Classical PID Controller

7/25/2019 Classical PID Controller

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Goodwin, Graebe, Salgado, Prentice Hall 2000Chapter 6

Chapter 6

Classical PID Control


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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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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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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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P"% Structure

Consider the simple S"S@ control loop shown inAigure 6!/&

Aigure 6!/a"ic feed$ac% control loop


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The"tandard formP"% are&

Proportional onl&&

Proportional plu" Integral&

Proportional plu" derivative&

Proportional, integral and

derivative&


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Bn alternative"erie"#orm is&

?et another alternative #orm is the, so called,

parallel #orm&


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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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'/( 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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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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Table 6!/& 'iegler!Nicol" tuning u"ing teo"cillation metod


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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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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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)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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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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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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%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!/

Goodwin Graebe Salgado Prentice Hall 2000Chapter 6


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Aigure 6!& PID '!N "etting" applied to "erie""tructure tic% line/ and conventional"tructure tin line/



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@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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'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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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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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!


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!

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Classical PID Controller