Characterization of symmetric send-on-delta PI controllers

ContentslistsavailableatSciVerseScienceDirect

Journal

of

Process

Control

jo u r n al h om ep a ge :w w w . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

Characterization

of

symmetric

send-on-delta

PI

controllers

Manuel

Beschi

a

_{, Sebastián}

_Dormido

b

_{, José}

_Sanchez

b

_{, Antonio}

_Visioli

a,∗ a_{DipartimentodiIngegneriadell’Informazione,UniversityofBrescia,Italy}

b_{DepartamentodeInformáticayAutomática,UNED,Spain}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received9May2012 Receivedinrevisedform 11September2012 Accepted12September2012 Available online 22 October 2012 Keywords:

Event-basedPIcontrollers Send-on-deltasampling Stability

a

b

s

t

r

a

c

t

Anewevent-basedproportional–integralcontroller,basedonaspecificsend-on-deltasamplingstrategy,

isanalyzedinthispaper.Inparticular,necessaryandsufficientconditionsonthecontrollerparameters

fortheexistenceofequilibriumpointswithoutlimitcyclesaregivenforafirst-order-plus-dead-time

process.Theseconditionscanbeusefullyexploitedforthetuningofthecontroller,thusmakingtheoverall

designeasier.Practicalissuesrelatedtothecontrollerimplementationarealsoaddressed.Simulationand

experimentalresultsareprovidedasillustrativeexamples.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Proportional–integral-derivative(PID)controllersarethemost employed controllersin industry becauseof their capabilityto provideasatisfactoryperformanceformanyprocesseswitha rela-tivelyeasydesign.Theiruseisalsomadeeasierbythepresenceof alargenumberoftuningrulesthatareavailable[1].However,the advancementofthetechnologyinindustrialplantsrequiresnew developmentsinPIDcontrol.Oneofthemostsignificantrecent issues istheintroductionof wireless sensorsandactuators [2], whichposes constraintsonthecommunicationrateinorderto minimizethepowerconsumption(andthereforetoincreasethe batterylife)andtherisk oflostdataand stochastictime delays [3–5].Inthiscontext,twoofthemostconvenientstrategiesthat canbeemployedtoreduce thecommunication loadare event-basedcontrolandself-triggeredcontrol,wherethereductionof communicationscanbeobtainedattheexpenseofasmall steady-stateerror,whichoftenisnotofmainconcerninpracticalcases. Intheformer,thecommunicationisdoneonlywhenalogic condi-tionbecomestrue,whileinthelatterthecontrollerhastopredict thenexttime instantwhen thelogic conditionwillbeverified [6].During thelastyears many researchershaveaddressedthe self-triggeredapproach(see, forexample [7–9])and the event-basedsamplingandcontrolapproach(see,forexample[10–12]),

∗ Correspondingauthor.Tel.:+390303715460;fax:+39030380014. E-mailaddresses:manuel.beschi@ing.unibs.it(M.Beschi),

sdormido@dia.uned.es(S.Dormido),jsanchez@dia.uned.es(J.Sanchez), antonio.visioli@ing.unibs.it(A.Visioli).

inparticularinthecontextofPIDcontrollers(see[13]forthe self-triggeredstrategiesand[14–21]for theevent-basedstrategies). Oneofthemostimportantadvantagesoftheself-triggered strate-giesis thepossibility ofusing themaster-slaveprotocol in the communicationsbetweencontrolagentsandinthesetechniques itis notnecessarytocontinuouslymonitorthesystemoutputs. However,thedisturbancerejectiontaskismorecomplicatedand theperformanceoftheseapproachesdependsonthemodelofthe system.Conversely,inanevent-basedcontrolstrategy,whena con-trolagentdetectsanevent,itsendstherelatedinformation,viathe communicationmedium,toanotheragent.In thisway,the dis-turbancerejectiontaskstartsatthetimeinstancewhenthelogic conditionisverifiedandthetechniquecanbemodel-free.

Inthis context,manylogicalconditionshavebeenproposed. Oneofthemostemployedistheso-calledsend-on-delta(SOD) sampling(alsoknownasdeadbandsampling[22]orlevelcrossing sampling[23])wherethemeasuredvalueoftheprocessvariableis senttothecontrollerwhenthecontrolerror(orsomefunctionof it)crossespredefinedquantizationlevels[24].

However,ithastoberecognizedthatinevent-basedcontrolthe eventsoccurasynchronouslyandthereforethetuningofthePID controllerparametersisingeneralmorechallenging,asthetiming oftheeventsinfluencesthesystemperformanceandlimitcycles mayarise(notethatthepresenceoflimitcyclesisatypicalproblem ingeneralevent-basedcontrolsystems[25]andsee,forexample [26]wheresufficientconditionsonthecontrollerparametersfor theexistenceofequilibriumpointsaregivenforaSOD-PIcontrol system).Further,inadditiontothePIDgains,thereareingeneral otherparameters(thresholdvalues)employedinthecontrol algo-rithmthathavetobetuned,thusmakingtheoverallcontroldesign 0959-1524/$–seefrontmatter © 2012 Elsevier Ltd. All rights reserved.

morecomplex.Indeed,thetuningofaPIDcontrollerwith dead-bandsamplinghasnotbeenexplicitlyaddressedintheliterature untilnow,atleasttotheauthors’knowledge.

Inthispaperweproposeanevent-basedstrategywhichresults fromamodificationoftheSODtechnique.Inparticular,the sam-pledsignalisquantizedbyaquantitymultipleofsothatthe relationshipbetweentheinputandoutputoftheevent-generator blockissymmetricwithrespecttotheorigin.Thus,wecallthis strategysymmetricsend-on-delta(SSOD)sampling.Inthisway,the valueofthesampledsignalisindependentontheinitialconditions. Wewillexploitthispropertytodeterminenecessaryconditions onsysteminstabilityandnecessaryandsufficientconditionson thecontrollerparametersfortheexistenceofequilibriumpoints withoutlimitcycleswillbegivenforafirst-order-plus-dead-time (FOPDT)process.Althoughthiscanbeconsideredasaspecialcase, itissignificantfromapracticalpointofviewasmanyindustrial processescanbemodeledeffectivelyinthisway.Event-basedP,I, andPIcontrollersareconsidered.

Thecase where theSSOD strategy is applied tothe control variable(thusreducingthenumberoftransmissionsbetweenthe controllerandtheactuator)isalsoaddressed.Bothcases(denoted respectivelyasSSOD-PIandPI-SSOD)aretreatedconvenientlyina unifiedframework.Itisbelievedthattheprovidedstability con-ditionscan be usefully employed for the tuning of the overall controller.Itisalsoshownthatthechoiceoftheparameterdoes notinfluencethestabilityofthesystem(seeSection3.2)and there-forecanbeselectedjustinordertohandlethetrade-offbetween thereductionofthedesirednumberofeventsandthereductionof thesteady-stateerror.Thepaperisorganizedasfollows.InSection 2theevent-basedcontrolstrategiesarepresented.InSection3, thetwostabilityandlimitcyclesproblemsareinvestigated. Neces-saryandsufficientconditionsforthepresenceofequilibriumpoints withouttheoccurrenceoflimitcyclesandnecessaryconditions onthesysteminstability(andthereforesufficientconditionsfor thesystemstability)aregiven.InSection4practicalissuesonthe controllerimplementationarepresented. Simulationresultsare showninSection5whileexperimentalresultsrelatedtoalevel controlproblemarepresentedinSection6.Conclusionsaredrawn inSection7.

2. Controlarchitecture

Inevent-basedcontrolstrategies,thecontrollercanbedivided intofourlogicalblocks:thesensorunit,thecontrolunit,the actu-atorunitandagovernor(forshortSU,CU,AUandG,respectively), asshowninFig.1.Theunitsandtheirtaskscanbedescribedas follows:

• Thesensorunitiscomposedofthesensoranditson-board intel-ligence.Itstaskistomeasuretheprocessoutputandtocalculate theerrorbetweenthemeasuredsignalandaconstantset-point valuereceivedfromthegovernor.

CU

SU AU

G

Plant

Fig.1.Schemeofagenericeventbasedcontrolstrategies.Thedashedarrows indi-catethepossibilityofanevent-triggereddatatransmission.

• Thecontrolunitimplementsthecontrolalgorithm,which deter-minesthecontrolactionbytakingintoaccountthelastreceived samplederrorandsendsittotheAU.

• TheactuatorunitreceivesthecontrolactionsignalfromtheCU andappliesittotheactuator.

• Thegovernor,which,inpractice,canbeimplementedtogether withoneoftheprevioustwoblocks,receivesthedesired set-pointvaluefromauserinterfaceorfromahigherhierarchical controller,andsendsittothesensorunit.

Theseblockscanbeimplementedinauniquemachineorintwo ormorephysicalentities.Inthislastcase,thedatahastobesent fromonetoeachotheronanetwork.Itisclearthatthe commu-nicationbetweentwoentitiesimpliesmoreeffortsthanthedata exchangingintoasinglemachine,especiallywhentheyare battery-powered.Forthisreason,itisrecommendedtouseevent-triggered dataexchangingforallthesignalswhicharesentbetweentwo machinesandnormaltime-drivensamplingforthedatawhichare elaboratedbyanuniquemachine.

Inthispaper,weconsidertwospecialcaseswheretwoofthe threecontrolagents(namely,SU,CUandAU)areplacedina phys-icalentityandtheotheroneislocatedinanothermachine.Wedo notaddressthecasewhereSUandAUareinthesamemachine, becauseinthiscaseitissufficienttoimplementalsotheCUtaskin thismachineinordertoobtainastandardcontroller.Theremaining casesare:SUseparatedfromCUandAU(whichareinthesame machine)and AU separatedfromSU and CU (whichare inthe samemachine).Inbothsituations, onlyanevent-triggereddata exchanginginthecontrolloopisrequired.Thecommunications betweenthegovernorandtheotherunits(consideringonlythe transmissionsconcerningthecontrolaspects)aredoneonlywhen theset-pointsignalchanges.

2.1. Symmetricsend-on-deltatriggering

Inthispaper,theevent-triggereddataexchangingisdoneby applyingaspecialcaseofthesend-on-deltasamplingmethod(see [4,15]),whichcanbeseenalsoasageneralizationofarelaywith hysteresis.

Wecallthistechniquesymmetricsend-on-delta(SSOD) samp-ling.Denoteas

v

(t)theinputsignaltothesamplingblockandas

v

∗(t)thesampledoutputsignal,whichcanassumeonlyvalues mul-tipleofapredefinedthresholdmultipliedbyagainˇ>0,namely

v

∗(t)=jˇwithj∈Z.Thesampledsignalchangesitsvaluetothe upperquantizationlevelwhentheinputsignal

v

(t)increasesmore than,ortothelowerquantizationlevelwhen

v

(t)decreasesmore than.Thisbehaviorcanbemathematicallydescribedas:

v

∗(t)=ssod(

v

(t);,ˇ) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

(i+1)ˇ if

v

(t)≥(i+1)and

v

∗(t−)=iˇ iˇ if

v

(t)∈[(i−1),(i+1)]and

v

∗(t−)=iˇ (i−1)ˇ if

v

(t)≤(i−1)and

v

∗(t−)=iˇ

. (1)

Therelationshipbetween

v

(t)and

v

∗(t)canbeconsideredasa gen-eralizationofarelaywithhysteresis,wherethereareaninfinite number ofthresholds [23],as shown inFig. 2.The mechanism canalsobeanalyzedbymeansofastate-machinerepresentation, wherejisthestatenumber,

v

(t)≥(j+1)istheconditiontojump totheupperstatej+1and

v

(t)≤(j−1)istheconditiontojump tothelowerstatej−1,asshowninFig.3.

TheSSODalgorithm,describedin(1),canberewrittenas:

v

∗(t)=ˇssod



_v

_(t)

;1,1



v t

( )

*

v t

( )

Δβ

−2Δ

−Δ

−3Δβ

β

β

2Δβ

−3Δ −2Δ

−Δ

Δ

2Δ

3Δ

3Δβ

β

Fig.2. Relationshipbetweenv(t)andv∗_(t).

Thisrepresentationsuggeststhatthestabilitypropertiesofa sys-temwhichcontainsaSSODeventgeneratordonotdependonthe parametersandˇ.Infact,itisalwayspossibletodoavariable changeandtoreducetheSSODblocktoitsnormalizedversion ssod(

v

(t);1,1).

Alsointhestandardsend-on-delta(SOD)technique,itis pos-sibletodefineastatemachinerepresentation,butinthiscasethe valueassumedby

v

∗inthestatejisequalto(j+q)ˇ(see[26]), whereq∈[0,)isanunquantizablequantitywhichdependson theinitialvalueof

v

(t).Thus,withtheSSODtechniquethe sys-temalwayshasastatewith

v

∗=0andthisfacthasanimportant influencetothecontrolledsystembehavior,asshowninSection3.

2.2. Modelingthecommunicationsdelaysandthepacketlosses In the SSOD (or SOD) sampling communication delays and packetlosingeffectscanbemodeledasadisturbanceaddedtothe SSOD(orSOD)output

v

(t),asshowninFig.4.Howthisadditive disturbanceaffectstheperformancedependsonitspositioninthe controlloop.Infact,iftheSSODsamplingisappliedontheerror signal(asinSSOD-PItechnique)itispossibletohaveadeviation betweenthelastreceivedoutputandthetrueSSODoutput.Away toavoidthissituationistoforcethesensortoresendthedatato thecontrollerifastatej /= 0iskeptformorethanamaximumtime intervaltmax.

2.3. PIcontroller

Thecontrollerusedinthisworkisa(discretizedversionof)a continuoustimePIcontroller,namely:

C(s)=Kp+Ki

s (3)

whereKp≥0istheproportionalgainandKi≥0istheintegralgain.

*

v = j−1

( )

Δ

_{v =j}

*

Δ

_{v = j+1}

*

_{( )}

Δ

( )

Δ

Δ

( )

Δ

( )

Δ

Δ

( )

Δ

Δ

( )

( )

Δ

v> j−

1

v> j

v> j+

1

v> j+2

v< j+1

v< j

v< j−1

v< j−2

j

j+1

j−1

Fig.3. Statemachinerepresentationofthesymmetricsend-on-deltasampling tech-nique.

Fig.4.Modelizationofcommunicationdelaysandpacketlossesasadditive disturb-ance,wherev∗_{(t)isthesentSSODoutput,v}∗

D(t)isthereceivedsignalandDNETisthe disturbanceduetothecommunicationsdelaysandthepacketlosses.

Remark 1. Because thecontroller is implementedin a single

machine,itispossibletoimplementastandardanti-windup tech-niquewithoutadditionalefforts.

2.4. SSOD-PIcontroller

Asexplainedbefore,weconsidertwodifferentarchitectures, whichpresentascommoncharacteristicthepresenceofonlytwo event-triggereddataexchanging.Thefirstistheset-pointvalue sending,whichisnotacomplextaskfromacontrolpointofview, thesecondisimplementedwiththeSSODtechnique.

Inthefirstarchitecture,showninFig.5,theSUis locatedin amachinewhileCUandAUareplacedinanotherphysicalentity (forexample,thecommunicationbetweenthetwo components couldbewireless).WecallthisarchitectureSSOD-PIcontroller, becausetheSSODblockisplacedbeforethePIcontrollerinthe controlloop.Notethatthecontrollercomputesthecontrolaction ataregularsamplingratebytakingintoaccountthelastreceived samplederror.

Remark2. Thisarchitectureisveryinterestingbecause,ina

net-worked control system,the sensor unit can be poweredusing abattery(thus,areductionofthecommunicationscanincrease thebatterylife).Conversely,theAUnormallyrequiresanexternal powersupply,thereforethepowerconsumptionreductionisnota criticalissueinthiscontrolagent.

Remark 3. In many control strategies thesensor hastosend

theerrorusinganevent-triggeredalgorithm.Thisfactcanbean important constraintwhen theSU cannotreceive theset-point information,whichissenttotheCU.WhentheSSODmethodis used,awaytosolvethisproblemistoapplytheSSODalgorithm totheprocessoutputandtosendit tothecontroller.Withthis data,theCUelaboratestheerrorsignalandroundsittothe near-estmultipleofˇ.Notethat,asmentionedinSection1,asmall steady-stateerrordoesnotconstituteaharddesignconstraintsfor manyindustrialprocesses.

P(s) ZOH e SSOD e * r Governor SU y D u C(s) CU and AU + − + +

Fig.5. ControlschemeoftheSSOD-PIcontrolledsystem.Thedashedarrowsindicate datasendingviathecommunicationmedium.

ZOH e r C(s) u SSOD u * y P(s) SU and CU AU D Governor + + + −

Fig.6.ControlschemeofthePI-SSODcontrolledsystem.Thedashedarrowsindicate datasendingviacommunicationmedium.

2.5. PI-SSODcontroller

Inthesecondarchitecture,showninFig.6,theSUandtheCU arelocatedinthesamemachine,whiletheAUisplacedinanother entity.ThissolutioniscalledforshortPI-SSODcontrollerbecause thecontrollerisbeforetheSSODblockinthecontrolloop.Notethat theAUholdsthelastreceivedcontrolactionvalueuntilthenext dataexchanging.

Remark4. Thecommunicationdelaysandthepacketlosingeffect

canbemodeledasdisturbances(see Section2.2).Usingthe PI-SSODcontroller,thisdisturbancecanbeseenasapartoftheload disturbance.

3. StabilityandlimitcyclesanalysisforaFOPDTprocess

Inthissection,thestabilitypropertiesofthetwoevent-based controlsystemsarestudied.Inparticular,theconsideredprocess isaFOPDTsystem,describedbythefollowingtransferfunction: P(s)= K

s+1e

−Ls ₍₄₎

whereKistheprocessgain(whichisassumedtobepositive with-outlossofgenerality),>0isthetimeconstantandL≥0isthe apparentdeadtime.Then,wecanwrite

Y(s)= K s+1e −Ls



_U(s)+D s



(5) whereY(s)istheLaplacetransformoftheprocessoutputy(t),U(s) istheLaplacetransform ofthe controlaction u(t)and Disthe amplitudeofaconstantloaddisturbance.

Itisimportanttonoticethat,becauseofthenonlinearnatureof thecontrollers,thesystemcanreachanequilibriumpoint,orcan presentalimitcyclearoundanequilibriumpointorcanbeunstable. Ingeneral,thebehaviorofeachequilibriumpointcanbedifferent. 3.1. Normalizationofthesystems

Inthissubsection,theSSOD-PIandPI-SSODcontrollersystems aremathematicallydescribed.Then,weintroduceanormalization inorder toshowthat thetwo systemshave thesamestability propertieswhentherearenotexogenoussignals.

TheSSOD-PIcontrolledsystemcanbedescribedinthe state-spaceformas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=− 1 x1(t)+ K (x2(t)+Kpe∗(t)+D) ˙x2(t)=Kie∗(t) y(t)=x1(t) e(t)=r−y(t) e∗(t)=ssod(e(t−L);,ˇ) x1(0)=x10=y0 x2(0)=x20=IE0 (6)

whereristheconstantreferencesignal,x1(t)istheprocessstate,

x2(t)istheintegraltermofthePIcontroller,y0istheinitialoutput

value,andIE0istheinitialvalueoftheintegralterm.

Itisimportanttonotethattheconstantloaddisturbancecanbe groupedwithx20withoutlossofgenerality,thereforetheSSOD-PI

controllercanexactlycompensateaconstantloaddisturbance. ThePI-SSODcontrolledsystemcanbedescribedinthe state-spaceformas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=− 1 x1(t)+ K (u∗(t)+D) ˙x2(t)= Kp  (x1(t)−K(u∗(t)+D))+Ki(r−x1(t)) y(t)=x1(t) u(t)=x2(t) u∗(t)=ssod(u(t−L);,ˇ) x1(0)=y0 x2(0)=x20=Kp(r−y0)+KiIE0 (7)

whereristheconstantreferencesignal,x1(t)istheprocessstate,

x2(t)isthecontrolaction,y0istheinitialoutputvalueandx20is

theinitialvalueofthePIcontrolaction.

By studying theequilibrium points of (7) and remembering thatu*_{(t)=jˇ,itistrivialtofindthatanequilibriumexistonly}

ifr=K(jˇ+D).Thus,thePI-SSODcontrollercannotexactly com-pensatea genericconstant loaddisturbance. In Section3.3,we presentamethodtocompensatethe“unquantizable”partofthe loaddisturbance,thereforeforthestabilityanalysisweconsider thattheexogenoussignalsareequaltozerointhePI-SSOD con-trolledscheme.

Thesystem(6)canbenormalizedin(8)bychoosing: ˜t =(t/), ˜y(˜t)=((y(t)−r)/),˜

v

∗(˜t)=(e∗(t)/), ˜x1(˜t)=((x1(t)−Kx2(t))/),

˜x2(˜t)=(K(x2(t)/)),K1=KKpˇ,K2=KKiˇ,a=K1−K2, ˜x10=((y0−

r−K(x20+D))/), ˜x20=((K(x20+D))/) and l=(L/) which is

callednormalizeddeadtime.

Also the system (7) can be normalized in (8) by choos-ing: ˜t =(t/), ˜y(˜t)=((y(t)−r)/),

v

˜∗(˜t)=(Ku∗(t)/), K1=KKpˇ,

K2=KKiˇ, a=K1−K2, ˜x1(˜t)=−(ax1(t)/), ˜x2(˜t)=−((ax1(t)+

x2(t))/), ˜x10=−(ay0/), ˜x20=−((ay0+x20)/) and l=(L/).

Thus,forthetwoevent-basedcontroller strategieswehavethe followingnormalizedsystem(seeAppendixA):

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙˜x1(˜t)=−˜x1(˜t)+a˜

v

∗(˜t) ˙˜x2(˜t)=K2˜

v

∗(˜t) ˜

v

(˜t)= ˜x2(˜t)+ ˜x1(˜t) ˜

v

∗(˜t)=ssod(−˜

v

(˜t−l);1,1) ˜x1(0)= ˜x10 ˜x2(0)= ˜x20 (8)

Thenormalizedsystem(8)representsboththetwopresented con-trollers,andthereforetheirstabilitypropertiesarethesame,with theexception of thedisturbanceexact compensation. The nor-malizedsystemschemeisshowninFig.7.TheSSOD-PIandthe PI-SSODcasesandtheirnormalizationformulaearesummarizedin Table1.

3.2. Stabilityanalysisofthenormalizedsystem

Inthissubsection,wedemonstratethatitispossibletofinda regionoftheparameterspaceK1−K2wherethesystemiscertainly

marginallyorasymptoticallystableforalltheequilibriumpoints. Thisregioncanthenbedividedintotworegions:thefirstwherethe

Fig.7. Schemeofthenormalizedsystem.

systemtrajectorycantendtoalimitcycleandthesecondwhere thesystemsurelydoesnotpresentalimitcycle.

Inparticular,weprovethatifthesystem(8)isunstablethen alsothecorrespondingcontinuoustimePI-controlledsystemwith thesameproportionalandintegralgainsisunstable,thereforethe instabilityregionofSSOD-PIandPI-SSODcontrollersisthesame thatthecontinuous-timePIcontroller.

Toprovethisnecessarycondition,wehavefirsttodemonstrate thatifthesignal

v

˜(˜t)diverges,thenalsoitsderivativedivergesfor allthepositiveproportionalandintegralgains(whicharetheonly oneswithaphysicalmeaning).

Proposition1. Inthesystem(8),forallthepositiveproportional

gains,iftheabsolutevalueof

v

˜(˜t)tendstoinfinitythenalsotheabsolute valueofitsderivativetendstoinfinity.

Proof. Theproofconsistsofdemonstratingthefollowing

impli-cation: lim

˜t→∞|˜

v

(˜t)|=∞⇒˜t→∞lim| ˙˜

v

(˜t)|=∞

Bycontradiction,wesupposethat: lim ˜t→∞|˜

v

(˜t)|=∞ and

v

˙˜(˜t)∈[˛,ˇ], with˛<ˇ, ˛,ˇ∈R, therefore, lim t→∞ ˙˜

v

(˜t) ˜

v

(˜t)=0. (9)

Itisimportanttonotethatifthederivative ˙˜

v

(˜t)islimited,then thedifferencebetween

v

˜∗(˜t)and˜

v

(˜t)isaboundedsignal,called f(˜t).Infact,thedifferencesbetween

v

˜(˜t)and˜

v

(˜t−l)andbetween ˜

v

∗(˜t)and−˜

v

(˜t −l)arebounded.Thus,using(8)and(9)and

v

˜∗(˜t)= −˜

v

(˜t)+f(˜t)=−(˜x2(˜t)+ ˜x1(˜t))+f(˜t),itispossibletowrite: lim ˜t→∞ ˙˜x2(˜t)+ ˙˜x1(˜t) ˜x2(˜t)+ ˜x1(˜t) =0, lim ˜t→∞ (K2+a)(f(˜t)− ˜x2(˜t)− ˜x1(˜t))− ˜x1(˜t) ˜x2(˜t)+ ˜x1(˜t) = 0 Table1

SummaryoftheSSOD-PIandPI-SSODcases.

SSOD-PIcontroller(Fig.5) PI-SSODcontroller(Fig.6)

Systemequations

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=−1 x1(t)+ K (x2(t)+Kpe ∗_(t)+D) ˙x2=Kie∗_(t) y(t)=x1(t) e(t)=r−y(t) e∗(t)=ssod(e(t−L);,ˇ) x1(0)=y0 x2(0)=x20

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=−1 x1(t)+ K (u ∗_(t)+D) ˙x2(t)=Kp (x1(t)−K(u∗(t)+D))+Ki(r−x1(t)) y(t)=x1(t) u(t)=x2(t) u∗(t)=ssod(u(t−L);,ˇ) x1(0)=y0 x2(0)=x20 Normalizedvariables ˜t =t  ˜y(˜t)= y(t)−r  l=L  ˜v ∗_(˜t)=e∗(t)  K1=KKpˇ ˜x1(˜t)=x1(t)−Kx2(t)  K2=KKiˇ ˜x2(˜t)=Kx2(t)  a=K1−K2 ˜x10=y0−r−K(x20+D)  ˜x20=K(x20+D)  ˜t= t  ˜y(˜t)= y(t)−r  l=L  v˜ ∗_(˜t)=Ku∗(t)  K1=KKpˇ ˜x1(˜t)=−ax1(t)  K2=KKiˇ ˜x2(˜t)=−ax1(t)+x2(t)  a=K1−K2 ˜x10=−ay0  ˜x20=−ay0+x20  Normalized

_⎧

scheme(Fig.7)

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙˜x1(˜t)=−˜x1(˜t)+a˜v∗(˜t) ˙˜x2(˜t)=K2˜v∗(˜t) ˜ v(˜t)= ˜x2(˜t)+ ˜x1(˜t) ˜ v∗(˜t)=ssod(−˜v(˜t−l);1,1) ˜x1(0)= ˜x10 ˜x2(0)= ˜x20

andfinally, lim t→∞ ˜x1(˜t) ˜x2(˜t)+ ˜x1(˜t)=−(a+K 2) (10)

wheref(˜t)/(˜x2(˜t)+ ˜x1(˜t))tendstozero.

From Eq. (10) we can state that ˜x1(˜t) and −˜

v

(˜t) diverge in

thesameway.Thus, ˜x2(˜t)=

v

˜(˜t)− ˜x1(˜t)candivergeas2˜

v

(˜t)orbe

bounded,andtherefore: lim

t→∞

˙˜x2(˜t)

˜

v

(˜t) =K2=0

whichisanabsurdifK2 /=0,otherwisewehavethat ˜x2(˜t)is

con-stantandequalto ˜x20,andtherefore(10)becomes:

lim

t→∞

˜x1(˜t)

˜x20+ ˜x1(˜t)

=1=−a

whichisanabsurdbecause,ifK2=0,a=K1≥0.

UsingProposition1itispossibletostatethefollowingnecessary condition.

Proposition2. If theclosed-loopsystem(8)isunstablethenthe

samesystemcontrolledbyacontinuous-timePIcontrollerwiththe samevalueofK1andK2isalsounstable.

Proof. Thetimeinterval˜t betweentwoeventscanbecalculated

astheratiobetween1andthemeanvalueofthederivativeof

v

˜(˜t), called ˙˜

v

,intheinterval,infact:

1=







+t  ˙˜

v

(˜t)d˜t







=| ˙˜

v

|˜t.

FromProposition1weknowthatiftheabsolutevalueoftheoutput tendstoinfinityalsotheabsolutevalueofitsderivativediverges toinfinity.Thus,thetimeinterval ˜tdecreasestozeroand (8) becomesacontinuous-timecontroller,whichprovesthe proposi-tion.

Thefollowingpropositionsdealwiththeparametersregionfor whichtherearenotsurelylimitcyclesandthesystemconvergesto theequilibriumstate.Theideaoftheproofconsistsinfindingthe parametersforwhichifthesystemisinthestatej=jeq±n,where

jeqistheequilibriumstateandn>0,itcannotevolvetothestate

j=jeq∓nandthereforeithastotendtotheequilibriumstate.The

nextpropositionaddressesthecasesofIandPIcontrollers.The goalofthepropositionistofindthevaluesofK1–K2parametersfor

whichtherearenotperiodictrajectorieswhichinvolvemorethan theequilibriumstate.

Proposition3. Inasystemdescribedby(8),withK1≥0andK2>0,

thenlimitcyclescannotoccurifthenormalizedgainsK1andK2are

insidetheregionofthefirstquadrantdelimitedbythefollowing para-metriccurve:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

K1(˜t1)=

⎧

⎪

⎨

⎪

⎩

˜t1−2l+2el_−el+˜t1_(2l− ˜t1+₂₎+_4el_sinh(l) 2lel_−2lel+˜t1+_{2 ˜t1e}l_sinh(l) ifl≤ ˜t1<2l 2 ˜t1− 2l− ˜t1 ˜t1(el−˜t1−1) if2l≤ ˜t1 ≤t1 K2(˜t1)=

⎧

⎪

⎨

⎪

⎩

2−2e2l_+2el+˜t1−_2el ˜t1− ˜t1e2l₊2lel+˜t1−2lel ifl≤ ˜t1<2l 2 ˜t1 if2l≤ ˜t1 ≤t1 (11) where ˜t1∈[l,t1]withK1(t1)=0.

Proof. To demonstrate the proposition, we find the locus of

parametersK1−K2whichallowsthesystemtohavethesmallest

limitcyclesthatinvolvethreestates(namelyj=_{−1,0,1_}).Inthis way,allthesetsof{K1,K2}insideofthepartofthefirstquadrant

delimitedbythatlocusdonotallowthesystemtohavealimitcycle.

Athree-levellimitcyclewithaperiod ˜Thastoverifythe follow-ingconditions • ˜

v

(k ˜T −l)=0whichimplies

v

˜∗(k ˜T)=0; • ˜

v

(k ˜T −l+ ˜t1)=1whichimplies˜

v

∗(k ˜T+ ˜t1)=1; • ˜

v

(k ˜T −l+ ˜t1+ ˜t2)=0whichimplies˜

v

∗(k ˜T + ˜t1+ ˜t2)=0; • ˜

v

(k ˜T −l+ ˜t1+ ˜t2+ ˜t3)=−1whichimplies

v

˜∗(k ˜T + ˜t1+ ˜t2+ ˜t3)= 0; • ˜

v

(k ˜T −l+ ˜t1+ ˜t2+ ˜t3+ ˜t4)=−1whichimplies ˜

v

∗(k ˜T + ˜t1+ ˜t2+ ˜t3+ ˜t4)=0;

whereT= ˜t1+ ˜t2+ ˜t3+ ˜t4.Becauseofthesymmetricnatureofthe

SSODtechnique,wecansupposethatthelimitcycleissymmetric. Thus: ˜x(˜t)=−˜x(˜t −T/2), ˜t3= ˜t1and ˜t4= ˜t2.

Duringatimeinterval ˜ti:=[˜ta, ˜tb]betweentwoevents,the

sig-nal

v

˜∗(˜t)isconstant,andthereforeitiseasytofindtheevolutionof thestatevariablesasthesumofthefreeandforcedresponsetoa constantinput: ˜x(˜tb)=R(˜tb− ˜ta)˜x(˜ta)+F(˜tb− ˜ta;K2,a)˜u(˜t) (12) where: R(˜t)=

e−˜t 0 0 1



and F(˜t;K2,a)=

a(1−e−˜t) K2˜t



,

Using(12)itispossibletowritethefollowingequations,which describetheevolutionofthesystemduringatimeperiod:

˜x1_{= ˜x(k ˜T+ ˜t}

1)=R(˜t1)˜x0+F(˜t1;K2,a)

˜x2_{= ˜x(k ˜T+ ˜t}

1+ ˜t2)=R(˜t2)˜x1=−˜x0.

where ˜x0_{= ˜x(k ˜T).Theseequationscanberewrittenas:}

˜x1_=(I+R(˜t

2)R(˜t1))−1F(˜t1;K2,a)

˜x0_=−(I+R(˜t

2)R(˜t1))−1R(˜t2)F(˜t1;K2,a)

(13)

Itispossibletowritealsothefollowingswitchingconditions: ˜xa_{= ˜x(k ˜T+ ˜t} 1−l) ˜

v

a=˜

v

(k ˜T + ˜t1−l)= ˜x2a+ ˜x1a=0 ˜xb_{= ˜x(k ˜T + ˜t1+ ˜t2−l)} ˜

v

b_=˜

_v

_{(k ˜T+ ˜t} 1+ ˜t2−l)= ˜xb2+ ˜xb1=1 (14)

Conditions(13)and(14)describeallthepossiblethree-levellimit cyclesinaSSOD-PIcontrolledFOPDTprocess.Toprovethe proposi-tionwewanttofindthesmallestlimitcycleandthecorrespondent normalizedgains.Thus,wecanaddthefollowingcondition sup

v

˜(˜t)=sup(˜x2(˜t)+ ˜x1(˜t))=1. (15)

Tofindthesuperiorof˜

v

(˜t)wehavetostudythesignofits deriva-tiveintheintervals[0, ˜t1]and[˜t1, ˜t1+ ˜t2](notethat,becauseof

symmetryitispossibletostudyonlyahalfoftheperiod). Inthefirstinterval,thestateevolutionis

−2 −1 0 1 2 −2 −1 0 1 2

Fig.8.Exampleoflimitcyclewitha<0wherethearcarrowsrepresentthetime intervals.

where ˜x0_{canbeeasilyfoundfrom(13).Deriving}

_v

_{˜(˜t)withrespect}

tothetime,weobtain: ˙

v

(˜t)= K2e˜t(1+e˜t1+˜t2)+ae˜t1(1+e˜t2)

e˜t_(e˜t1_e˜t2+₁₎ ,

wherethenumeratorispositiveifa>−K2e(˜t−˜t1)(1+e(˜t1+˜t2))/(1+

e(˜t2)_{)andbyrememberingthatK1=K2+a}≥_{0,itispossibletoobtain} that

v

˙(˜t)whichisalwayspositivein ˜t∈[0, ˜t1].

Inthesecondinterval,thestateevolutionis: ˜x(˜t)=R(˜t − ˜t1)˜x1,

therefore˜

v

(˜t)=e−(˜t−˜t1)_˜x1

1+ ˜x12anditsderivativeisequalto

v

˙(˜t)=

−e−(˜t−˜t1)_˜x1

1.Thevalueof ˜x 1

1canbeeasilyfoundbysolving(13)and

itresults: ˜x11=ae˜t2 e

˜t1−₁ e˜t1+˜t2+₁,

therefore

v

˙(˜t)in ˜t∈[˜t1, ˜t1+ ˜t2]hasaninversesignwithrespectto

a.

Summarizing,ifa<0(thecorrespondinglimitcycleisshown inFig.8)thenthederivative

v

˙(˜t)ispositivein ˜t∈[0, ˜t1+ ˜t2]and

negativein ˜t ∈[˜t1+ ˜t2, ˜t1+ ˜t2+ ˜t3+ ˜t4] and therefore the

maxi-mumis

v

˜(˜t1+ ˜t2), else ifa>0 (the correspondinglimit cycle is

showninFig.9)thenthederivative

v

˙(˜t) ispositivein ˜t ∈[0, ˜t1]

and ˜t ∈[˜t1+ ˜t2, ˜t1+ ˜t2+ ˜t3],negativein ˜t∈[˜t2, ˜t1+ ˜t2]and ˜t∈[˜t1+

˜t2, ˜t1+ ˜t2+ ˜t3+ ˜t4]andthereforethemaximumis

v

˜(˜t1).

Consideringthecasea<0,thevariable

v

˜(˜t)assumesthevalue 1int= ˜t1+ ˜t2,wherethestatecoincideswiththepoint−˜x0,but

alsointhepoint ˜xb_{(namely,linstantbefore−˜x}0_)thesignal

_v

_˜(˜t)is

equalto1.Thissituationispossibleifandonlyifthetimeinterval ˜t2isinfiniteand−˜x0isastationarypoint(namely,−˜x0=[0,1]T)

orlisamultipleoftheperiod.Consideringthefirstcase(namely, ˜t2→∞),since ˜x02=−1andthesecondvariablechangeslinearly

duringthetimeinterval[0, ˜t1]anditisconstantinthetimeinterval

[˜t1, ˜t1+ ˜t2],wecanwrite: ˜x22=−˜x02= ˜x02+K2˜t1 −2 −1 0 1 2 −2 −1 0 1 2

Fig.9. Exampleoflimitcyclewitha>0wherethearcarrowsrepresentthetime intervals.

andobtain: K2=

2 ˜t1.

Toobtaintheexpressionofa,itisimportanttonotethat ˜t1>l

becausethetrajectoryintersectstheswitchingsurface ˜x2+ ˜x1=1

whenitassumesthevalue ˜xb_{andlinstantsafterthetrajectorydoes}

nothaveyetintersectedtheotherswitchingsurface ˜x2+ ˜x1=0

(becauseitisinastationaryvalue).Thus,theexpressionof ˜xa_(see

(13)and(14))becomes: ˜x1₌ R(l)˜xa+F(l;K2,a) orequally: ˜xa=R(l)−1(˜x1−F(l;K2,a)). Finally, imposing ˜xa

2+ ˜xa1=0 it is easy to find therelationship

betweenaand ˜t1,andtherefore(rememberthatK1=K2+a)the

relationbetweenK1and ˜t1,presentedin(11).

Consideringnowthesecondcase(l=N ˜T=2n(˜t1+ ˜t2))withn∈

N+_{(thus ˜t1<land ˜t2<l),itispossibletonotethat ˜x}a_{= ˜x(˜t1−l)=}

˜x1_{and ˜x}b_{= ˜x(˜t}

1+ ˜t2−l)= ˜x2.Rememberingthat˜

v

a=0and˜

v

b=1,

itispossibletofind: K2= 2 ˜t1 1 1−e−˜t1

andthisconditionislessstringentthantheothercase(˜t2→∞),

notingthat ˜t1inthesecondcaseislowerthanlandinthefirstcase

isgreaterthanl.

Consideringthecasea≥0,thevariable

v

˜(˜t)assumesthevalue 1int= ˜t1,wherethestatevectorisequalto ˜x1,andalsointhis

caseitcorrespondstothepoint ˜xb_{.Inthissituationthedelaylhas}

tobeequalto ˜t2+n ˜T (with n∈N+), infact when

v

˜∗(˜t) isequal

to1(namelyintheinterval[0, ˜t1])thesignal

v

˜(˜t)monotonically

increases.Similarlytothecasea<0,itispossibletodemonstrate thatthecasewithn=0isthemoststringentcase.

Thus,inthetimeinstant ˜t1thesystemhastoswitchintothe

instant ˜t1−lthevariable˜

v

(˜t)hastobeequalto0.Becauseofthese

conditions,wecanwrite: ˜x1_{=(I+R(l)R(˜t} 1))−1F(˜t1;K2,a) ˜x1_{= ˜x}b ˜x1_=R(l)˜xa_+F(l,K2,a) ˜

v

b 1=1 ˜

v

a1=0.

Solving this system of equations we can find the relationship betweenK2,a(andthereforeK1)and ˜t1shownin(11).

ThecaseofSSOD-PandP-SSODcontrollersisaddressedinthe followingproposition.Alsointhiscasethegoalistofindthevalues ofK1forwhichtherearenotperiodictrajectorieswhichinvolve

morethantheequilibriumstate.

Proposition4. Inasystemdescribedby(8),ifK1<_1−e1−lthenlimit cyclescannotoccur.

Proof. Alsointhiscase,theproofofthepropositionmainlyconsist

infindingthevalueofK1(namely,a=K1whenK2=0)whichallows

thesystemtoattainthesmallestlimitcycle(namely,j=− 1,0,1). Becauseofthesymmetryofthemapbetween˜

v

(˜t)and

v

˜∗(˜t),the limitcyclesaresymmetrical,thus ˜

v

(˜t)=−˜

v

(˜t −T/2).Thesignals ˜

v

∗(˜t)and˜

v

(˜t)aredefinedas:

˜

v

∗(˜t)=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

1 if ˜t∈[0, ˜t1] 0 if ˜t ∈[˜t1, ˜t1+ ˜t2] −1 if ˜t ∈[˜t1+ ˜t2, ˜t1+ ˜t2+ ˜t3] 0 if ˜t ∈[˜t1+ ˜t2+ ˜t3, ˜t1+ ˜t2+ ˜t3+ ˜t4] (16) ˜ v(˜t)=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

˜ v(0)e−t_+K1(1−e−t_{) if ˜t∈[0, ˜t1]} ˜ v(˜t1)e−(t−˜t1) _{if ˜t ∈[˜t1, ˜t1}+ ˜t2] −˜v(0)e−t−K1(1−e−(t−˜t1−˜t2)_{) if ˜t}∈_[˜t1+ ˜t2, ˜t1+ ˜t2+ ˜t3] −˜v(˜t1)e−(t−˜t1−˜t2) _{if ˜t∈[˜t1+ ˜t2+ ˜t3, ˜t1+ ˜t2+ ˜t3+ ˜t4]} (17)

where ˜t1= ˜t3 and ˜t2= ˜t4. Imposing thecondition on

periodic-ityand symmetry

v

(0)=−

v

(˜t1+ ˜t2), it ispossibletofind

v

(0)=

−K1((e−˜t2(1−e−˜t1))(1+e−˜t1−˜t2)).

Inordertofindthesmallestlimitcycle,wehavetoimposethat themaximumof

v

˜(˜t)equalsto1.Themaximumof

v

˜(˜t)isfoundby studyingthesignof ˙˜

v

(˜t),whichis:

˙˜

v

(˜t)=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

K1 1+e−˜t2 1+e−˜t1−˜t2e −˜t _{if ˜t∈[0, ˜t} 1] −K1 1−e−˜t1 1+e−˜t1−˜t2e −(˜t−˜t1) _{if ˜t}∈_[˜t 1, ˜t1+ ˜t2]

Thus,themaximumisin ˜t= ˜t1anditisequaltomax˜

v

(˜t)=K1((1−

e−˜t1₎₍₁+_e−˜t1−˜t2_{)).Themaximumisequalto1onlyif:} K1=

1+e−˜t1−˜t2 1−e−˜t1

(18) Thevaluesof ˜t1and ˜t2canbefoundusingthefollowingswitching

conditions: ˜

v

(˜t1−l+nT)=0 (19)

˜

v

(˜t1+ ˜t2−l+nT)=1 (20)

whereT= ˜t1+ ˜t2+ ˜t3+ ˜t4 isthetimeperiod,andn∈Z.Because

max ˜

v

(˜t)=˜

v

(˜t1)=1,thecondition(20)implies˜

v

(˜t1+ ˜t2−l+nT)=

˜

v

(˜t1)orequallyl−nT= ˜t2.

Consideringfirstthecasewithn=0,wecanobtaint2=landthe

condition(19)cannotbeverifiedinthefourthtimeinterval(˜

v

(˜t)is alwayspositive),therefore ˜t1≥ ˜t2anditispossibletoobtainthat

˜t1=log(1+el−e−l)andK1=(1/(1−e−l))whichprovesthe

propo-sition.

Consideringnowthecaseswithn>0,therefore ˜t1<(l/2)and

˜t2<(l/2).Using(18),wecanstate:

1+e−˜t1−˜t2 1−e−˜t1 ≥ 1 1−e−˜t1 ≥ 1 1−e−l.

Thus,limitcycleswithatimeperiodlowerthanlcannotoccurusing K1≤(1/(1−e−l)).

NotealsothatPropositions3and4arenecessaryandsufficient conditionstoavoidthepresenceoflimitcycles.Inotherwords,in theregionoftheplaneK1−K2delimitedbytheseconditionsthere

aresurelynolimitcycles,whereasoutofthisregionthereissurely atleastapossiblelimitcycle.Actually,forsomevaluesofinitial conditions,referencesignalandloaddisturbanceitispossiblethat thesystemstatedoesnotreachthelimitcycletrajectorybuttends totheequilibriumpoint.

Anotherimportantconsiderationisthattheparameterdoes nothaveinfluenceonthestabilityanalysisandthereforeitcanbe chosen,inprinciple,usingonlyconsiderationsaboutthedesired precision,namely,themaximumadmissiblesteady-stateerror.In practicalcases,measurementnoiseshouldbetakenintoaccount. Actually,itisadvisabletoselectavalueofgreaterthanthenoise band,thatisthepeak-to-peakamplitudeofthenoisesignal(see Section5.1).

3.3. Disturbanceestimation

Asalreadymentioned (seeSection3.1),becauseofits quan-tizednature,thePI-SSODcontrolstrategy(7)cannotcompensate theunquantizedpartoftheloaddisturbance,thereforeabimodal limitcyclesurelyarises.ThislimitcyclewithperiodTinvolvestwo consecutivestatesjl+1andjl,wheret*istheintervaltimewhere

j=jl+1.Thus,thecontrolactionassumesonlythevaluesjlˇand

(jl+1)ˇ,asshownin(21). u(t)=



(jl+1)ˇ ift∈[0,t∗] jlˇ ift∈[t∗,T] (21) Byapplyingthedefinitionoflimitcycle,thetrendsoftheprocess andthecontrollerquantitiesduringacycleareperiodic. Consider-ingnowtheintegratederrorIE(t)anditsLaplacetransformIE(s), whichcanbecalculatedas

IE(s)= R(s)

s −

K

s(s+1)(U(s)+d)

whichcorrespondstothefollowingdifferentialequation:  ¨IE(t)+ ˙IE(t)=r(t)−K(u(t)+d+ ˙r(t)).

Byintegratingoveraperiodandrememberingthat,byhypothesis, randdareconstantandIE(t)isperiodic,weobtain:

( ˙IE(T)− ˙IE(0))+(IE(T)−IE(0))=rT−KdT−K

T 0 u(t)dt=0 andfinally: u=jlˇ+ˇt ∗ T = r K−d (22)

whereu isthemeanvalueofu(t)duringaperiodand ˆd:=ˇ(t∗/T) isthe“unquantized”partofthecontrolactionnecessaryto com-pensatetheloaddisturbance.

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

Fig.10.Planewiththenecessaryconditiononstability(dashedline),andcondition onabsenceoflimitcycles(solidline).Themarker◦denotesthesetofparameters usedtoobtaintheresultsshowninFig.11,themarker∗denotesthesetofparameters usedtoobtaintheresultsshowninFig.12.

In real plants, the trajectory tends to the limit cycles but doesnotreachit.Hence,thehypothesisofperiodicityisonlyan approximation, therefore an exact compensation of the distur-bancesisvirtuallyimpossible.Forthisreason,asmalldeadbandis introduced,thewidthofthisbandisupper-limitedbyanditcan

besetbytakingintoaccountthenumericalerrorsinthecalculus oft∗_{andTandthepresenceofnoise.}

4. Practicalissues

The SSOD-PI and PI-SSOD control techniques can be easily implementedinverysimple industrialcontrollers,becausethey donotrequirealargecomputationaleffortorcomplexroutines. Thealgorithmsforthesensorandthecontrolunits,canbe out-linedusingthepseudocodesshowninTable2(forsakeofbrevity theinitializationisnotshown),wherehisthecontrolunit samp-ling period (note that the two algorithms do not require any synchronization and theycanhave differentsamplingperiods). Note that the actuator unit task of PI-SSOD controller has to onlykeepthelastreceivedvalueuntilthenextevent.Notealso thatthesteps8–9ofthePI-SSODcontroller’sSensorandControl UnitTaskverifyifabimodalcyclesisoccurred(namely,thelast threestateswerej,j−1andj)andintroducetheloaddisturbance compensation.

5. Simulationresults

Inthissection,simulationresultsaregiven.Inparticular, illus-trativeexamplesonthepresenceoflimitcyclesandtheexogenous signalarepresented,thenthetwoSSODcontrolstrategiesare com-paredwithotherevent-basedsamplingtechniquesandadiscrete time,usingasahigh-orderplusdeadtimesystemapproximated withthehalf-rulemethodandusingtheSIMCtuningrules[27].

0 16 32 48 64 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 16 32 48 64 80 −0.5 0 0.5 1 1.5 2 0 16 32 48 64 80 0 16 32 48 64 80

Fig.11.Responsesforprocess(23)usingSSOD-PIandPI-SSODcontrollerswiththefirstsetofparameters.Firstplot(fromthetop):processvariablefortheSSOD-PIcontroller (solidline)andPI-SSODcontroller(dashedline).Secondplot:controlvariablefortheSSOD-PIcontroller(solidline)andPI-SSODcontroller(dashedline).Thirdplot:events oftheSSOD-PIcontrollersystem.Fourthplot:eventsofthePI-SSODcontrollersystem.

Table2

PseudocodesoftheSSOD-PIandPI-SSODcontrollers.

SSOD-PIcontroller PI-SSODcontroller

Sensorunit 1. calculatee;

2. if|e−e∗_{|≤thengoto5;} 3. ife>e∗_thene∗_=e

elsee∗= e; 4. sendˇe∗_{tothecontrolunit;}

5. end.

Controlandactuatorunit

1. ifanewˇe∗_{isreceivedthenupdatethevalueofe}∗_; 2. calculateui,k=ui,k−1+Kiˇe∗handuk=ui,k+Kpˇe∗; 3. senduktotheactuator;

4. setui,k−1=ui,k; 5. end.

Sensorandcontrolunittask 1. calculatee;

2. calculateui,k=ui,k−1+Kiehanduk=ui,k+Kpe; 3. setui,k−1=ui,k;

4. if|u−u∗_{|≤thengoto11;}

5. setu∗,2=u∗,1,t∗,2=t∗,1,u∗,1=u∗andt∗,1=t∗;

6. saveint∗thetimeintervalbetweenthelastandthecurrentevent. 7. ifu>u∗_thenu∗_=u

elseu∗= u; 8. ifu∗,2_=u∗_{thengoto9elsegoto10;} 9. ifu∗_>u∗,1_{then ˆD =ˇ} t∗

t∗,1+t∗;

10. sendˇu∗+ ˆD totheactuatorunit; 11. end.

5.1. IllustrativeExample1

InthisexamplethefollowingFOPDTsystemiscontrolledbythe

SSOD-PIandPI-SSODcontrollers:

P(s)= 1

s+1e−s (23)

UsingProposition3itispossibletofindtheregionofK1−K2(shown

inFig.10)wherethelimitcyclesareavoided.Inordertoshowthe validityofProposition3,twosetsofparametersarechosen,thefirst (Kp=0.542andKi=0.792)slightlyinsidethenolimitcycleregion

(itisindicatedwithacircleinFig.10)andthesecond(Kp=0.542

andKi=0.840)whichdoesnotbelongtothisregion(itisindicated

witha‘*’inFig.10).Thevaluesofandˇaresetequalto0.1and 1,respectively.

Aunitset-pointstepsignalisappliedtothecontrolsystemat timet=0andaunitloaddisturbancestepisappliedattimet=40. Figs.11and12showthecontrolsystemresponseusingthefirstand thesecondset,respectively.ItispossibletonotethatProposition 3isverified,asthereisnotalimitcycleinFig.11whilethereisin Fig.12.

Anotherimportantresultof Propositions3 and4 isthatthe parameterdoesnotinfluencethestabilityandlimitcycle prop-erties,butonlythesystemprecisionandthenumberofevents.This aspectishighlightedinFig.13,wherethecontrolleristunedwith thesecondsetofparametersandissetequalto0.01.Itispossible tonote(seethezoomsinsidethefigure)thatthesystem trajecto-riestendtolimitcycleswiththesameperiodofFig.12butthe processvariablesareclosertotheset-point.However,thenumber ofeventsincreasessignificantly.

0 16 32 48 64 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 16 32 48 64 80 −0.5 0 0.5 1 1.5 2 0 16 32 48 64 80 0 16 32 48 64 80

Fig.12.ResponsesforProcess(23)usingSSOD-PIandPI-SSODcontrollerswiththesecondsetofparameters.Firstplot(fromthetop):processvariablefortheSSOD-PI controller(solidline)andPI-SSODcontroller(dashedline).Secondplot:controlvariablefortheSSOD-PIcontroller(solidline)andPI-SSODcontroller(dashedline).Third plot:eventsoftheSSOD-PIcontrollersystem.Fourthplot:eventsofthePI-SSODcontrollersystem.

0 16 32 48 64 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 16 32 48 64 80 −0.5 0 0.5 1 1.5 2 0 16 32 48 64 80 0 16 32 48 64 80 20 25 30 35 40 0.98 1 1.02 20 25 30 35 40 0.98 1 1.02 60 65 70 75 80 0.98 1 1.02 60 65 70 75 80 −0.02 0 0.02

Fig.13.Responsesforprocess(23)usingSSOD-PIandPI-SSODcontrollerswiththesecondsetofparametersandasmallervalueof.Firstplot(fromthetop):process variablefortheSSOD-PIcontroller(solidline)andPI-SSODcontroller(dashedline).Secondplot:controlvariablefortheSSOD-PIcontroller(solidline)andPI-SSODcontroller (dashedline).Thirdplot:eventsoftheSSOD-PIcontrollersystem.Fourthplot:eventsofthePI-SSODcontrollersystem.

Finally,theissueoftheselectionofinthepresenceof mea-surementnoiseisanalyzedbyconsideringthesamecaseofFig.11 butwitha measurementnoisewitha noisebandequalto0.08. Bychoosing=0.1(thatis,slightlygreaterthanthenoiseband), theresultsshowninFig.14areobtained.Itappearsthatthereis notasignificantperformancedecrementwhilethereisonlyaslight incrementofthenumberofevents.Actually,theSSOD-PIcontroller islesssensitivetothenoisethanthePI-SSODcontrollerbecausethe SSODquantizationreducestheeffectsofthenoiseonthePIcontrol unit.

5.2. IllustrativeExample2

Inthisexampletheexogenoussignalcompensationtaskofthe PI-SSODcontrollerisshown.Theconsideredprocessisthesame oftheExample1(see(23)).Thecontrollerparametersaresetas Kp=0.542,Ki=0.792,=0.1andˇ=1;theset-pointvalueisset

equalto1.05,inthiswaythecompensationisnecessary(namely, r /= jKˇ,

∀

j∈Z).InFig.15,itispossibletonotethatatthetime instantt=23,thecontrolvariableissetequalto1.052using(22). Becausethecompensationis exact, asmall deadband(equalto 0.1K=0.01)isappliedontheerrorsignalcalculus.

5.3. IllustrativeExample3

Inthisexample,thehighordersystem[27]

P(s)= (−0.3s+1)(.08s+1)

(2s+1)(s+1)(0.4s+1)(0.2s+1)(0.05s+1)3 (24) isconsidered.TheprocessisapproximatedasaFOPDTsystemwith K=1,=2.5andL=1.47usingthehalfrule.

ThecontrollergainsaretunedusingtheSIMCrules[27]andthey resultinthefollowingcontrollerparametersKp=0.85andKi=0.34

whichareinthenolimitcyclesregionaccordingtoProposition 3.ThepresentedcontrolstrategiesarecomparedwiththeSOD-PI controllerpresentedin[26],theevent-basedcontroller(denotedas ˚Arzén)presentedin[14],thePIDpluspresentedin[2]andadiscrete time-drivenPIcontroller(denotedasDT-PI)withasampleperiod equalto/5,assuggestedin[2].

Theparametersandˇaresetequalto0.1and1,respectively. Theparametertmax,necessaryin[14,2],issetequalto.TheSOD-PI

parametersaresetasfollows:p=,i=,



=.Thedeadband

presentinthePI-SSODcontrollerissetequalto0.1K.

Fig.16showsthesystemresponseswhentheexogenous sig-nalsareaunitset-pointstep,appliedatthetimeinstantt=0,anda loaddisturbancestep,withamplitudeequalto0.95andappliedat thetimeinstantt=30.Itisimportanttonotethattheperformance ofSSOD-PI,PI-SSODandSOD-PI controllersare thefastest ones butpresentmoreovershootwithrespecttothePIDpluscontroller, whichislessaggressivealsoduringthedisturbancerejectiontask. TheArzénPIcontroller presentsa peakonthecontrolvariable causedbythemethodusedtoelaboratetheintegralaction(which issolvedinthePIDpluscontroller).Theperformanceofthe con-trollers,showninTable3,confirmtheseconsiderations.Another importantconsiderationisthattheSSOD-PIandPI-SSODcontrol strategiespresentthesmallestnumbersofevents,becauseofthe absenceofthetmaxtriggeringcondition.Itispossibletoseethatthe

event-basedcontrolstrategies(withtheexceptionoftheArzénPI controller)havebetterperformancethanadiscretePItechnique witha highsampling period. Note that,reducing the sampling periodofDT-PIcontrollerincreasestheperformancebutalsothe numberoftheevents.

0 16 32 48 64 80 −0.5 0 0.5 1 1.5 2 0 16 32 48 64 80 −0.5 0 0.5 1 1.5 2 0 16 32 48 64 80 0 16 32 48 64 80

Fig.14.ZoomofFig.13.Firstplot(fromthetop):processvariablefortheSSOD-PIcontroller(solidline)andPI-SSODcontroller(dashedline).Secondplot:controlvariable fortheSSOD-PIcontroller(solidline)andPI-SSODcontroller(dashedline).Thirdplot:eventsoftheSSOD-PIcontrollersystem.Fourthplot:eventsofthePI-SSODcontroller system. 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 −0.5 0 0.5 1 1.5 0 5 10 15 20 25 30

Fig.15.Responsetoastepofamplitude1.05forProcess(23)usingPI-SSODcontroller.Firstplot(fromthetop):processvariable.Secondplot:controlvariable.Thirdplot: eventsofthePI-SSODcontrollersystem.

0 10 20 30 40 50 60 0 0.5 1 1.5 2 0 10 20 30 40 50 60 −0.5 0 0.5 1 1.5 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60

Fig.16.ResponsesforProcess(24).Firstplot(fromthetop):processvariablefortheSSOD-PIcontroller(thicksolidline),PI-SSODcontroller(thickdashedline), ˚Arzén controller(thickdot-dashedline),PIDpluscontroller(thickdottedline),SOD-PIcontroller(thindashedline)andDT-PIcontroller(thinsolidline).Secondplot:control variablefortheSSOD-PIcontroller(thicksolidline),PI-SSODcontroller(thickdashedline), ˚Arzéncontroller(thickdot-dashedline),PIDpluscontroller(thickdottedline), SOD-PIcontroller(thindashedline)andDT-PIcontroller(thinsolidline).Fromthirdtoeighthplot:eventsoftheSSOD-PI,PI-SSOD, ˚Arzén,PIDplus,SOD-PIandDT-PIcontroller systems,respectively.

Remark5. ItisimportanttonotethattheEventBased-PI(EB-PI)

strategiesdifferentfromtheSSOD-PIandPI-SSODmethodscanbe alsoappliedwhenthethreecontrolagentsareseparated.

6. Experimentalresults

In order toprove theeffectiveness of thepresented control strategiesinpracticalapplications,alaboratory-scalesetupmade byQuanserhasbeenemployed(seeFig.17).Specifically,the exper-imentalsetupisafourtankssystemwherejustthelowerlefttank hasbeenconsideredforlevelcontrol.

Becausetheapparentdead-timeofthesystemisverysmallwith respecttoitsdominanttimeconstant,andinordertoprovidea significantresult,atimedelayof2shasbeenaddedviasoftwareat theplantoutput.TheFOPDTmodelhasbeenobtainedbyapplying aleast-squaresproceduretotheresponseofaseriesofopen-loop steps,resultingin:

P(s)= 82.1 22.2s+1e

−2s_{. (25)}

Thepercentage oftheoutputvariationthat isexplainedbythe modelisaround90%.

Table3

Performanceofthecontrollers.IAE:integratedabsoluteerror(computedonly whentheerrorsignalisoutoftheband±).OV:overshoot.ne:numberofevents. st:settingtimetoabandof±aroundtheset-point.

IAE OV ne st SSOD-PI 8.510 15.185 32 9.032 PI-SSOD 7.821 13.360 30 9.135 ˚Arzén 10.143 23.465 49 17.730 PIDplus 8.342 8.957 46 5.996 SOD-PID 7.855 12.547 58 9.062 DT-PID 9.341 30.997 120 14.897

Theexperimentconsistsinaset-pointstepchangefrom10cm

to20cm,thenaloaddisturbanceisapplied,usinganadditional

pump.

The PIcontroller are tunedusing the following parameters:

Kp=0.04,Ti=16,whichareinsideofthenolimitcyclezone.The

tmax(whenitisnecessary)issetequalto.Theparameterisset

0 50 100 150 200 250 300 350 400 450 0 5 10 15 20 25 30 0 50 100 150 200 250 300 350 400 450 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450

Fig.18.ResponsesforprocessshowninFig.17.Firstplot(fromthetop):processvariablefortheSSOD-PIcontroller(thicksolidline),PI-SSODcontroller(thickdashed line), ˚Arzéncontroller(thickdot-dashedline),PIDpluscontroller(thickdottedline),SOD-PIcontroller(thindashedline)andDT-PIcontroller(thinsolidline).Secondplot: controlvariablefortheSSOD-PIcontroller(thicksolidline),PI-SSODcontroller(thickdashedline), ˚Arzéncontroller(thickdot-dashedline),PIDpluscontroller(thickdotted line),SOD-PIcontroller(thindashedline)andDT-PIcontroller(thinsolidline).Fromthirdtoeighthplot:eventsoftheSSOD-PI,PI-SSOD, ˚Arzén,PIDplus,SOD-PIandDT-PI controllersystems,respectively.

equal0.4(biggerthanthenoisebandofthesensor)forallthe con-trollersexceptthePI-SSODstrategy,whereitissetequalto0.02 (choseninordertoobtainthesamenumberofeventsofthe SSOD-PIcontroller).ThedeadbandofthePI-SSODcontrollerissetequal to0.1K=0.16.Theparametersiand



oftheSOD-PIcontroller

aresetequalto0.4and1,respectively.Fig.18andTable4showthe experimentalresultsandtheperformanceindexobtainedusingthe samecontrollersofExample3.

Alsointhiscase, thebestperformanceis achievedusingthe SSOD-PI,thePI-SSOD,theSOD-PIandthePIDpluscontrollers. How-ever,withthetwocontrolstrategiespresentedinthispaperitis possibletoreducethenumberofevents(andthereforethe num-beroftransmissions)withoutasignificantlossofperformancewith respecttotheSOD-PIandthePIDpluscontrollers.

Table4

Performanceofthecontrollers.IAE:integratedabsoluteerror(computedonly whentheerrorsignalisoutoftheband±).OV:overshoot.ne:numberofevents. st:settingtimetoabandof±aroundtheset-pointwhentheset-pointsignalis aunitarystep. IAE OV ne st SSOD-PI 166.826 12.802 30 36.100 PI-SSOD 174.432 10.744 45 35.700 ˚Arzén 222.763 26.515 77 111.600 PIDplus 185.070 13.038 64 45.200 SOD-PID 166.531 9.240 148 72.300 DT-PID 209.121 29.145 107 30.500 7. Conclusions

Inthispaper,wehaveaddressedthestabilityissueandthe

pres-enceoflimitcyclesfortwoevent-basedPIcontrolsystems,based

onthesymmetric-send-on-deltaevent-triggeringmethod.In

par-ticular,themainadvantagesofthiseventdetectionstrategyhave

beenoutlined.Then,stabilityandlimitcyclespropertiesare

inves-tigated.Inparticular,necessaryconditionsonthesysteminstability

havebeendetermined.

Limitcyclesaresurelyavoidedifthecontrollergainisinsidethe

setofcontrollerparametersdescribedusingtwosufficient

condi-tions.Boththestabilityandlimitcyclepropertiesdonotdepend

ontheevent-triggeringmethod,whichcanbetunedinorderto

handlethetrade-offbetweenthenumberofeventsandthesystem

precision.

Simulationandexperimentalresultshaveconfirmedthe

effec-tiveness of the approach which represents a valuable step in

the development of easy-to-use tuning rules for this kind of

controllers.

AppendixA. Normalizationformulae

A.1. SSOD-PIcontroller

TheSSOD-PIcontrolledFOPDTprocessisdescribedbyEq.(6).

exogenoussignals(namely,r=D=0).Thus,Eq.(6)canberewritten as:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=− 1 x1(t)+ K (x2(t)+Kpe ∗_(t)) ˙x2(t)=Kie∗(t) y(t)=x1(t) e(t)=−y(t) e∗(t)=ssod(e(t−L);,ˇ) Thesignale∗(t)canberewrittenas: e∗(t)=ˇssod



_e(t−L)

 ;1,1



.

In order to normalize the system, we scale the time variable ˜t=(t/) (thus, l=(L/)). In this way, the derivatives becomes: (dx1/dt)=(dx1/d˜t)(1/) and (dx2/dt)=(dx2/d˜t)(1/). Thus, we

canrewritethesystemequationas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(˜t)=−x1(˜t)+K(x2(˜t)+Kpe∗(˜t)) ˙x2(t)=Kie∗(˜t) y(˜t)=x1(˜t)

v

(˜t)=x1(˜t) e∗(˜t)=ˇssod(−

v

(˜t −l)  ;1,1)

where

v

(˜t)=−e(˜t).Wedefine˜

v

∗(˜t)=ssod



v(˜t−l)  ;1,1



,therefore thesystemequationsbecome:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(˜t)=−x1(˜t)+Kx2(˜t)+KKpˇ

v

∗(˜t) ˙x2(t)=Kiˇ

v

∗(˜t) y(˜t)=x1(˜t)

v

(˜t)=x1(˜t) ˜

v

∗(˜t)=ssod

−

v

(˜t −l)  ;1,1



Tomakethestabilityanalysiseasierwediagonalizethedynamic equationusingthefollowingtransformationmatrix:

=



1 −K

0 K



Finally, defining ˜x(˜t)=(x(˜t)/), ˜y(˜t)=(y(˜t)/), K1=KKpˇ,

K2=KKiˇanda=K1−K2,thesystemequationsbecome(8).

A.2. PI-SSODcontroller

ThePI-SSODcontrolledFOPDTprocessisdescribedbyEq.(7). Asmentioned, thestabilityanalysisis donewithoutexogenous signals.Thus,Eq.(7)canberewrittenas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(t)=− 1 x1(t)+ K u ∗_(t) ˙x2(t)= Kp  (x1(t)−Ku∗(t))+Ki(−x1(t)) y(t)=x1(t) u(t)=x2(t) u∗(t)=ssod(u(t−L);,ˇ)

Thesignalu∗(t)canberewrittenas: u∗(t)=ˇssod



_u(t−L)

 ;1,1



.

In order to normalize the system, we scale the time variable ˜t=(t/) (thus, l=(L/)). In this way, the derivatives becomes: (dx1/dt)=(dx1/d˜t)(1/) and (dx2/dt)=(dx2/d˜t)(1/). Thus, we

canrewritethesystemequationas:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(˜t)=−x1(˜t)+Ku∗(˜t) ˙x2(˜t)=Kp(x1(˜t)−Ku∗(˜t))−Kix1(˜t) y(˜t)=x1(˜t)

v

(˜t)=x2(˜t) u∗(˜t)=ssod(−

v

(˜t −l);,ˇ)

where

v

(˜t)=−u(˜t)=−x2(˜t).Wedefine˜

v

∗(˜t)=ssod



v(˜t−l)  ;1,1



, thereforethesystemequationsbecome:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙x1(˜t)=−x1(˜t)+Kˇ˜

v

∗(˜t) ˙x2(˜t)=(Kp−Ki)x1(˜t)−KKpˇ˜

v

∗(˜t) y(˜t)=x1(˜t)

v

(˜t)=−x2(˜t)

v

∗(˜t)=ssod(−

v

(˜t−l)  ;1,1)

Tomakethestabilityanalysiseasierwediagonalizethedynamic equationusingthefollowingtransformationmatrix:

=

Kp−Ki 0

Ki−Kp −1



Finally, defining ˜x(˜t)=(x(˜t)/), ˜y(˜t)=(y(˜t)/), K1=KKpˇ,

K2=KKiˇanda=K1−K2,thesystemequationsbecome(8).

References

[1]A.O’Dwyer,HandbookofPIandPIDTuningRules,ImperialCollegePress, Lon-don,UK,2006.

[2]T.Blevins,PIDadvancesinindustrialcontrol,in:PreprintsIFACConferenceon AdvancesinPIDControl,Brescia,Italy,2012.

[3]P.Otanez,J.Moyne,D.Tilbury,Usingdeadbandstoreducecommunicationin networkedcontrolsystems,in:ProceedingsofAmericanControlConference, Anchorage,USA,2002.

[4]M.Miskowicz,Send-on-delta:anevent-baseddatareportingstrategy,Sensors 6(2006)49–63.

[5]A.Pawlowskyy,J.L.Guzmán,F.Rodríguez,M.Berenguel,J.Sánchez,S.Dormido, Event-basedcontrolandwirelesssensornetworkfor greenhousediurnal temperaturecontrol:asimulatedcasestudy,in:Proceedings13thIEEE Inter-nationalConferenceonEmergingTechnologiesandFactoryAutomation,2008. [6]U.Tiberi,C.F.Lindberg,A.J.Isaksson,Aself-triggerPIcontrollerforprocesses withdeadtime,in:PreprintsoftheIFACConferenceonAdvancesinPIDControl, Brescia,Italy,2012.

[7]X.Wang,M.D.Lemmon,Self-triggeredfeedbackcontrolsystemswith finite-gainL2stability,IEEETransactionsonAutomaticControl54(3)(2009)452–467. [8] M.MazoJr.,A.AntaMartínez,P.Tabuada,AnISSself-triggeredimplementation

oflinearcontrollers,Automatica46(8)(2010)1310–1314.

[9]J.Araujo,A.Anta,M.Mazo,J.Faria,A.Hernández,P.Tabuada,K.H. Johans-son,Self-triggeredcontroloverwirelesssensorandactuatornetworks.,in: ProceedingsIEEEInternationalConferenceonDistributedComputinginSensor Systems,2011June,pp.1–9.

[10]K.J. ˚Aström,Eventbasedcontrol.,in:A.Astolfi,L.Marconi(Eds.),Analysisand DesignofNonlinearControlSystems:InHonorofAlbertoIsidori,2008. [11]V.Vasyutynskyy,K.Kabitzsh,Event-basedcontrolOverviewandgenericmodel,

in:Proceedingsof8thIEEEInternationalWorkshoponFactoryCommunication Systems,2010May.

[12] W.P.M.H.Heemels,J.H.Sandee,P.P.J.VanDenBosch,Analysisofevent-driven controllersforlinearsystems,InternationalJournalofControl81(4)(2008) 571–590.

[13]D.Lehmann,K.H.Johansson,Event-triggeredPIcontrolsubjecttoactuator sat-uration,in:PreprintsIFACConferenceonAdvancesinPIDControl,Brescia,Italy, 2012.

[14]K.E. ˚Arzén,Asimpleevent-basedPIDcontroller,in:Proceedingsof14thWorld CongressofIFAC,Beijing,China,1999.

[15]V.Vasyutynskyy,K.Kabitzsh,ImplementationofPIDcontrollerwith send-on-deltasampling,in:ProceedingsofInternationalControlConference,2006. [16] M.Rabi,K.H.Johansson,Event-triggeredstrategiesforindustrialcontrolover

wirelessnetworks,in:Proceedingsof4thAnnualInternationalConferenceon WirelessInternet,Maui,HI,USA,2008.

[17]S.Durand,N.Marchand,Furtherresultsonevent-basedPIDcontroller,in: Proceedingsof10thEuropeanControlConference(ECC’09),Budapest,Hungary, 2009.

[18] V.Vasyutynskyy,K.Kabitzsh,Firstorderobserversinevent-basedPID con-trols.,in:ProceedingsofIEEEConferenceonEmergingTechnologies&Factory Automation,2009September.

[19]V.Vasyutynskyy,K.Kabitzsh,AcomparativestudyofPIDcontrolalgorithms adaptedtosend-on-deltasampling,in:ProceedingsofIEEEInternational Sym-posiumonIndustrialElectronics(ISIE),Bari,Italy,2010.

[20] J.Sánchez,A.Visioli,S.Dormido,Atwo-degree-of-freedomPIcontrollerbased onevents,JournalofProcessControl21(2011)639–651.

[21]J.Sánchez,A.Visioli,S.Dormido,Event-basedPIDcontrol,in:R.Vilanova, A. Visioli (Eds.), PID Control in the Third Millennium, Springer, 2012, pp.495–526.

[22]V.Vasyutynskyy,K.Kabitzsh,Towardscomparisonofdeadbandsampling types.,in:InProceedingsofIEEEInternationalSymposiumonIndustrial Elec-tronics,2007June.

[23]E.Kofman,J.Braslavsky,Levelcrossingsamplinginfeedbackstabilizationunder datarateconstraints,in:Proceedingsof45thIEEEInternationalConferenceon DecisionandControl,2006.

[24] J.Sánchez,M.A.Guarnes,S.Dormido,A.Visioli,Comparativestudyof event-basedcontrolstrategies:anexperimentalapproachonasimpletank,in: ProceedingsofEuropeanControlConference,2009.

[25]A.Cervin,K.J. ˚Aström,Onlimitcyclesinevent-basedcontrolsystems,in: Proceedingsof46thIEEEInternationalConferenceonDecisionandControl, NewOrleans,LA,USA,2007,pp.3190–3195.

[26] M.Beschi,A.Visioli,S.Dormido,J.Sánchez,Onthepresenceofequilibrium pointsinPIcontrolsystemswithsend-on-deltasampling,in:Proceedingsof 50thIEEEInternationalConferenceonDecisionandControlandEuropean Con-trolConference,Orlando,USA,2011.

[27] S.Skogestad,ProbablythebestsimplePIDtuningrulesintheworld.InAIChE Annualmeeting,Reno,NV,USA,2001.