PID CONTROL. Presentation kindly provided by Dr Andy Clegg. Advanced Control Technology Consortium (ACTC)

Transcription

1 PID CONTROL Preentation kindly provided by Dr Andy Clegg Advaned Control Tehnology Conortium (ACTC)

2 Preentation Overview Introdution PID parameteriation and truture Effet of PID term Proportional, Integral and Derivative term Tuning PID Controller Equivalene of PID and Lead-Lag ontroller Implementation apet of PID

3 Introdution to PID Control Proportional-Integral-Derivative ontrol Predominant ontroller ued in indutry Reaon: it i adequate for mot appliation Initially: analogue implementation Now: motly digital

4 Motivation and Limitation Motivation Simple to get working Can be tuned to meet time-domain peifiation Readily available in PLC/DCS ytem Digital and analogue implementation eay Limitation For ingle-input ingle-output ytem only Diffiult to tune to meet preie peifiation Subtle differene in implementation aue problem

5 Generi Equation and Traking Error Generi Equation and Traking Error Cloed-loop TF from referene to error Type 0 ervo, unit-tep teady-tate error Type ervo, unit-ramp teady-tate error ) ( G()() ) ( r e p k G e ) ( ) ( 0 lim v k G e ) ( ) ( 0 lim 2

6 Idealied PID-Controller Configuration Set-Point r() e() () u() Output y() Controller Plant

7 Generi PID Control Equation Time domain: de ( t) k e t k e t dt k 23 ( ) 2 ( ) P 23 dt u 3 t I D Laplae domain: k2 U ( ) k E E k E 23 ( ) ( ) ( ) P D I However there are many different truture...

8 Ideal PID Parameteriation u ( t) t de e( t) edt Td T dt i 0 : proportional oeffiient T i : integral time ontant T d : derivative time ontant Common in PLC and DCS implementation

9 Parallel PID Parameteriation u ( t) pe( t) i edt t 0 d de dt p : proportional gain i : integral gain /T i d : derivative gain T d Ued in SW tool (e.g. Matlab) and ome indutrial ytem

10 Serie PID Parameteriation u : T i : T d : ( t) t de (.) dt e( t Td Ti dt 0 ) proportional oeffiient integral time ontant derivative time ontant I found in ome indutrial ytem Hitorially popular ine an be implemented with jut one op-amp

11 PID Parameteriation There are alo differene in unit for gain term Proportional gain: a a pure gain or proportional band ( 00%/gain) Integral gain: a reet (i.e. i, unit of repeat per eond or minute) or integral time (i.e T i, unit of eond or minute) Derivative gain: a derivative time (i.e T d, unit of eond or minute)

12 PID-Controller (Proportional-Integral Integral-Differential) integral fator integrator proportional fator et-point error p PID T i ommand variable plant proe value T d derivative fator meaurement The proportional fator p generate an output proportional to the error, it require a non-zero error to produe the ommand variable. Inreaing the amplifiation p dereae the error, but may lead to intability The integral time ontant T i produe a non-zero ontrol variable even when the error i zero, but make the ytem intable (or lower). The derivative time T d peed up repone by reating to an error hange with a ontrol variable proportional to the teepne of hange. Aknowledgement: Prof. Dr. H. irrmann, EPFL / ABB Reearh Center, Baden, Switzerland

13 PID Control Ation Traditionally the PID term at on the error ignal, a in previou equation. In proe appliation it i ommon to have derivative ating on the output rather than error Called PI-D or Derivative on PV In ome ae the proportional ontrol an at on the loop output a well Called I-PD or SP on I-only Note: if the et-point i ontant then all three form are equivalent, they only differ when the et-point hange..

14 PID Control Ation Repone of a PID ating on the error ignal Controller Output Set-Point The big pike in ontroller output i due to the et-point tep being fed diretly through derivative Loop Output

15 Derivative - on Loop Output Derivative often ued in feedbak path only i.e PI-D ontrol u ( ) ( e( ) T y( ) ) d P-D Controller r() - e() C u ( e( G() ) T y( )) ( ) - u C () Plant d y() C T d Eliminating e(), uing e() r() - y(), give..

16 Derivative - on Loop Output Derivative often ued in feedbak path only i.e PI-D ontrol u ( ) r( ) ( y( ) T y( ) ) d ff () r() P-D Controller C u ( e( G() ) T y( )) ( ) - u C () Plant d y() C (T d ) () CLTF G G ( T ) d

17 PI-D D Control Ation Repone of a PI-D (i.e. Derivative on PV) Controller Output Set-Point The pike in ontroller output i maller - derivative kik ha been eliminated Loop Output

18 I-PD Control Ation Repone of an I-PD (i.e. Proportional & Derivative on PV) Controller Output Set-Point No pike in ontroller output - both proportional and derivative kik eliminated Loop Output I-PID Control hould not be ued on integrating proee (e.g. level ontrol)

19 PID Parameteriation Thee many different form of PID do aue problem: eah ha different behaviour eah require different tuning rule you mut know whih form i ued before doing any tuning, deign and/or imulation However, it an provide an additional degree of freedom when deigning a ontrol ytem

20 Effet of P, I & D Term Proportional Ation Firt-order plant Seond-order plant Integral Ation Firt-order plant Derivative Ation Firt-order plant Seond-order plant Rate feedbak

21 Proportional - t Order Plant Firt-order plant: Pole at -a, teady-tate gain /a Cloed-loop TF: New pole -(a ) G( ) a y( ) r( ) G( ) ( ) G( ) ( ) Steady-tate error with tep input: a However, proportional a e lim gain annot jut limbe inreaed arbitrarily to 0 G( ) ( ) 0 remove teady-tate error. Problem arie with a atuator aturation a and intability with real, higher-order plant

22 Proportional - t Order Plant Repone to unit tep et-point hange: Step Repone Amplitude Time (e.)

23 Proportional - 2nd Order Plant Seond-order plant: G( ) 2 ω 2 n 2ζω ω n 2 n Cloed-loop TF: A inreae: y () ω r () 2 ( ) 2 n 2 2 ζωn ωn CL natural frequeny inreae CL damping ratio dereae overhoot inreae teady tate error dereae

24 Proportional Proportional - 2nd Order Plant 2nd Order Plant Steady Steady-State Error State Error If larger, error maller n n n G e 2 0 lim ) ( ) ( 0 lim ω ζω ω

25 Proportional - 2nd Order Plant Repone to unit tep et-point (ζ 0.4, ω n 3) Step Repone Amplitude Time (e.)

26 Integral Ation Integral Ation Objetive i to remove teady-tate error PI ontroller in time domain: Laplae domain: With a ontant error: t i edt T e t u 0 ) ( T i ) ( i t i t i T et e dt T e e edt T e t u 0 0 ) ( That i, if a ontant error exit the ontroller output will keep inreaing, until the error i zero...

27 Integral Ation - Contant Error

28 Integral Ation - t Order Plant Plant: G( ) a Controller: ( ) T i Cloed-loop TF: y( ) r( ) CL natural frequeny and damping ratio now funtion of T i Steady tate error now removed: e T i 2 at i 2 2ζ 0 lim lim a T a T at i i i ω l 2 nl ω nl ω 2 nl

29 Integral Ation - t Order Plant Repone to unit tep et-point hange: Step Repone Ti0. Ti Ti4 Amplitude Time (e.)

30 Derivative Ation Objetive: tabilie ytem; low down tranient PD ontroller: In Laplae domain: u ( t) e( t With ontant error, derivative ation 0 no ontribution to teady-tate behaviour ) ( ) With tranient on error, derivative ation large T rik atuator aturation from meaurement noie, tep hange in et-point d de dt ( T ) d

31 Derivative - t Order Plant Plant: G( ) a Controller: ff ( ) ( ) ( T ) d Cloed-loop TF: y( ) r ( ) ( Td ) a Cloed-loop pole: ( a ) ( Td a T d i inreaed, CL repone get lower ) Steady tate error not effeted by T d : e lim 0 a ( T ) d a

32 Derivative - t Order Plant Repone to unit tep et-point hange: 0.7 Step Repone Amplitude Td Td2 Td Time (e.)

33 Derivative - 2nd Order Plant Generalied Plant: G ( ) 2 2ζ ω p ω 2 np np ω 2 np Cloed-loop TF: y( ) r( ) CL natural frequeny: 2 ω 2 np ( 2 ) 2 2ζ ω ω T ( ) ω nl p np ω np d ω np np CL damping ratio: ζ l ζ p ωnp 2 T d Steady-tate error i not effeted by T d

34 Derivative Derivative - 2nd Order Plant 2nd Order Plant Steady-tate error: ( ) d np np p np d np np p np T T e lim ω ω ζ ω ω ω ζ ω

35 Derivative - 2nd Order Plant Repone to unit tep et-point hange: 0.7 Step Repone Amplitude Td0. Td0.3 Td Time (e.)

36 Derivative - a Rate Feedbak Controller Plant, G() r() - e() ω G * C - θ y() r angular veloity angle Implementation uing diret meaure of derivative Similar to P-D ontrol with r T d CLTF G G ( ) r

37 Tuning PID Controller Fundamental Trade-off Appliability of PID Proe Reation Tuning Sutained Oillation Tuning IMC PID Tuning Advaned Tuning Method

38 Fundamental Trade-off Set-point traking: Good traking performane Stability: eep ()G() away from - Diturbane rejetion: Redue effet of diturbane Noie immunity: Noie hould not exite u() ()G() large limit on () () large limit on ()

39 Appliability of PID PI ontroller adequate for mot appliation Derivative ation i rarely utilied often miundertood due to the many different form and the different way in whih they work For dominant eond (and higher) order plant PID ontrol may be more appropriate additional D term an provide damping to redue overhoot D an be benefiial for low thermal proee

40 Appliability of PID The following generaliation i from Shinky ("Proe Control Sytem", MGraw Hill, 979) Proe Proportional Band (%) Integral Derivative Flow eential no Preure, liquid eential no Preure, ga 0-5 no no Level 5-80 eldom no Vapour (T and p) 0-00 ye eential Chemial Compoition eential if poible There will alway be exeption, o need tuning rule...

41 Proe Reation Tuning Plant in open loop with and input u nom Apply a tep to input (u tep ) and reord the repone R Δy ΔT y tep ΔT y nom Determine L and R from the repone

42 Proe Reation Tuning Normalie R: R N u tep R u nom Ue L and R N in thi table to give PID etting Controller p T i T d P.0/(R N L) - - PI 0.9/(R N L) 3L - PID.2/(R N L) 2L /(2L)

43 Sutained Oillation Tuning Carried out with plant in loed-loop ueful for open-loop untable plant Uing a proportional-only ontroller, inreae the gain until utained oillation are attained need to be areful that oillation do not upet normal running Reord the gain ( ) and the period of the oberved oillation (T )

44 Sutained Oillation Tuning Ue and T in thi table to give PID etting Controller p T i T d P PI T - PID T 0.25T Thee are jut initial value to get the ontroller to work will need fine tuning

45 IMC Internal Model Control Require plant model IMC PID Tuning model truture and order (e.g. firt order with dead-time) parameter uh a gain, dead-time and time ontant uually obtained from plant tet and deired loed loop time ontant (β) IMC formulae give required PID gain for peified plant model and β

46 If the plant model i: firt order eond order IMC PID Tuning - ontroller PI - ontroller PID firt order dead-time - ontroller PID For higher order plant model, reulting PID i only an approximation of true IMC ontroller Tool are available that ontain the formulae e.g. EZYtune (

47 Advaned Tuning Method Atröm and Hagglund ( Relay ) Auto-tuning: imilar to utained oillation method, but ue a relay with dead-zone to etablih oillation tuning rule built into ontroller o that it an perform thi tet and automatially re-tune itelf PID tuning uing ytem identifiation and loed loop pole plaement alled elf-tuning if arried out on-line and automatially

48 Loop Tuning Chek Lit define deired performane rie time, peak overhoot, tability margin, et. identify the dominant plant dynami look at interation with neighbouring loop whih PID onfiguration i being ued? onider aturation examine et-point traking and diturbane rejetion of CL ytem ha the deired performane been met?

49 PID and Lead-Lag Lag Controller Charateriti PI imilar to lag-ompenator PD imilar to lead-ompenator. hown in the following bode plot

50 PI & Lag Compenator Bode plot: Bode Diagram Magnitude: PI: riing for dereaing PI frequeny lag Lag: level out at low frequenie Phae (deg); Magnitude (db) Phae: Frequeny (rad/e) PI: 90 phae lag for frequenie below breakpoint Lag: phae lag only between breakpoint

51 PD & Lead Compenator Bode plot: PD lead Bode Diagram Magnitude: PD: riing for inreaing frequeny Lead: levelling out Phae (deg); Magnitude (db) Phae: Frequeny (rad/e) PD: 90 phae lead for frequenie above breakpoint Lead: phae lead only between breakpoint

52 Implementation Apet of PID Bumple Tranfer Derivative Filtering Integral Windup Digital Implementation

53 Bumple Tranfer Prevent piky demand ignal to atuator when withing between different ontrol mode Our when withing from manual to automati Caue: ontroller output different from urrent ignal Solution: et referene to follow urrent plant output value alled a traking ontroller Derivative ation on PV alo help

54 Derivative Filtering Mot indutrial PID ontroller have ome filtering on derivative ation Prevent noiy meaurement giving noiy ontroller output Uually applied a: T αt α derivative time multiplier typially in range d d

55 Integral Windup Caued by integral ation and atuator aturation Can lead to intability

56 Graphial Repreentation Thi extra ontrol i a time delay - whih an detabilie the plant.

57 Anti-Windup Mehanim Analogue implementation: In pratie it i diffiult to et-up gain

58 Anti-Windup Mehanim Logi/digital implementation eaier: * * if Δu 0 0 if Δu<>0 Δu Anti-windup preent in mot indutrial ontroller

59 Digital Implementation Digital Implementation A ontinuou time PID ontroller i given by: Thi an be at into thi direte time form: where k i the ample number and τ i the ampling period dt de dt t e t e t u d t i p 0 ) ( ) ( ) ( [ ] ) ( ) ( ) ( ) ( ) ( 0 k e k e j e k e k u d k j i p τ τ

60 Digital Implementation Digital Implementation Thi may be re-written a: or: The hange in ontroller output (Δu ) i alulated and the inherent digital integrator form u (k) Called an inremental PID ontroller [ ] [ ] 2) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( k e k e k e k e k e k e k u k u d i p τ τ ) ( ) ( ) ( k u k u k u Δ

61 Why Doe It Work and i o Sueful? Ha high gain at LF for diturbane rejetion Ha only ingle pole o mall phae lag introdued With D term an have phae advane offet lag in plant (amplifie noie of oure). Can be tuned jut with plant data no model eential. No advaned theory o a tehniian may eaily try to tune. Eaily often related to phyial plant parameter. by pole plaement for example.

62 Conluding Remark. One of the mot popular ontrol tehnique. 2. Unrealiti high frequeny gain. 3. Simple to tune. 4. Motly for ingle input ingle output ytem. 5. Auto-tuning now ued for ome proee. 6. Integral wind-up a problem. 7. Derivative kik mut alo guard againt.