classical pid controller
TRANSCRIPT
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Chapter 6
Classical PID Control
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
This chapter examines a particular control structurethat has become almost universall used in industrial
control! "t is based on a particular #ixed structurecontroller #amil, the so$called P"% controller #amil!These controllers have proven to be robust andextremel bene#icial in the control o# man
important applications!P"%stands #or& P 'Proportional(
I 'Integral(
D 'Derivative(
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Historical )ote
*arl #eedbac+ control devices implicitl orexplicitl used the ideas o# proportional, integral and
derivative action in their structures! However, it wasprobabl not until inors+-s wor+ on ship steering.published in /22, that rigorous theoreticalconsideration was given to P"% control!
This was the #irst mathematical treatment o# the tpeo# controller that is now used to control almost allindustrial processes!
. inors+ '/22( 1%irectional stabilit o# automaticall steeredbodies,
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
The Current Situation
%espite the abundance o# sophisticated tools, includingadvanced controllers, the Proportional, "ntegral,
%erivative 'P"% controller( is still the most widelused in modern industr, controlling more that 7 o#closed$loop industrial processes.
.8str9m :!;! < H=gglund T!H! /, 1)ew tuning methods #or P"%controllers,Proc. 3rd European Control Conference, p!246$62> and
?amamoto < Hashimoto //, 1Present status and #uture needs& The view
#rom ;apanese industr, Chemical Process Control, CPCIV, Proc. 4t Inter!national Conference on Cemical Proce"" Control, Texas, p!/$25!
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
P"% Structure
Consider the simple S"S@ control loop shown inAigure 6!/&
Aigure 6!/a"ic feed$ac% control loop
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
The"tandard formP"% are&
Proportional onl&&
Proportional plu" Integral&
Proportional plu" derivative&
Proportional, integral and
derivative&
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Bn alternative"erie"#orm is&
?et another alternative #orm is the, so called,
parallel #orm&
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Tuning o# P"% Controllers
ecause o# their widespread use in practice, wepresent below several methods #or tuning P"%
controllers! Bctuall these methods are Duite old anddate bac+ to the /0-s! )onetheless, the remain inwidespread use toda!
"n particular, we will stud! 'iegler!Nicol" ("cillation )etod 'iegler!Nicol" *eaction Curve )etod
Coen!Coon *eaction Curve )etod
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
'/( Eiegler$)ichols 'E$)( @scillationethod
This procedure is onl valid #or open loop stableplants and it is carried out through the #ollowing
steps Set the true plant under proportional control, with a
ver small gain!
"ncrease the gain until the loop starts oscillating! )ote
that linear oscillation is reDuired and that it should bedetected at the controller output!
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Fecord the controller critical gain +p+c and the
oscillation period o# the controller output, Pc!
Bdust the controller parameters according to Table6!/ 'net "lide(> there is some controvers regardingthe P"% parameteriIation #or which the E$) methodwas developed, but the version described here is, tothe best +nowledge o# the authors, applicable to the
parameteriIation o# standard #orm P"%!
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Table 6!/& 'iegler!Nicol" tuning u"ing teo"cillation metod
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
General Sstem
"# we consider a general plant o# the #orm&
then one can obtain the P"% settings via Eiegler$)ichols tuning #or di##erent values o# and 0! The
next plot shows the resultant closed loop stepresponses as a #unction o# the ratio
0>/(' 00
0
0 >+
=
"
e+
"-
"
!0=
G d i G b S l d P i H ll 2000
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Aigure 6!3& PI '!N tuned o"cillation metod/ controlloop for different value" of te ratio !
0
0
=
G d i G b S l d P ti H ll 2000C 6
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
)umerical *xample
Consider a plant with a model given b
Aind the parameters o# a P"% controller using theE$) oscillation method! @btain a graph o# theresponse to a unit step input re#erence and to a unitstep input disturbance!
G d i G b S l d P ti H ll 2000Ch t 6
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Solution
Bppling the procedure we #ind&
+c 5 and 0c 3!
Hence, #rom Table 6!/, we have
The closed loop response to a unit step in there#erence at t 0 and a unit step disturbance at t /0are shown in the next #igure!
Good in Graebe Salgado Prentice Hall 2000Ch t 6
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Aigure 6!4& *e"pon"e to "tep reference and "tepinput di"tur$ance
Goodwin Graebe Salgado Prentice Hall 2000Ch t 6
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
%i##erent P"% StructuresJ
B +e issue when appling P"% tuning rules 'such asEiegler$)ichols settings( is that o# which P"%
structure these settings are applied to!To obtain an appreciation o# these di##erences weevaluate the P"% control loop #or the same plant in*xample 6!/, but with the E$) settings applied to the
series structure, i!e! in the notation used in '6!2!(,we have
+" 4!5 I" /!5/ D" 0!4 s 0!/
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Aigure 6!& PID '!N "etting" applied to "erie""tructure tic% line/ and conventional"tructure tin line/
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
@bservation
"n the above example, it has not made muchdi##erence, to which #orm o# P"% the tuning rules are
applied! However, the reader is warned that this canma+e a di##erence in general!
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
'2( Feaction Curve asedethods
B lineariIed Duantitative version o# a simple plantcan be obtained with an open loop experiment, using
the #ollowing procedure&/! Kith the plant in open loop, ta+e the plant manuall to a
normal operating point! Sa that the plant output settles at&'t( &0#or a constant plant input u't( u0!
2! Bt an initial time, t0, appl a step change to the plant
input, #rom u0to u'ti" "ould $e in te range of 12 to
2 of full "cale(!
ContL!!!
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
3! Fecord the plant output until it settles to the new operatingpoint! Bssume ou obtain the curve shown on the nextslide! This curve is +nown as theproce"" reaction curve!
"n Aigure 6!6, m!s!t! stands #or maimum "lope tangent!
4! Compute the parameter model as #ollows
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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6
Aigure 6!6& Plant "tep re"pon"e
The suggested parameters are shown in Table 6!2!
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, , g ,Chapter 6
Table 6!2& 'iegler!Nicol" tuning u"ing te reactioncurve
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, , g ,Chapter 6
General Sstem Fevisited
Consider again the general plant&
The next slide shows the closed loop responsesresulting #rom Eiegler$)ichols Feaction Curvetuning #or di##erent values o#
/('
0
0
0 +=
"
e+"-
"
!0
=,
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, , g ,Chapter 6
Aigure 6!M& PI '!N tuned reaction curve metod/control loop
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C p e 6
@bservation
Ke see #rom the previous slide that the Eiegler$)ichols reaction curve tuning method is ver
sensitive to the ratio o# dela to time constant!
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p
'3( Cohen$Coon Feaction Curveethod
Cohen and Coon carried out #urther studies to #indcontroller settings which, based on the same model,
lead to a wea+er dependence on the ratio o# dela totime constant! Their suggested controller settingsare shown in Table 6!3&
Table 6!3& Coen!Coon tuning u"ing te reaction curve.
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p
General Sstem Fevisited
Consider again the general plant&
The next slide shows the closed loop responsesresulting #rom Cohen$Coon Feaction Curve tuning#or di##erent values o#
/('
0
0
0 +=
"
e+"-
"
!0
=
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p
Aigure 6!5& PI Coen!Coon tuned reaction curvemetod/ control loop
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Nead$lag Compensators
Closel related to P"% control is the idea o# lead$lagcompensation! The trans#er #unction o# these
compensators is o# the #orm&
"# /O 2, then this is a lead net5or%and when / 2,
this is a lag net5or%!
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Aigure 6!& Approimate #ode diagram" for leadnet5or%" 1612/
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@bservation
Ke see #rom the previous slide that the lead networ+gives phase advance at /L/ without an increase
in gain! Thus it plas a role similar to derivativeaction in P"%!
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Aigure 6!/0&Approimate #ode diagram" for lagnet5or%" 6121/
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@bservation
Ke see #rom the previous slide that the lag networ+gives low #reDuenc gain increase! Thus it plas a
role similar to integral action in P"%!
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"llustrative Case Stud&Di"tillation Column
P"% control is ver widel used in industr! "ndeed,one would we hard pressed to #ind loops that do not
use some variant o# this #orm o# control!Here we illustrate how P"% controllers can beutiliIed in a practical setting b brie#l examiningthe problem o# controlling a distillation column!
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*xample Sstem
The speci#ic sstem we stud here is a pilot scaleethanol$water distillation column! Photos o# the
column '5ic i" in te Department of CemicalEngineering at te 7niver"it& of S&dne&, Au"tralia(are shown on the next slide!
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Condenser Feed-point Reboiler
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Aigure 6!//& Etanol ! 5ater di"tillation column
B schematic diagram o# the column is given below&
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odel
B locall lineariIed model #or this sstem is as#ollows&
where
)ote that the units o# time here are minutes!
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%ecentraliIed P"% %esign
Ke will use two P"% controllers&
@ne connecting 8/ to 7/
The other, connecting 82 to 72!
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"n designing the two P"% controllers we will initiallignore the two trans#er #unctions -/2and -2/! This
leads to two separate 'and non$interacting( S"S@sstems! The resultant controllers are&
Ke see that these are o# P" tpe!
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Simulations
Ke simulate the per#ormance o# the sstem with thetwo decentraliIed P"% controllers! B two unit step in
re#erence / is applied at time t 0 and a one unitstep is applied in re#erence 2 at time t 20! Thesstem was simulated with the true coupling 'i!e!including -/2 and -2/(! The results are shown on
the next slide!
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9igure :.1; Simulation re"ult" for PI control of
di"tillation column
"t can be seen #rom the #igure that the P"% controllers give Duite acceptableper#ormance on this problem! However, the #igure also shows something that isver common in practical applications $ namel the two loops interact i!e! achange in re#erence r/ not onl causes a change in &/'as reDuired( but alsoinduces a transient in &2! Similarl a change in the re#erence r2 causes a changein &2'as reDuired( and also induces a change in&/! "n this particular example,these interactions are probabl su##icientl small to be acceptable! Thus, incommon with the maorit o# industrial problems, we have #ound that two simpleP"% 'actuall P" in this case( controllers give Duite acceptable per#ormance #or this
problem! Nater we will see how to design a #ull multivariable controller #or this
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Summar
P" and P"% controllers are widel used inindustrial control!
Arom a modern perspective, a P"% controller issimpl a controller o# 'up to second order(containing an integrator! Historicall, however,P"% controllers were tuned in terms o# their P, I
and Dterms!
"t has been empiricall #ound that the P"%structure o#ten has su##icient #lexibilit to ield
excellent results in man applications!
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The basic term is the proportional term, P, whichcauses a corrective control actuation proportional
to the error! The integral term, Igives a correction proportional
to the integral o# the error! This has the positive#eature o# ultimatel ensuring that su##icient
control e##ort is applied to reduce the trac+ingerror to Iero! However, integral action tends tohave a destabiliIing e##ect due to the increased
phase shi#t!
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The derivative term, D, gives a predictivecapabilit ielding a control action proportional to
the rate o# change o# the error! This tends to havea stabiliIing e##ect but o#ten leads to large controlmovements!
Qarious empirical tuning methods can be used to
determine the P"% parameters #or a givenapplication! The should be considered as a #irstguess in a search procedure!
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Bttention should also be paid to the P"% structure!
Sstematic model$based procedures #or P"%
controllers will be covered in later chapters! B controller structure that is closel related to P"%
is a lead$lag networ+! The lead component acts
li+e D and the lag acts li+e I!
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Rse#ul Sites
The #ollowing internet sites give valuablein#ormation about PNC-s&
www!plcs!net
www!plcopen!org
Aor example, the next slide lists the manu#acturersDuoted at the above sites!
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BBl#a NavalBllen$radleBNST@LCegelecBromatButomation %irectLPNC %irectL:ooL
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HimaHitachiHonewellHorner *lectric
"dec"%TLCutler Hammer
;etter gmbh
:eence:irchner So#t:loc+ner$oeller:ooLButomation %irectLPNC %irect
icroconsultantsitsubishiodiconLGouldoore Products
@mron@pto22
PilIPNC %irectL:ooLButomation %irect
FelianceFoc+well ButomationFoc+well So#tware
Cont
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SB"B$urgessSchleicherSchneider ButomationSiemensSigmate+So#tPNCLTele$%en+enSDuare %
Tele$%en+enLSo#t PNC
TelemecaniDueToshibaTriangle Fesearch
E$Korld
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Bdditional )otes& Eample"commerciall& availa$le PID controller"
"n the next #ew slides we brie#l describe some o# thecommerciall available P"% controllers! There are, o#course, a great man such controllers! The examples we
have chosen are selected randoml to illustrate the +indso# things that are available!
There are several variations in algorithms, with the threemain tpes being series, parallel and ideal #orm!
Some controllers are con#igured to act on the error andsome appl the Dterm to the #eedbac+ onl! ost havespecial #eatures to deal with saturation and slew ratelimits on the plant input! '>i" topic i" di"cu""ed inCapter 11(!
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Bllen radle PNC$ P"% loc+
The P"% #unction in this controller is an outputinstruction that must be executed periodicall atspeci#ied intervals determined b the external code!
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There are 4 di##erent #orms o# the controller eDuation&
'/( Kith derivative action on the output
'2( Kith derivative action in the error
$ia"&e+u"
">
">c d>d
i+
+
+=
+/6
/
//
$ia"e+u"
">
">c d>d
i+
+
+=
+/6
/
//
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'3( Similar to '/( but with di##erent gains
'4( Similar to '2( but with di##erent gains
$ia"&e+up+"d+
di "+
"
+
p +
+
+= +/6/
$ia"e+u
p+
"d+di "+
"
+p +
++=+
/6/
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G* 50 P"%BS loc+
The G* #amil o# PNC-s have a P"% bloc+ whichmust be executed periodicall at speci#ied intervalsdetermined b the external code! This #unction isimplemented b a velocit tpe algorithm, with thecontroller being converted to an absolute controller
b adding the previous output value! Thus the
controller output is o# the #orm&( ) ( )
/002 2//
/
++++= tttctcttctt
eeeDeIeePuu
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The reader can convert the above discreteimplementation to approximate continuous time#orm b noting that
where is the sampling interval! Thus the controllaw is roughl eDuivalent to the #ollowing&
dtdett ee
/
2
2
22/2
dt
edttt eee
+
+
+= e"DeI
"eP"u cc
c2
/00
/
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Two comments regarding this eDuation are&
'/( uch more will be said on the relationship
between and and Chapters /2, /3and /4!
'2( )ote that to achieve approximatel the same
per#ormance with di##erent sampling rates, Icand Dc need to be scaled!
= /ttee
e dtde
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?o+ogawa %CS Aunction loc+
This %CS o##ers nine tpes o# regulator control bloc+s $ PID
Sampling PI
PID 5it $atc "5itc
t5o po"ition on
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The basic P"% controller has variations! The main3 structures being&
'/(
'2(
'3(
)ote that the parameters in these controllers are
'rougl&( invariant w!r!t! !
( ) ( ) ++++= 2/// 2 tttdti
ttp"tt eee>e>ee++uu
( ) ( )
+
+++= 2/// 2 ttt
dt
ittp"tt &&&
>e
>ee++uu
( ) ( )
+
+++= 2/// 2 ttt
dt
ittp"tt &&&
>e
>&&++uu
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Bdditional #eatures o# these controllers are Selection o# the tpe o# eDuation, including the #acilit to invert
the output>
Butomatic or manual mode selection, with an option #or trac+ing> umpless trans#er>
Separate input and output limits, including rate and absolute limits>
Bdditional non$linear scaling o# the output>
"ntegrator anti$windup 'called re"et!limiter(> Selectable execution interval as a multiple o# scan time>
Aeed #orward, either to the #eedbac+ or controller output>
B dead$band on the controller output!
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Ai h C t l 4/: G
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Aisher Controls 4/: GaugePressure Controller
This pressure controller is a pneumatic device, withmechanical lin+ages, that is coupled to a controlvalve, speci#icall #or providing pressure regulation!@ne advantage o# pneumatic controllers is that, asthe are powered b instrument air, there is noelectrical power emploed!
The controller can be con#igured as a P, P" or P"%controller, which can be con#igured as direct orreverse acting! Aeatures such as anti$windup areoptional!