. This can be avoided by proper coding.Example h 0.05 sExample: Ti 5000 sThis implementation givesbumpsZk ti e(s)dsTi k 1kh 10 5 TiThis implementation doesnot give bumpsZ tki e(s)dsTiIf the error has 3 digits the integral need to be updated with 8digits (28 bits) to avoid rounding off the errors! PSfrag replacementsThe basic issue is that multiplication with a time function doesnot commute with differentiation or integration.PID ControlRequirementsnd1. Introduction2. Derivative FilterrFΣeCuΣPxΣy3. Set Point Weighting4. Integrator Windup 15. Computer Implementation6. Tuning7. Summary Reduce the effect of load disturbances Do not inject too much measurement noise Low sensitivity to process variations Good response to set point changesc K. J. Åström, October 2002&9

IntroductionZiegler-Nichols’ Step Response MethodA wide range of methods have been developed to design andtune PID controllers Special methods for PID controllers Application of general techniques for control system designlike pole placement that you have learned in the class.The methods differ with respect to Models Switch controller to manual. Make a step in the control variable. Log process output. Normalize the curve so that it corresponds to a unit step. Determine intercepts of tangent with steepest slopei.e. parameters a and L. The controller parameters areobtained from a table.3 Model acquisition2 Criteria1 Design techniques0We will present a selection 1Ziegler-Nichols’ Step Response MethodData: apparent time delay L and intercept a. Controller parameters are given byControllerkPPI1/ aPID0.9/a1.2/aTiTd2L5.7LL/224683.4LParameter Tp is an estimate of the response time of the closedloop system.101214161820Ziegler-Nichols’ Frequency Response Method Switch the controller topure proportional. Adjust the gain so that theclosed loop system is atthe stability boundary.Tp4L3L0 Determine the gain ku(the ultimate gain) andthe period Tu (the ultimateperiod) of the oscillation. Suitable controller parameters are obtained from atable.c K. J. Åström, October 2002&0.50 0.5 1 0.500.5110