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DisturbanceObserverforAdvancedMotionControlwith MATLAB/Simulink

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JeffreyReed DiomidisSpinellis

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TomRobertazzi

AhmetMuratTekalp

DisturbanceObserverforAdvancedMotion ControlwithMATLAB/Simulink

AkiraShimada

ShibauraInstituteofTechnology,Tokyo,Japan

IEEEPressSeriesonControlSystemsTheoryandApplications

MariaDomenicaDiBenedetto,SeriesEditor

Copyright©2023byTheInstituteofElectricalandElectronicsEngineers,Inc.Allrights reserved.

PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada.

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Author’sNote

Itisourgreatpleasuretohaveachancetopublishthebook“DisturbanceObserver forAdvancedMotionControlwithMATLAB/Simulink”andintroduceittoreadersworldwide.

Disturbanceobserver(DOB)isanalgorithmorafunctionforestimatingdisturbanceswell-knowntocontrolengineers.Still,nobookhasbeenpublishedthat systematicallyandcomprehensivelyexplainsitscontents.Forexample,wecan representDOBwithatransferfunctionorinastate-spacerepresentation.

Moreover,wecanalsodesignitasadigitalsystemoruseitinvibrationor communicationdelaysystems.Furthermore,weshouldconsidertheeffectsof noiseandmodelingerrorswhendesigningthesystem.Thisbookincludesall suchproblemsandexplainshowtounderstandandhandletheseissuesinan easy-to-understandmannerwithmanyexamplesusingMATLAB/Simulink.

WeinitiallypublishedthisbookinJapaneseinAutumn2021sincewewantedto postitforJapanesecontrolengineersandstudents.However,afterthepublishing, manyfriends,professors,andengineersrecommendedthatIshouldpublishitin Englishforengineersandstudentsworldwide.Thispublicationisaresponseto theirstrongencouragement.

ThecontentsofthisEnglishversionarethesameasthoseoftheJapanese version,butIrevisedallprogramsandtheSimulinkmodelusingtheMATLAB/SimulinkR2022aversiontomakethemclearforreaders.Thereaderscan downloadallsampleprogramsfromWiley’shomepage.

Moreover,IinitiallyselectedmanyreferenceswritteninJapanese.However, theyarenotconvenientforreadersworldwide.ThereforeIreselectedbooksand articleswritteninEnglish.

Wesincerelyhopethisbookbecomeshelpfulforyou.

BooksintheIEEEPressSeriesonControlSystems TheoryandApplications

SeriesEditor: MariaDomenicaDiBenedetto,Universityofl’Aquila,Italy

Theseriespublishesmonographs,editedvolumes,andtextbookswhicharegeared forcontrolscientistsandengineers,aswellasthoseworkinginvariousareasof appliedmathematicssuchasoptimization,gametheory,andoperations

1. AutonomousRoadVehiclePathPlanningandTrackingControl LeventGüvenç,BilinAksun-Güvenç,ShengZhu,SükrüYarenGelbal

2. EmbeddedControlforMobileRoboticApplications LeenaVachhani,PranjalVyas,andArunkumarG.K

3. MergingOptimizationandControlinPowerSystems:PhysicalandCyberRestrictionsinDistributedFrequencyControlandBeyond FengLiu,ZhaojianWang,ChanghongZhao,andPengYang

4. Model-BasedReinforcementLearning:FromDatatoContinuousActionswitha Python-basedToolbox MiladFarsiandJunLiu.

5. DisturbanceObserverforAdvancedMotionControlwithMATLAB/Simulink AkiraShimada

Contents

AbouttheAuthor xv

Preface xvii

AbouttheCompanionWebsite xxi

1IntroductionofDisturbanceObserver 1

1.1TypesofDisturbanceObservers 1

1.1.1Introduction 1

1.1.2ObserverandControlSystemDesignConcepts 3

1.2FormatofExampleandUseofMATLAB 4

1.2.1FormatoftheExampleProblem 4

1.2.2UsingMATLAB/Simulink 5

1.3HowThisBookIsOrganized 5

1.3.1TheStructureofThisDocument 5

1.3.2HowtoReadThisBook 6 References 7

2BasicsofDisturbanceObserver 9

2.1WhatIsDisturbance 9

2.2HowDisturbanceEstimationWorks 11

2.3DisturbanceRejectionandAccelerationControlSystem 13

2.3.1ConceptofDisturbanceRejectionandAcceleration 13

2.3.2DifferentDisturbanceObserversDependingonHowtheDisturbance IsCaptured 15

2.3.3BasicControlSystemDesign 16

2.4ReactionForceObserver(RFOB) 18

2.4.1ReactionForceObserverDesign 18

2.4.2CombinedUseofDOBandRFOB 20

2.5InternalModelandTwo-degrees-of-freedomControl 24

2.5.1InternalModelPrinciple 24

x Contents

2.5.2FeedforwardControl 28

2.5.3ControlSystemwithDisturbanceObserverandFeedforward 29

2.6EffectofObservationNoiseandModelingError 31

2.6.1EffectofObservationNoise 31

2.6.2EffectofModelingError 31

2.6.3EffectofViscousFriction 32

2.6.4EffectofVaryingMass 33

2.7RealSystemModeling 37

2.7.1DCMotorTorqueControlModel 37

2.7.2WithoutCurrentFeedback 38

2.7.3RelationshipBetweentheCartModelandRotary-typeMotor 38

2.8IdeaofRobustControl 39 References 41

3StabilizedControlandCoprimeFactorization 45

3.1CoprimeFactorizationandDerivationofStabilizingController 45

3.1.1DerivationofParametersforCoprimeFactorization 46

3.1.2StabilizingControllerandFreeParameters 50

3.1.3DoubleCoprimeFactorizationInvolving Q(s) 52

3.2RelationshipwithDisturbanceObserver 52

3.3CoprimeFactorizationandStructureofTwo-degrees-of-freedom ControlSystem 53 References 56

4DisturbanceObserverinStateSpace 59

4.1IdentityInputDisturbanceObserver 59

4.1.1HowtoDesigntheIdentityInputDisturbanceObserverinContinuous System 59

4.1.2ControllabilityandStateFeedback 68

4.1.3Continuous-timeServoSystemwithIdentityDisturbance Observer 69

4.2IdentityReactionForceObserver 72

4.3IdentityOutputDisturbanceObserver 75

4.4IdentityHigherOrderDisturbanceObserverDesign 79

4.5MinimalOrderDisturbanceObserver 82

4.6DesignofPeriodicDisturbanceObserver 89

4.7ObservabilityandNoninput/OutputDisturbances 94

4.7.1MathematicalModelofaDCMotor 94

4.7.2DCMotorObservableMatrixandRank 95

4.7.3ObservabilityofDisturbanceEstimation 97

4.7.4Noninput/OutputDisturbanceObserverandControl 97 References 100

5DigitalDisturbanceObserverDesign 101

5.1IdentityDigitalDisturbanceObserverDesign 101

5.2ConfirmationofSeparationTheorem 108

5.3MinimalOrderDigitalDisturbanceObserver 109

5.4IdentityHigh-orderDigitalDisturbanceObserver 119 References 122

6DisturbanceObserverofVibratingSystems 123

6.1ModelingoftheTwo-inertiaSystem 123

6.2VibrationSuppressionControlinTransferFunction Representation 126

6.3DisturbanceObserverandStabilizationforTwo-inertiaSystems 129

6.3.1ObservertoEstimateInputShaftDisturbance ��d1 129

6.3.2ObservertoEstimateOutputShaftDisturbance ��d2 132

6.4ServoSystemwithDOBforTwo-inertiaSystems 135

6.4.1InputShaftServoSystemConsideringInputShaftDisturbance ��d1 136

6.4.2OutputShaftServoSystemConsideringOutputShaftDisturbance ��d2 138 References 140

7CommunicationDisturbanceObserver 141

7.1SmithMethodOverview 141

7.2CommunicationDisturbanceObserver 142

7.3ControlwithCommunicationDOBUnderDisturbance 146 References 149

8MultirateDisturbanceObserver 151

8.1MultirateSystemModeling 151

8.2MultirateDisturbanceObserver(Method1) 153

8.2.1DisturbanceObserverDesign(Method1) 153

8.2.2ControllerDesignUsingMultirateObserver(Method1) 154

8.3MultirateDisturbanceObserver(Method2) 158 References 161

9ModelPredictiveControlwithDOB 163

9.1ModelPredictiveControl(MPC) 163

9.1.1OverviewofMPC 163

9.1.2FormulationandObjectiveFunctionfortheMPCDesign 165

9.2ConstraintDescriptions 167

9.2.1TreatmentofConstraintsontheControlInput û(k) 168

9.2.2ConstraintsontheControlVariable ̂ z(k) 169

9.2.3Constraintson Δû(k) ChangeintheControlInput 169

9.2.4ConstraintsontheControlInputsandQuantities 170

9.3MPCSystemDesign 170

9.4DesignofDisturbanceObserver-MergedMPCSystem 174

References 178

10KalmanFilterwithDisturbanceEstimation(KFD) 179

10.1DesignofKalmanFilterwithDisturbanceEstimation 179

10.2DesignofStationaryKalmanFilterwithDisturbanceEstimation (SKFD) 190

10.3DesignofExtendedKalmanFilterwithDisturbanceEstimation (EKFD) 193

References 200

11AdaptiveDisturbanceObserver 201

11.1StructureofanAdaptiveObserver 201

11.2DerivationofObservableCanonicalSystemforAdaptiveDOB 202

11.3CreatingStateVariableFilter 203

11.4DesignofKreisselmeier-TypeAdaptiveDisturbanceObserver 208 References 214

12MethodsforMeasuringandEstimatingVelocities 217

12.1ImportanceofVelocityMeasurement 217

12.2VelocityMeasurementandEstimationMethods 219

12.2.1Pseudo-derivative 219

12.2.2CountingandTimekeepingMethods 220

12.2.3 M ∕T Method 222

12.2.4SynchronousCountingMethod 223

12.2.5InstantaneousVelocityObserver 225 References 227

AppendixAMathematicalFoundationsandControlTheory 229

A.1Mathematics 229

A.1.1DefinitionandCalculusofMatrixExponentialFunctions 229

A.1.2PositiveDefiniteMatrix 229

A.1.3MatrixRank 230

A.2BasicClassicalControlTheory 230

A.2.1PolesandZeros 230

A.2.2PIVelocityControl 231

A.2.3PIDPositionControlSystem 232

A.2.4FinalValueandInitialValueTheorems 232

A.3BasicModernControlTheory 233

A.3.1StateandOutputEquations 233

A.3.2SolutionoftheStateEquationfortheContinuousSystem 234

A.3.3EquationofStatetoTransferFunction 234

A.3.4PolesandZerosofContinuousSystems 234

A.3.5ControllabilityandObservabilityofContinuousSystems 235

A.3.6DualityTheorem 236

A.3.7StateFeedbackControlofContinuousSystems 236

A.3.8ServoSystemDesign 243

A.4Doyle’sNotationandDoubleCoprimeFactorization 244

A.4.1Doyle’sNotation 244

A.4.2ConfirmationofDoubleCoprimeFactorization 245

A.5FoundationsofDigitalControlTheory 245

A.5.1DigitalControlandStateandOutputEquations 245

A.5.2PolesandZerosofDigitalSystems 247

A.5.3ReachabilityandObservabilityofDigitalSystems 247

A.5.4DigitalStateFeedbackControlSystemDesign 248

A.5.5DigitalServoSystemDesign 248

A.6RepresentationandMeaningofOptimalProgramming 250

A.6.1WhatIsOptimalProgramming? 250

A.6.2fminconFunction 250

A.6.3ExampleofaDrawingProgram 252 References 254 Index 255

AbouttheAuthor

Dr.AkiraShimada wasborninChiba,Japan,in1958.HereceivedB.S.degreein electronicsengineeringfromtheUniversityofElectro-Communications,Japan, in1983andreceivedPhDinengineeringfromKeioUniversity,Japan,in1996. Aftergraduation,heworkedasaroboticsengineeratSeikoInstrumentsfrom1983 to2001.Hedevelopedsomeindustrialrobotcontrollers.Hismaincontribution wastocreatedigitalservocontrolsystems,includingdisturbanceobservers,and designDCandACservomotordrivers.Concurrently,hewasaguestprofessor atChibaUniversity.HewasanassociateprofessoratthePolytechnicUniversity, Japan,from2001to2009.HehasbeenafullprofessoratShibauraInstituteof Technology,Japan,since2009.Hiscurrentinterestsincludemotioncontrol, robotics,controlengineering,andfreeclimbing.Thepresentstudythemesare motioncontrolandpathplanningandcollisionavoidanceforhumanoidclimbing robots,wheeledmobilerobots,invertedpendulumrobots,autonomousdrones, etc.Hisphilosophyforthestudyistohaveactualandpracticalexperience.When hestudiescookingrobots,hecooksavarietyoffoods.Todevelopclimbingrobots, heclimbsexistingwallsinthemountainsorclimbinggymseveryweek.Heisa memberofIEEE,SICE,andRSJandaseniormemberofIEEJ.

Preface

Thereisanestimationmethodcalledthe“disturbanceobserver.”Whensome mechatronicsystemsmove,friction,gravity,andexternalforcesmaydisturbtheir motion.Werefertothemcollectivelyasdisturbances.Thedisturbancesareoften unmeasurable,andthecontentsareunknown.Ifwecanestimatethetotalvalue ofthesedisturbances,wecanimprovethestabilityandtrackingperformanceof thecontrolsystemorusethemininformationprocessing.Generally,theterm “observer”referstoaconferenceobserver,butitistranslatedas“stateobserver” incontrolengineeringandmeansafunctionforestimatingstatevariables.The observerwasproposedin1964byD.G.Luenberger,saidtobeadoctoralstudent atStanfordUniversitythen[1,2].Thedisturbanceobserverisanobserverthat estimatesdisturbances.SincethepublicationofthepapersbyKiyoshiOhishi, KouheiOnishi,etal.[3]andKouheiOhnishiandToshiyukiMurakami[4], theyhaveattractedwidespreadattentionandhavebeenstudiedandappliedby researchersandengineersworldwide[5–9].MeditchandHostetter[10]reported thedesignofa0-observerforestimatingunknownstationaryinputsanda k-observerforestimatingunknowninputsrepresentedby k-degreepolynomials. Theextendedsystemforobserverdesignisdefinedusingtheunknowninput u(t) as x =[x T (t), uT (t)]T inadditiontotheoriginalstatevariable x (t).Both x (t) and u(t) canbeestimatedaccordingtothegeneralobservertheoryifobservabilityholds. Thisisnotdifferentfromhowwedesigndisturbanceobserversintoday’sstate spacerepresentation.Additionally,manypapershavereportedvariousunknown inputestimationmethods[11–16].However,thename“disturbanceobserver” wasinventedbyfocusingintenselyonthedisturbance.Overall,“disturbance observer”includesalmostcompletedisturbancesuppressioncontrolbyfeeding backthedisturbanceestimatetocancelthedisturbance,suppressparameter fluctuation,andcontrolacceleration.Itmeansthatthe“disturbanceobserver technology”isconsideredtohavestartedwiththepapers[3,4],andmany others.Thisbookaimstosystematicallydescribethedesignprocess,application methods,andvariouspropertiesof“disturbanceobservers”sothattheycanbe

helpfultomanypeoplewhostudycontrol.Inthedesignofdisturbanceobservers formechatronicssystemcontrol,itisnecessarytoobserveorcalculatethe position(orangle)orvelocity(orangularvelocity)informationusingsensors.Itis essentialtoobtainhighlyaccuratevelocityinformationnotaffectedbynoise.We expressoursincerethankstoProf.ToshiakiTsujiofSaitamaUniversityandMr. HiroyukiNagatomifromOhnishiLab.Theycooperatedinwritingthepaperon velocitymeasurementandestimationtechniques.Thecontentsofthispaperwere testedinundergraduateandgraduateclasses,andthenmanysuggestionswere givenbythemembersoftheMotionControlLaboratory(ShimadaLaboratory), especiallyMr.KentaMatsuo,Mr.KazukiTokushige,Mr.KatsumichiTakase, Mr.RyoyaNakajima,andMs.YukaKimura.Inaddition,Prof.TakashiOhhira ofChuoUniversitypointedoutinadequaciesinthedescriptionsandmade manysuggestions.CoronaPublishingCo.,Ltd.publishedthisbookasoneof thenewbookssolicitedbytheSocietyofInstrumentandControlEngineers (SICE)in2021.WeexpressoursinceregratitudetoProf.ShiroMasudaofTokyo MetropolitanUniversitywhowasinchargeofthisbookandthePublication Committeefortheirsupportinitscompletion.Finally,wethankProf.Kouhei Ohnishiandmanyfriendsfortheircontinualmeetingsandguidance.

References

1 DavidG.Luenberger:Observingthestateofalinearsystem,IEEE TransactionsonMilitaryElectronica,Vol.8,No.2,74–80,1964.

2 GeorgeEllis:ObserversinControlSystems:APracticalGuide,AcademicPress, 2014.

3 KiyoshiOhishi,KouheiOhnishi,KunioMiyachi:Torque-SpeedRegulationof MotorBasedonLoadTorqueEstimationMethod(IPEC-Tokyo’83),1209–1218, 1983.

4 KouheiOhnish,ToshiyukiMurakami:Advancedmotioncontrolinrobotics, 15thAnnualConferenceofIEEEIndustrialElectronicsSociety(IECON’89), 356–359,1989.

5 AsifSabanovic,KouheiOhnishi:MotionControlSystems,Wiley-IEEEPress, 2011.

6 EmreSariyildiz,RobertoOboe,KouheiOhnishi:Disturbanceobserver-based robustcontrolanditsapplications:35thanniversaryovervew,IEEETransactionsonIndustrialElectronics,Vol.67,No.3,2024–2053,2020.

7 ShihuaLi,JunYang,Wen-HuaChen,XisongChen:Disturbance Observer-BasedControl,CRC-Press,2014.

8 AkitaShimada,KiyoshiOhishi,MasaakiShibata,OsamuIchikawa: EEtextmotioncontrol,IEEJ&Ohmsha,2004,118–119,157–164,192–202 (InJapanese).

9 AkiraShimada:Recentadvancesandoutlookinindustrialinstrumentation andMECHATRONICScontrol,IEEJTransactionsofElectricalandElectronic Engineering,Vol.11,No.52,S100–S107,2016.

10 J.S.Meditch,G.H.Hostetter:Observersforsystemswithunknownand inaccessableinputs,InternationalJournalofControl,Vol.19,No.3,473–480, 1974.

11 Shih-HoWang,E.JournalDavison,PeterDorato:Observingthestatesof systemswithunmeasurabledisturbances,IEEETransactionsonAutomatic Control,Vol.20,No.5,716–717,1975.

12 C.D.Johnson:Optimalcontrolofthelinearregulatorwithconstant disturbances,IEEETransactionsonAutomaticControl,Vol.13,No.4, 416–421,1968.

13 JohnO’Reilly:Minimal-orderobserversforlinearmultivariablesystems withmeasurabledisturbances,InternationalJournalofControl,Vol.28, No.5,743–751,1978.

14 TsutomuMita:Onthesynthesisofanunknowninputobserverforaclassof multi-input/outputsystems,InternationalJournalofControl,Vol.26,No.6, 841–851,1977.

15 NobuakiKobayashi,TakayoshiNakamizo:Anobserverdesignforlinear systemswithunknowninputs,InternationalJournalofControl,Vol.35,No.4, 605–619,1982.

16 S.P.Bhattacharyya:Observerdesignforlinearsystemswithunknowninputs, IEEETransactionsonAutomaticControl,Vol.23,No.3,483–484,1978.

AbouttheCompanionWebsite

Thisbookisaccompaniedbyacompanionwebsite: www.wiley.com/go/disturbanceobserver

ThewebsiteincludessampleprogramswithMATLAB/Simulink.

IntroductionofDisturbanceObserver

The disturbanceobserver iscalledasthepronounof motioncontrol andhas beenhighlyevaluatedworldwide[1,2].Disturbancescanbeaddedtotheinputof thecontrolplant,theoutput,orevenanypartofthe internalstate.Therearetwo typesofdisturbanceobservers(DOBs):thosethatonlyestimatedisturbancesand thosethatalsoestimatestatevariables,suchaspositionandvelocity.1 Theyare collectivelycalledDOBsunlessonewantstoemphasizesomethinginparticular. Nomatterhowgoodthecontrolorestimationmethodis,itisnotapanacea. Whenanewcontrolmethodisproposed,itisoverhyped,leadingtoaboom. Examplesinclude H∞ control,slidingmodecontrol,andmodelpredictive control.However,astheboomcontinuesforawhile,theirdisadvantagesbecome apparent,suchastheneedfortheskillandtechniqueinusingthemandtheir compatibilitywiththecontrolplant.DOBsarenoexception.Itisessentialto understandtheirprosandconstousethemwell.

Theintendedreadersarestudentsandprofessionalslearningthetheoriesand techniquesrelatedtocontrolengineeringandmotioncontrol.Theyareassumedto havesomeknowledgeof classicalcontroltheory and moderncontroltheory. Aminimalexplanationisprovidedintheappendixforreaderswithoutsufficient knowledge.

1.1TypesofDisturbanceObservers

1.1.1Introduction

ThisbookintroduceseighttypesofDOBsfordesigningaDOB,asshownin Table1.1. Kalmanfilter intheeighthlineisnotanobserverbutisincludedin

1Exampleshavebeenproposedsuchasreferringtoitasdisturbanceandvelocityestimation observer[3,4]becauseitestimatesvelocityaswellorreactionforceestimationobserver[5] becauseitestimatesreactionforcespecifically.

DisturbanceObserverforAdvancedMotionControlwithMATLAB/Simulink,FirstEdition.AkiraShimada. ©2023TheInstituteofElectricalandElectronicsEngineers,Inc.Published2023byJohnWiley&Sons,Inc. Companionwebsite:www.wiley.com/go/disturbanceobserver

1IntroductionofDisturbanceObserver

Table1.1 Typesofdisturbanceobserverdesign.

Systems DOBdesignforms Objecttoestimate

(1)Transferfunction Onlydisturbances

Continuous (2)Identityobserver Allstatevariablesanddisturbances system (3)Minimalorderobserver Allstatevariablesexceptoutputs (4)Adaptiveobserver Parameters,statevariablesanddisturbances (5)Transferfunction Onlydisturbances

Digital (6)Identityobserver Allstatevariablesanddisturbances system (7)Minimalorderobserver Allstatevariablesexceptoutputs (8)Kalmanfilter Allstatevariablesanddisturbances thistablebecauseitisdesignedforestimatingdisturbances.TheKalmanfilter andtheadaptiveobservercanbedesignedasbothcontinuousanddigitalsystems, buttheyarelimitedtotheabove.

Manyreferencesusethe transferfunction ofthecontinuoussystemtorepresentDOBs,specializingonlyinitsestimation,whichcorrespondstothefirstline ofTable1.1.Thismaybebecauseitissimpleandeasyforustounderstandand implement.However,wecansimultaneouslyestimatethecontrolplant’soriginal statevariables,suchasvelocityandcurrentotherthandisturbances,usingthe generalobservertheory.Thesecorrespondtolines2–4and6–8ofthetable.

Supposethatreaderswanttousevariousphysicalestimateseffectivelyin designingthecontrolsystem.Inthatcase,theycandesigntheDOBusingthe identityobserver designingmethod.However,iftheyemphasizethatthey donotneedtoestimatetheobservedoutput,theycanusethe minimalorder observer designmethodasanexcellentchoice.

Theoretically,DOBsexpressedintransferfunctionsareequivalenttothose derivedfromthedesignprocessforminimalorderobservers,asshowninrows 3and7.Manystudiesusetheformofthetransferfunctionfromthebeginning becauseonlythedisturbanceestimationfunctionisextractedandreexpressedin theformofatransferfunctioninthedesignprocess.

Thedifferencebetweenacontinuousandadigitalsystemiswhetherthedesign isbasedoncontinuouscontroltheoryordigitalcontroltheory.2 Ifthereisnoneed toconsidertheeffectofthelengthorshortnessofthecontrolperiod,itisbetter todesignacontinuoussystemwherethephysicalmeaningiseasytograsp.However,itismoredesirabletodesignitasadigitalsystemifthecontrolcycleandthe

2Notethatthisbookdoesnotdistinguishbetweenthewordsdiscretesystemsanddigital systems.

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