PID Controllers: Theory, Design And Tuning
PID Controllers:Theory, Design and Tuning
IntroductionBasics of PID controllersTuning of PID controllersOptimization in MatlabAuto tuningLecture content
By far the most popular controller In process control 95 controllers are ofPI(D)-type Good for linear process control Relatively easy to understand (importantreason for wide popularity) Still many of the PID-control loops arepoorly tuned.PID-controllers: introduction
– 30% poor tuning– 30% valve problems– 20 % variety of problems (e.g sensorproblems, bad choice of sampling rates.) Over 2000-500 control loops 97 % PI-controllers Only 20% of PI-controllers work welldecreasing process variability Reason for poor performance:Typical paper mill
Today most of the PID controllers aremicroprocessor based DAMATROL MC100: digital single-loop unitcontroller which is used, for example, as PIDcontroller, ratio controller or manual controlstation. Often PID controllers areintegrated directly into actuators(e.g valves, servos)PID-controller
yspeControlleru-1lProcessxny”Increase the manipulated variable when processvariable is smaller than the setpoint and decreasethe manipulated variable when the process variableis larger than the setpoint” Principle of negative feedback:Simple idea of feedback.
refTo Workspace1rPID ControllerPIDloadClock1Zero-PoletTo Workspace2Band-LimitedWhite Noise(s 1)(s 1)(s 1)Simulink modelTo WorkspaceyScope
u (t )t1K e(t )e( ) dTi o– K proportional gain– Ti integral time– Td derivative time Controller parameters:– P-term (proportional to e)– I-term (proportional to integral of e)– D-term (proportional to derivatice of e) ”textbook” version of PID Control variable u is a sum of:PID controlde(t )Tddt
High value of gainmakes the systemmore insensitive toload disturbance Too large a gainmakes the systemmore sensitive tomeasurement noise Steady-state errordecreases when gainincreases Oscillation howeveroften increases00.20.40.60.811.21.402Proportional action4s 1136K 5810K 212K 114Closed loop system with proporional controlG161820
Integral termremoves steady stateerror Short integration timeoften leads tooscillation Long integration timecommon in processcontrolIntegral action00.20.40.60.811.21.41.602468Ti 210Ti 5Ti 112Ti inf14Closed loop system with proportional and integral control161820
Derivative term canpredict output Fast and stableresponse Noise can makederivative controlproblematic Also long delays areproblematic whenusing derivative termDerivative action00.20.40.60.811.21.40246Td 0.7Td 0.1810Td 2.512Derivative control (K 3, Ti 2)14161820
– computing derivatives from process output– using filtered derivative term (this is usedoften in real applications) Fast changes in reference signal resulthigh derivatives control signal saturates Fixes:Derivative action
––––Gs ,2 ( s )Gs ,1 ( s )ss , lim Gs ,2 ( s )ss 1sGs ,1 ( s )easy to implement in practisemore insensitive to noise than normal derivative termcorresponds with derivation of low pass filtered signalby choosing tau smallsystem has same response as by using normal derivation Benefits:Filtered derivative term:
control signal u0510152025303540024ref68infinite signal peak1s10time(s)12IntegratorControl signale16Gain1TdGain31/Ti14Transfer Fcntau.s 1sPID .50.60.70.80.91Filtered derivativeoutput y0ul2uf46ProcessGTo Workspace2Scope810time (s)12Step response (tau 0.1)y14To Workspaceyf161820
Yref- EKi/s- Kp KdsG(s)Y Use process output for derivation and gain No zeros to controlled closed-loop system(prevents overshoots)Output derivation – Tachometer feedback
eG (s)DIP1K 1sTisTduG (s)'D'1K 1sTi 'PI1 sTd' Interacting common in commercial controllers(said to be easier to tune manually) Non-interacting more general Non-interacting & interactingAlternative representations
refeIntegrator1sDerivativedu/dtGain31/TiGain1TdPID CONTROLLERGain2Kpu Simulink PID-block is1of form:K p KiTd ss ”text book” version is:ProcessGyPID ControllerPIDSIMULINK PID-controllerScope
Integrator windup
If the control variable reaches actuatorlimitsfeedback loop is broken! Still error will continue to integratevery large integral term (”wind up”) Large transients when the actuatorsaturates– limited speed– valve opening All actuators have limitations:Integrator windup
– setpoint limitation (limit performance,windup caused by disturbances?)– incremental algorihms– back calculation and tracking– conditional integration Integrator action must be stopped whenoutput saturates! Solutions:Integrator windup
When tight control not neededPI-control adequate– level control in single tank– stirred tank with perfect mixing. Often derivative action switched off Dominant dynamics are of the 1. order For example:When is PI control sufficient?
– temperature control For example:– damping improved– higher gain can be used to speed uptransient response Dominant dynamics are of the 2. order PID speeds up the response versus PIWhen is PID control sufficient?
System has high order dynamicsSystem is time variantLong delaysNon-linear processMIMO/MISO system with strong crossdepenciesWhen PID control is insufficient?
– load disturbance attenuation– effects of measurement noise– robustness to process variations– response to setpoint change– model requirements– (computational requirements) Problem: how to determine theparameters? Tuning is a trade-offs between:Controller design
00.10.20.30.40.50.60.70.8024rise time6overshootTime (sec)8settling timeStep Response10Performance criteriaAmplitude12steady state1416
u (t ) 1(t )G(j )y (t )Open loopIdentify plant dynamicsLet u(t) be a unit step (’good’ test signal)Measure outputOpen loop tuning
KaLTOpen loop tuningK 1L 1.3 sT 4.4 sa 0.4From figureKG(s)e1 sTsL
1.2/aPIDL0.9/aPIa1/aPController KTiZiegler-Nichols2L3LTdL/24L3.4L5.7LTp
hols response141618Ziegler-Nichols: response20
Decay ratio is close to one quarterOvershoot is quite largeSimple and widely usedOften insufficient necessary to havemore data about process dynamics Gives a starting point for fine tuning Also frequency response method can beused Ziegler-Nichols analysis
Ziegler-Nichols
– setpoint response– load disturbance Different parameters for tuning:– quickest response without overshoot– quickest response with 20% overshoot Modification of Ziegler-Nichols method Better dampled closed-loop stepresponse Parameter tables for:Chien, Hrones & Reswick Method
PPIPID0.3/a0.35 a 1.2T0.6/a T0.5L0.7/a0.6 a T0.95/a 1.4T0.47LPID Controller parameters;Chien, Rhones Reswick setpoint response method0%20 % overshootControl KTiTdKTiTd T time constantCHR for setpoint response
PPIPID0.3/a0.6 a 4L0.95/a 2.4L0.42L0.7/a0.7 a 2.3L1.2/a 2L0.42LPID Controller parameters;Chien, Rhones Reswick load disturbance response method0%20 % overshootControl KTiTdKTiTdCHR for load disturbance resp.
00.20.40.60.811.21.41.60246810time(s)12141618ZNCHR 20%CHR 0%20G ( s)1( s 1)3väärin Overshoot exist but response is much betterwhen CHR parameters are usedZ-N vs. CHR
GcG0(closed loop transfer function)G01(closed loop transfer function)G p 1 G01 G p GcG p GcGc (controller transfer function)G p (process transfer function)Analytical tuning methods
e(t)PIDu(t)G(s)c(t)u (t )Kp1e(t )Ti0te( ) ddeTddt1. Set Ti and Td 0.2. Increase Kp until the system oscillates to obtain criticalgain Kcr and critical period Tcr (or frequency)-r(t) 0 Linear systemClosed-loop tuning
u (t )Kp1e(t )Ti0te( ) ddeTddt0.8Tcr 1.80.5Tcr 1.1 0.125Tcr 0.30.5Kcr 2.30.4Kcr 1.90.6Kcr 2.8PPIPIDTdTiControl KPID Controller parameters;Ziegler-Nichols frequency response methodClosed-loop tuning
Frequency response method
Frequency response interpretation
Frequency response – phase margin
Comparison
e(t)PIDMu(t)K p e (t )1TiG(j )Linear systemu (t )e( ) dc(t)0tTdFirst relay on, to obtain critical gain Kcr and critical period Tcr-r(t) 0 MRelay tuningdedt
Increasing proportional gain decreasesstability Error decays more rapidly if integration timeis decreased Decreasing integration time decreasesstability Increasing derivative time improves stabilityRule-based empirical tuning
Tuning maps
Tuning map
Counterintuitive
Process must be modelled Define the desired closed -loop poles PID-controller gives desired closed-looppoles Process parameters max. 3 poles canbe set freely using PID-controllerPole Placement
TTiTs1 KpKsKpK1 Gc G p Characteristic eq:20G (s)1 G p GcG p Gc Closed loop system:1K (1)sT1(1 sT )GcGpKp Controller (PI): Process:Pole Placement: example0
Desired close-loop polescharacterized by relativedamping ( ) and frequency( ):By making coefficient equal incharacteristics equations andsolving controller parametersgives:s2TiK2T 1T020200T 1Kps202Pole placement: example0
Largest value of transfer function from loaddisturbance to process outputGang of six
o More sensitive whenChoosingT is smallo
First order approxApproximate models
Approximate models
Minimize the cost function by simulatingthe controlled model Use fminsearch function to findoptimized parameters for the PIDcontroller Choose appropriate cost functioncarefully Don’t forget local minimaOptimization in Matlab
Gc 1 faster response (smallertime constant) controller transfer function:1 s TesL1 sTK p (1 s T e –Tuning processes with long dead time desired closed-loop tf:G0Analytical tuning methodssL)
4.3.1.2.11K inK dnK dn correction (depends on optimization algorithm)K in correction (depends on optimization algorithm)K pn correction (depends on optimization algorithm)1JntoleranceIf not, set n n 1 and go back to 2.JnCheck if minimum is reached1K pnGuess initial values for K p0 , K i0 , K d0 , Set n 0Solve y(t) using simulation (SIMULINK) and alsothe value of cost function J K pn , K in , K dnUse optimization algorithm (FMINSEARCH orFMINUNC) to update the control parameters:Optimization in Matlab
-ePIDcontrollerParameter vector pK p , Ki , K dPID controllerParameters Kp, Ki, and KdInputyref ControllerTu (t )uth x t ,u t ,tK p , Ki , K d0te(t )dedte( ) d0K p e( t ) K i ey tx f x t ,u t ,tSystemOptimization in MatlabpdTt0ydedte( ) de(t )deKddt
ITSEITAE00te(t ) dt2t e(t ) dtCost criteriaISTEITE00t 2 e(t ) 2 dtte(t )dt
refOut22To Workspace3IntegratorMathFunction1uISELTI System1sTransportDelayGsmeasurement noise2PID ControllerPIDload disturbanceTo Workspace1rExample modelClocktTo Workspace2Out11To WorkspaceyScope
-0.200.20.40.60.811.21.40510152025Open loop response303540
%assign new parameters toworkspace for ('base','Ti',inputs(2));%simulate[t,x,y] sim(model,Tsim);%y(2) corresponds with costJ max(y(:,2)); fminsearch('cost',[110])IN MATLAB: %evaluate cost function from thesimulink modelfunction J cost(inputs)%paramatersTsim 40; %simulation timemodel 'pidmodel'; %simulinkmodel 101515Cost function in Matlab20202525303035354040
yspControlleruController parametersGain schedulingProcessTableySchedulingvariable
increasesreducesincreasesK increasesTi increasesTd le-Based Methods
Multiple Input Multiple Output (MIMO)MIMO PID control- Decentralized
MIMO PID control- Decentralized
Amount ofinteractionUnit step in ref1K 0.83, Ti 2, I 0.5LieslehtoMIMO PID control- Decentralized
Amount ofinteractionUnit step in ref2K 1.5, Ti 2, I 0.5LieslehtoMIMO PID control- Decentralized
CxyG(s) C( sI - A) 1 BTransfer functionAx BuxState-space equationMIMO PID control - Centralized
G(0)-1Decoupling at steady stateG(s)MIMO PID control - Centralized
KiG (0)G (0)0.8111.2 0.91.2 0.90.8 112.11.71.92.5MIMO PID control – Integral gain
Kps0lim sG ( s )10.6 0.90.2 0.331.20.92s 1 s 1lim sG ( s ) lim ss 0s 00.814s 1 3s 1118.3110.2 0.30.6 0.95033MIMO PID control – Proportional gain
Setpointy mSPPID controllerHpid (s)uControl variableController with dead-time compensatorProcess modelHu,dc (s)Hu (s)ProcessSmith-predictory mlHv (s)v(tdc)(t)epProcess measurementym
K. Åström and T. HägglundPID Controllers: Theory, Design, andTuning (3rd ed) (2005)References
r(t)- e(t)PIDQin(t)G(j )Linear systemQout(t)
r(t)Input- KpKI / se(t)Kp KI/sQin(t)e-sTQout(t)Output
r(t)Input- e(t)KpKI / sQin(t)esTQout(t)Output
yspControlleruController parametersGain schedulingProcessTableySchedulingvariable
PID-controller Today most of the PID controllers are microprocessor based DAMATROL MC100: digital single-loop unit controller which is used, for example, as PID controller, ratio controller or manual control station. Often PID controllers are integrated directly into actuators (e.g valves, servos)File Size: 1MBPage Count: 79Explore furtherWhen not to use PID-controllers - Control Systems .www.eng-tips.comPID Controller-Working and Tuning Methodswww.electronicshub.org(PDF) DC MOTOR SPEED CONTROL USING PID CONTROLLERwww.researchgate.netTuning for PID Controllers - Mercer Universityfaculty.mercer.eduLecture 9 – Implementing PID Controllerscourses.cs.washington.eduRecommended to you b
controllers utilise PID feedback. The importance of PID controllers has not decreased with the adoption of advanced control, because advanced controllers act by changing the setpoints of PID controllers in a lower regulatory layer.The performance of the system depends critically on the behavior of the PID controllers. 2016: Sun Li
typical unit negative feedback control system [4]. PID control theory is widely used in the field of industrial automation control. The basic principle is clear and concise. As a useful complement to PDCA theory, PID control theory has good feasibility.PID control theory uses PID control ideology to supplement and improve the correction
control theory, according to the literature, more than 95% of industrial controllers are still PID, mostly PI controllers. PI (PID) control is sufficient for a large number of control processes, particularly when dominant process dynamics are of first (second) order and there design re
PID-controllers is derived. A systematic synthesis procedure to obtain such PID-controllers is presented with numerical . and from a tuning point-of-view, it also presents a major restriction that only certain classes of plants can be controlled by using PID-con
Logic Controllers), DCS (Distributed Control System) or single loop or stand alone controllers. The PID principle is also the basic for many advances control strategies. In this paper a novel optimal PID controller tuning approach based on the HC12 is proposed. The optimal PID parameters design
PID Control Proportional-Integral-Derivative (PID) controllers are one of the most commonly used types of controllers. They have numerous applications relating to temperature control, speed control, position control, etc. A PID
Standard PID Control A5E00204510-02 Finding Your Way Chapter 1 provides you with an overview of the Standard PID Control. Chapter 2 explains the structure and the functions of the Standard PID Control. Chapters 3 helps you to design and start up a Standard PID Control. Chapters 4 explains the signal processing in the setpoint .File Size: 1MB
Por Alfredo López Austin * I. Necesidad conceptual Soy historiador; mi objeto de estudio es el pensamiento de las sociedades de tradición mesoamericana, con énfasis en las antiguas, anteriores al dominio colonial europeo. Como historiador no encuentro que mi trabajo se diferencie del propio del antropólogo; más bien, ignoro si existe alguna conveniencia en establecer un límite entre .
