<%BANNER%>

High Performance Control Theory, Design, and Applications

Permanent Link: http://ufdc.ufl.edu/UFE0022096/00001

Material Information

Title: High Performance Control Theory, Design, and Applications
Physical Description: 1 online resource (146 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: engineering, online, pid, predictive, process, recovery, steady, swarm, tracking
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Online virtual labs offer potential for students to participate in laboratory practice without access to physical laboratory equipment. Existing online labs show a tradeoff between extensibility for different systems and interactivity with the user. A Virtual Control Lab that overcomes this hurdle has been developed using LabView. A modular structure that allows a developer to simulate a new system is presented to the user by a system of panels and tabs which provides the user a variety of system analysis and real-time simulation data. Classical PID controllers cannot track a ramp change in a set point or disturbance for most systems. In order to achieve offset-free behavior a controller with an additional integral is required. A double integral controller which could be practically implemented as two standard PI controllers connected in series is suggested. Tuning correlations are developed to minimize the integral of the time-weighted absolute error for tracking of a set point ramp. Model Predictive Control is a common control strategy for industrial ultivariable systems; however, it suffers from many of the same offset related problems as proportional control. MPC methods are described which use integral states and input-velocity weights to track steps, and in some cases ramps, without offset, while avoiding the complexity of previous tracking methods based on targets and disturbance estimation. A controller is also proposed to use the property of flatness for control of a swarm of wheeled mobile robots. The controller is augmentented with formation tracking and collision avoidance routines.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Crisalle, Oscar D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022096:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022096/00001

Material Information

Title: High Performance Control Theory, Design, and Applications
Physical Description: 1 online resource (146 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: engineering, online, pid, predictive, process, recovery, steady, swarm, tracking
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Online virtual labs offer potential for students to participate in laboratory practice without access to physical laboratory equipment. Existing online labs show a tradeoff between extensibility for different systems and interactivity with the user. A Virtual Control Lab that overcomes this hurdle has been developed using LabView. A modular structure that allows a developer to simulate a new system is presented to the user by a system of panels and tabs which provides the user a variety of system analysis and real-time simulation data. Classical PID controllers cannot track a ramp change in a set point or disturbance for most systems. In order to achieve offset-free behavior a controller with an additional integral is required. A double integral controller which could be practically implemented as two standard PI controllers connected in series is suggested. Tuning correlations are developed to minimize the integral of the time-weighted absolute error for tracking of a set point ramp. Model Predictive Control is a common control strategy for industrial ultivariable systems; however, it suffers from many of the same offset related problems as proportional control. MPC methods are described which use integral states and input-velocity weights to track steps, and in some cases ramps, without offset, while avoiding the complexity of previous tracking methods based on targets and disturbance estimation. A controller is also proposed to use the property of flatness for control of a swarm of wheeled mobile robots. The controller is augmentented with formation tracking and collision avoidance routines.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Crisalle, Oscar D.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022096:00001


This item has the following downloads:


Full Text

PAGE 1

HIGHPERFORMANCECONTROLTHEORY,DESIGN,ANDAPPLICATIONSByCHRISTOPHERS.PEEKADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2008 1

PAGE 2

c2008ChristopherS.Peek 2

PAGE 3

ACKNOWLEDGMENTS IthankProfessorDenisGillet,PiyawayKaekward,andSamuelDeprazofEcolePolytechniqueFederaledeLaussanne(EPFL)fortheircollaborationinthevirtualcontrollabandswarmcontrolwork. 3

PAGE 4

TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2VIRTUALCONTROLLABORATORY ...................... 11 2.1Introduction ................................... 11 2.1.1Background ............................... 11 2.1.2ProblemStatement ........................... 12 2.2ArchitecturalRequirementsforaVirtualControlLaboratory ........ 13 2.2.1GeneralPrinciples ............................ 13 2.2.2SpecicArchitecturalRequirements .................. 15 2.3RealizationofaVirtualControlLaboratory ................. 19 2.3.1SoftwarePlatformforDevelopment .................. 20 2.3.2InterfaceDesignBasedonThreeSpecializedPanels ......... 20 2.3.3AnimationandInteractionPanels ................... 21 2.3.4NavigationPanelanditsTabbedWindows .............. 23 2.3.5PublicationandDeploymentontheWorldWideWeb ........ 27 2.3.6ModularityOtherSoftwareDesignDetails .............. 28 2.4PedagogicalScenarios .............................. 30 2.5Conclusions ................................... 32 3DOUBLEINTEGRATORCONTROLDESIGN .................. 39 3.1Introduction ................................... 39 3.1.1Background ............................... 39 3.1.2ProblemStatement ........................... 41 3.2Approach .................................... 42 3.3OptimalTuningRelationships ......................... 43 3.4AnalysisandDiscussion ............................ 45 3.4.1ComparisonofthePI2SchemesConsidered .............. 45 3.4.2ComparisonwithLiterature ...................... 45 3.4.3ValidationoftheOptimizationResults ................ 46 3.5Conclusions ................................... 47 4

PAGE 5

4OFFSETFREEMODELPREDICTIVECONTROL ............... 53 4.1Introduction ................................... 53 4.2ProblemStatement ............................... 55 4.3Oset-FreeMPCMethodsProposed ..................... 60 4.3.1MethodI:IntegralStates ........................ 60 4.3.2MethodII:Input-VelocityCost .................... 66 4.3.3MethodIII:IntegralStatesandVelocityControl ........... 67 4.3.4MethodIV:DoubleIntegralStates .................. 68 4.4Estimators .................................... 70 4.5SimulationStudy ................................ 71 4.6Conclusion .................................... 75 5COLLECTIVEMOTIONUSINGFLATNESS-BASEDCONTROLMETHODS 98 5.1Introduction ................................... 98 5.1.1LiteratureReview ............................ 99 5.1.2ProposedDesign ............................. 99 5.2Flatness-BasedControllerforaSingleVehicle ................ 100 5.3CollectiveMotion ................................ 102 5.3.1NotationandDenitions ........................ 103 5.3.2ControlDesignforTrajectoryTracking ................ 104 5.3.2.1FormationSchemes ...................... 104 5.3.2.2LeaderController ....................... 106 5.3.2.3NonleaderController:NonrotationalFormation ...... 107 5.3.2.4NonleaderController:RotationalFormation ........ 108 5.3.2.5NumericalSimplications .................. 110 5.3.3ControlDesignforCollisionAvoidanceandRecovery ........ 111 5.4SimulationStudies ............................... 113 5.5Conclusions ................................... 115 6CONCLUSION .................................... 121 APPENDIX AALGEBRAICBACKGROUNDFORRAMPTRACKINGCONTROLLERS .. 122 A.1ComparisonofPI2ControllerswithBelangerandLuyben'sDesign .... 122 A.2ConstraintsRelatingtheDierentPI2schemes ............... 123 A.3ProofofOsetEliminationforSeveralFormsofPI2TrackingRamps ... 123 BEXAMPLEMATLABCODE ............................ 126 B.1DoubleIntegralControllerTuning ....................... 126 B.1.1TuneCorMain.m ............................. 126 B.1.2itae.m .................................. 128 B.2OsetFreeMPC ................................ 129 B.2.1Example1.m ............................... 129 5

PAGE 6

B.2.2MPCMain.m ............................... 131 B.2.3MPCLoadPlant.m ............................ 134 B.2.4MPCGenerateSP.m ........................... 137 B.2.5MPCGenerateRampSP.m ........................ 137 B.2.6MPCDeltas.m .............................. 138 B.2.7MPCMatsI.m .............................. 139 B.2.8MPCMatsII.m .............................. 140 B.2.9MPCGains.m .............................. 142 REFERENCES ....................................... 143 BIOGRAPHICALSKETCH ................................ 146 6

PAGE 7

LISTOFTABLES Table page 3-1Tuningrelationshipsforallthreeschemes ..................... 44 4-1EquationsandsimulationresultsforvariousMPCmethods ............ 76 5-1Parametersusedinthesimulationstudies ..................... 114 7

PAGE 8

LISTOFFIGURES Figure page 2-1ADCmotorVirtualControlLabaccessedbystudentsusingawebbrowser. .. 34 2-2AnimationpanelandinteractionPanel ....................... 35 2-3Thecontrollertabofthenavigationpanel. ..................... 36 2-4Theanalysistabofthenavigationpanel. ...................... 37 2-5Thesimulationtabofthenavigationpanel. .................... 38 3-1Closedloopcontrolstructure ............................ 48 3-2Scheme1optimalITAEtuningrelationship. .................... 48 3-3Scheme2optimalITAEtuningrelationship. .................... 49 3-4Scheme3optimalITAEtuningrelationship. .................... 49 3-5TimeresponsesofthethreePI2schemesforvariousvaluesofplantparameters. 50 3-6ComparisonofLuyben'sPI2controllerwithscheme2proposedhere. ...... 51 3-7Contourplotsforonetimeconstantcalculations. ................. 52 4-1AblockdiagramillustratingthevariouselementsofanMPCsystem. ...... 93 4-2ResultsofMethodIMPC .............................. 94 4-3ResultsofMethodIIMPC .............................. 95 4-4ResultsofMethodIIIMPC ............................. 96 4-5ResultsofMethodIVMPC ............................. 97 5-1Thecoordinatesystemofthewheeledmobilerobot. ................ 116 5-2Anon-rotatingformation. .............................. 116 5-3Arotatingformation. ................................. 117 5-4Arobotusingcollisionavoidance. .......................... 117 5-5Arobotusingsmoothcollisionavoidance. ..................... 118 5-6Simulationofaswarmwithnorecoverystrategy .................. 118 5-7Simulationofaswarmwithadiscontinuousrecoverystrategy .......... 119 5-8Simulationofaswarmwithasmoothrecoverystrategy .............. 120 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyHIGHPERFORMANCECONTROLTHEORY,DESIGN,ANDAPPLICATIONSByChristopherS.PeekMay2008Chair:OscarD.CrisalleMajor:ChemicalEngineeringOnlinevirtuallabsoerpotentialforstudentstoparticipateinlaboratorypracticewithoutaccesstophysicallaboratoryequipment.Existingonlinelabsshowatradeobetweenextensibilityfordierentsystemsandinteractivitywiththeuser.AVirtualControlLabthatovercomesthishurdlehasbeendevelopedusingLabView.Amodularstructurethatallowsadevelopertosimulateanewsystemispresentedtotheuserbyasystemofpanelsandtabswhichprovidestheuseravarietyofsystemanalysisandreal-timesimulationdata.ClassicalPIDcontrollerscannottrackarampchangeinasetpointordisturbanceformostsystems.Inordertoachieveoset-freebehavioracontrollerwithanadditionalintegralisrequired.AdoubleintegralcontrollerwhichcouldbepracticallyimplementedastwostandardPIcontrollersconnectedinseriesissuggested.Tuningcorrelationsaredevelopedtominimizetheintegralofthetime-weightedabsoluteerrorfortrackingofasetpointramp.ModelPredictiveControlisacommoncontrolstrategyforindustrialmultivariablesystems;however,itsuersfrommanyofthesameosetrelatedproblemsasproportionalcontrol.MPCmethodsaredescribedwhichuseintegralstatesandinput-velocityweightstotracksteps,andinsomecasesramps,withoutoset,whileavoidingthecomplexityofprevioustrackingmethodsbasedontargetsanddisturbanceestimation.Acontrollerisalsoproposedtousethepropertyofatnessforcontrolofaswarmofwheeledmobilerobots.Thecontrollerisaugmententedwithformationtrackingandcollisionavoidanceroutines. 9

PAGE 10

CHAPTER1INTRODUCTIONControlseducationisimprovedbytheuseofavirtualcontrollaboratory.Thelaboratoryallowsaccesstosimulatedplantsandcontrollersoverawebinterfaceforanalysisandsimulationofcommoncontrolproblems.ThelabmakesuseofLabVIEWsoftwareandthenewsimulationandcontroldesigntoolkitsforittooersubstantialadvantagesoverexistingsimulationtoolsincludingagoodbalancebetweenmodularityforthedevelopmentofnewsimulationsusingthelabframeworkandsimulationinteractivityfortheenduser.Thisimplementationalsoprovidespotentialforswitchingbetweensimulationandremotecontrolofrealexperimentalapparatus.Thevalueofsuchalaboratoryisprimarilypedagogical,forteachingprinciplesofprocesscontrol,butitalsohaspotentialasatestrigfornewcontroldesignsorlabscaleapparatusduetoitsextensiblenature.Thetendencyofmanycontrollerstodrivethesystemundercontroltonalvaluesthatarenotexactlythosedesiredisknownasoset.Theoset-freetrackingoframpsinsetpointscanbeaddressedviaasimpledoubleintegralcontroller.Whilethebenetsofmultipleintegratorsarewellknown,asimpleimplementationofthisideaandtuningguidelinesareoered,whichcantransformtheknowntheoryintoapracticalandeectivecontrolscheme.Steadystateosetisalsoconsideredinthemoreadvanced,multivariable,ModelPredictiveControlcontrolstrategy.Severaloset-freemethodsareproposedwhichavoidthecomplexityofimplementationandcalculationthatarecharacteristicofthosealreadyavailableintheliterature.Theproblemofcontrollingaswarmofwheeledmobilerobotsisaddressedbyextendingaatness-basedcontrolstrategypreviouslydevelopedforasinglerobot.Thecontrolstrategyexaminedprovidesmethodsformovingtherobotsinformationandforavoidingcollisionsandrestoringtheformationintheeventthattheformationisdisrupted. 10

PAGE 11

CHAPTER2VIRTUALCONTROLLABORATORY 2.1Introduction 2.1.1BackgroundEngineeringeducatorsareincreasinglyinterestedindevelopingtoolsthatallowstudentstoacquirehands-onlaboratoryexperiencewithoutnecessarilyprovidingphysicalaccesstoabuildingthathousesspecicexperimentalequipment[ 1 ][ 2 ].Thesetoolsoftentaketheformofsoftwareenvironmentscommonlyreferredtoasvirtualorremotelaboratories[ 1 ][ 3 ][ 4 ][ 5 ].Thesoftwareeitherallowsausertointeractwithanexperimentalsetuplocatedinanothergeographicallocation(i.e.,aremotelaboratory)orusesnumericalsimulationtoolstoemulatethebehaviorofexperimentalsystem(avirtuallaboratory).Theappealofvirtualandremotelaboratorytoolsislargelyduetotheincreasingdemandforactivelearningandexibleeducation,andfortheappealofimplementingtechniquesoflearningviadiscovery.Activelearningseekstoprovidestudentswithopportunitiestobetterintegrateandreinforceknowledgepresentedintheclassroom,aswellastoacquirethepracticalknow-howthatissuchanessentialcomponentinengineeringeducation.Problem-basedlearningandhands-onlaboratoryactivitiesaregoodexamplesofactive-learningvehicles.Intraditionaleducationalsettings,aneectiveactive-learningenvironmentcallsforintenselevelsofinteractionwithexperimentalresources,andrequiresthecoordinationoftheeortsandschedulesofmultipleparties,includingnumerousstudentsandpedagogicalsupportsta.Asaconsequence,theeortmaybechallengedbysignicantobstaclesstemmingfromlogistical,organizational,andcostconcerns.Ontheotherhand,suchconstraintscanbeovercomebymakingavailableexiblelearningresourcesthatmeetthestudents'individualneedsandschedules.Inparticular,virtualandremotelaboratoriesoersubstantialexibleeducationbenetssincetheycanbemadeaccessibletothestudentsatanytimeandfromanylocation,featuresthatcannotbeeasilymatched 11

PAGE 12

bytraditionallearningenvironments[ 6 ].Furthermore,virtualandremotelaboratoryresourcescanbecombinedwithotherexible-educationtools,suchaslecturesoeredviadigitalstreaming-video,tomaximizethedegreeofexibilityofthelearningenvironmentprovidedtothelearners.Inaddition,anothersubstantialadvantageofvirtualandremotecontrollaboratorytoolsisthattheycanbeusedtosupplementtraditionalteachingmethodologies.Forexample,theycanbeusedduringatraditionallecturetoshowthestudentshowtoapplyconceptspresentedinclasstoasimulatedorremoteexperimentalsystem.Thisapproachimbuesaclassicallecturewithanactiveandexiblelearningcomponent,thusstrengtheningthepedagogicalvalueofthelecture.Anotherbenetofvirtualandremotelaboratoriesineducationisthattheypromotediscoverylearning.Inthisapproachstudentsaregivenaccesstothelaboratorywithminimalinstructions,andareallowedtoexplorethesystemsforthemselves[ 7 ].Avirtuallaboratoryorawell-designedremotelaboratorycanoerstudentsachancetoexploresafelyandeasilythebehaviorofasystemthatmaybephysicallyinaccessible. 2.1.2ProblemStatementGiventhesignicantappealoftheengineering-educationbenetsrealizedbyvirtualandremotelaboratories,itisnotsurprisingthatseveralsystemsarealreadyavailableontheWorldWideWeb(cf.[ 3 ][ 8 ][ 9 ][ 10 ]and[ 11 ],amongothers).Anearlyvisionforavirtualcontrollaboratoryarchitecturecanbefoundin[ 12 ].Theavailablesystemsaddresstheexiblehands-onlearningparadigmatdierentfunctionalitylevels,andwithdierentdegreesofstudentinteractivity.Somevirtuallaboratories,especiallythosedevelopedusingcommercialsimulationtools,canbeeasilycustomizedtosuittheuserneeds,buttendtoshowlimitedpotentialforinteractivity.Forexample,itisquitecommonthatasimulationrunhastobestoppedforthepurposeofchangingcontrollersettingsorprocessparameters.Softwareplug-insorexpensivespecializedthird-partysoftwarearealsooftenrequiredtorunthesimulations.AnotherclassofWeb-basedlaboratoriesattainasubstantialdegreeofinteractivitythroughtheuseofproprietary 12

PAGE 13

softwarewrittenbythelaboratorydevelopersinaformalprogramminglanguagesuchasJava.Asaconsequence,thesetoolsrequireknowledgeofthespeciclanguageadopted,andattemptstocustomizetheenvironmenttoaddressspecicneedsofthelearnermayinvolvesubstantialmodicationsofsourcecode.Therefore,theselaboratoriesaretypicallyonlydevelopedforaspecicandrestrictedsetofsimulatedorphysicalsystems.Anotherconcernwhenusingproprietarycodeisthatinstancesofthesamelaboratorysoftwaredevelopedfordierentphysicalsystemsoftenhavesubstantiallydierentinterfaces,requiringausertogetre-familiarizedwiththetoolsmadeavailableforeachtask.Insummary,thetypicalWeb-basedlaboratoriesthatarecurrentlyavailableseektostrikeareasonabletradeobetweenuserinteractivityandusageversatility.TheobjectiveofthispaperistodeneageneralarchitectureforaVirtualControlLaboratory(VCL)andtodemonstratearealizationofthatarchitectureintermsofaconcreteexample.TheVCLisasoftwareenvironmentspecializedforcontrolengineeringeducation,capableofavoidingthetrades-opresentinexistingWeb-basedlaboratories,andoeringasingleframeworkthatcanbeusedtoeasilydevelopWeb-based,interactivevirtualandremotelaboratoriesforthedeploymentinawidevarietyofsystemsandscenarios.Thetoolshouldbeabletoprovideeducationalbenetsinbothexibleandtraditionaleducation,beyondthoseattainedbyexistingsoftwarecontrolslaboratories.Itshouldalsofacilitatethedeveloper'staskofgeneratingnewlaboratoryexperimentsthroughamodulardesign. 2.2ArchitecturalRequirementsforaVirtualControlLaboratory 2.2.1GeneralPrinciplesTherelativeshortcomingsofvirtualcontrollaboratoriesthathavebeenproposedtodatecanbeovercomebyconceivingaVCLarchitecturethatmeetsthreegeneralprinciples:interactiveWeb-accesscapability,modularityofdesign,andanintuitiveanduser-friendlyinterface. 13

PAGE 14

WearguethataVCLtoolshouldbecapableofdeploymentintheWorld-Wide-WebenvironmentusingstandardWebbrowsers.ThealternativeofusingspecializedclientsoftwarerenderstheVCLlessexible,sincetheusercannotaccesstheresourcesusingonlystandardsoftwareavailableinmostcomputers.Hence,WebaccessisthekeyingredientinimbuingtheVCLwithexiblelearningfeatures,makingitpossibletoaccommodateeachstudent'spersonalschedule.Therefore,thesoftwaredevelopmentenvironmentshouldsupporteasypublicationoftheVCLontheWeb.Furthermore,theWebVCLtoolshouldbehighlyinteractive,allowingthestudenttochangeselectedparametersandimmediatelyobservetheeectofthechanges.SuchinteractivecapabilityiscrucialforthesuccessoftheVCLbecausethetoolseekstoemulateexperimentationactivitiesthatwouldtraditionallybeconductedinaphysicallaboratory.AnothercrucialrequirementfortheVCLenvisionedisthatthesourcecodeshouldbehighlymodular[ 13 ].AninstructorshouldbeabletousetheVCLframeworktoquicklyandeasilydevelopasimulationandanalysisenvironmentforanyplantorcontrollerthatisofinterest.Thismodularityisrealizedbydevelopingaframeworkwheretheinterfaceusedtosetuptheanalysisandsimulationforonephysicalsystemcanbeeasilymodiedtosetuptheanalysisandsimulationforadierentsystem.AthirdrequirementfortheVCListhatisshouldhaveauser-friendlyinterface.ThestudentmustnditeasytolearnhowtousetheWeb-baselaboratoryifthetoolisexpectedtohavesubstantialpedagogicalvalue[ 14 ].Anexcessivenumberofinstructionscaninterferewiththediscovery-learningprocess.Itisthereforeessentialthattheinterfacebeintuitive.Itmaybedesirabletomaketheinterfaceresemblethatofothersoftwareandhardwareinterfacesthatarealreadyfamiliartotheuser,tosimplifyandacceleratethelearningprocess.Anotherprincipleregardingtheintuitivenessoftheinterfaceisthatthesameinformationshouldbedeliberatelypresentedinseveraldierentformats,forinstanceviagraphicalanimation,throughdisplaysonsliderbars,andalsoinachart.Consistent 14

PAGE 15

useofcolorisrecommendedtohelptheuserquicklyidentifydierentrepresentationsofthesamedata. 2.2.2SpecicArchitecturalRequirementsSpecicarchitecturalrequirementsthatareofcentralconcerninthedesignofaVCLincludethefollowing:(1)theabilitytodeneandmodifytheplantandthecontroller,(2)easytransferfromopen-looptoclosed-loopoperationbyswitchingthecontrollerfrommanualtoautomaticmode,(3)theinclusionofadvancedanalysistools,(4)theabilitytosupportcontrolsynthesistasks,(5)theabilitytoproduceopen-loopandclosed-loopsimulationoftime-domainresponses,(6)theinclusionofversatilesignalgeneratorstoproducestandardinputsignals,(7)theinclusionofanimationtoprojectavisualperceptionoftheevolutionoftheplant,(8)theabilitytoincorporatethedynamicsofsensorsandactuators,asneeded,and(9)embeddeddocumentationandhelp-guides,and(10)reportingcapabilities.Asatoolspecializedforcontrolengineering,theVCLmustincludetwokeycomponents,namelyaphysicalsystemofinterest(theplant),andacontroller.ExamplesofplantstypicallyconsideredforpedagogicalpurposesinstandardtextbooksincludeaDCmotor,aninvertedpendulum,ahelicopter,andachemicalreactor,amongothers.Theplantmodelmaycontainsignicantnonlineardynamics.Ontheotherhand,unlesstheVCLisspecicallydesignedtoillustratenonlinearcontroltheory,fromaneducationalperspectiveitisparticularlyimportanttoincludealinearizedversionoftheplantmodeltoenablethestudentstodevelopexperiencewiththesystematicandwell-understoodtoolsoflinearcontrolanalysisanddesign.Inaddition,theVCLshouldallowtheusertochangethevalueofselectedphysicalparametersoftheplant,suchastheinertiaofaDCmotorortheheat-transferareainachemicalreactor,thusallowingfortheinvestigationofthedierentdynamic-responseregimes.Examplesofcontrollersincludetheubiquitousproportional-integral-derivative(PID)controlscheme,lead-lagcontrol,on-ocontrol,cascadecontrol,andoptimalcontrol, 15

PAGE 16

amongothers[ 15 ][ 16 ][ 17 ][ 18 ][ 19 ][ 20 ][ 21 ][ 22 ].Theusershouldbeabletochangekeyparametersofthecontroller.Forexample,toadjustthegainofaPIDcontrollerforthepurposeofcarryingouttuningexperiments.Whenpedagogicallysuitable,theusershouldbeabletoeasilyselectfromalistofrelevantcontrolschemes.Forexample,theusermaychoosetoselectasinglePIDcontrollerandproceedtoevaluateitsperformance,andthenselectadual-PIDcascadeschemeforevaluationandsubsequentcomparisonwiththesingle-PIDchoice.Inthecaseofmultiloopcontrolschemes,theVCLshouldeasilyallowtheusertomodifytheselectionofappropriateinput-outputpairings.TheVCLshouldmakeavailabletotheuserconciseandfactualinformationabouteachcontrolschemeavailable.AnotherrequiredfeatureofaVCListhatitshouldenabletheusertosetthecontrollerinmanualmode,allowingtheusertotesttheresponseoftheplanttodiagnosticinputsignals,suchasastepforcingorasinusoidalexcitation.Thissituationisknownasopen-loopcontrolbecausetheoutputoftheprocessiseectivelydisconnectedfromthecontroller.Alternatively,theusermayselectaspeciccontrolstructurefordeployment,inwhichcasetheplantismanipulatedinautomaticmodebythecontroller,deningaclosed-loopcontrolconguration.AkeyrequirementisthattheVCLshouldclearlyandunambiguouslyshowtotheuserthatthesystemisineitheropen-looporclosed-loopmode.Thiscanbeachievedbydisplayingamessageinalltherelevantlocationswhereitisimportanttodistinguishbetweenthetwomodes.Anothersolution,whichisattractivefromapedagogicalviewpoint,wouldbetopresenttheuserwithaspecicgraphicalrenditionoftherelationshipbetweenthemanualcontrollerandtheplantunderopen-loopoperation,clearlyshowingthattheoutputoftheprocessisnolongeravailabletothecontroller.TheVCLwouldthenswitchtoadierentgraphicalrepresentationduringclosed-loopoperation,perhapsshowingasignal-owdiagramthatclearlyconnectstheoutputsoftheplanttotheautomaticcontrollerandtheoutputsofthecontrollertotheinputportsoftheplant. 16

PAGE 17

Animportantconcepttaughtincontrolengineeringistheprocessofanalysisofboththeopen-loopandtheclosed-loopsystem.Underopen-loopcontroltheanalysisstudiesconsistofrevealingallthefeaturesoftheplantthatareofdynamicrelevance.Forlinearsystemstheanalysistaskofteninvolvesthedeterminationoftheplantpolesandzeros,thecharacterizationofthefrequencyresponseoftheplantthroughmagnitudeandphasediagramsintheformofBodeorNicholsplots,aswellasacharacterizationofthetime-domainresponseoftheplanttosteporsinusoidalinputsignalsimplementedviathemanual-modecontroller.TheVCLshouldupdatethepoles,zeros,Bodeplots,andtheotheranalysisimmediatelyaftertheuserchangesthevaluesoftheplantparameters.Theopen-loopanalysistoolsincludedintheVCLshouldallowtheusertoclassifytheplantasstableorunstable,toassessitslevelofdamping,andtodeterminethevalueofitsdominanttimeconstantsanditsbandwidth,etc.Underclosed-loopcontroltheanalysistaskalsoconsistsofproducingamapofpolesandzeros,generatingBodeorNicholsplots,andtracingtime-domainresponsecurves;however,inthiscasetheresultsdescribetheentireloopcreatedbetweentheplantandtheautomaticcontrollerratherthantoonlytheplant.Forexample,aclassicalclosed-loopanalysistaskisthedeterminationofwhetherthecontrollersucceedsincreatingaloopthatbeisstable,anissuethatcanbeassessedbyinspectingthelocationoftheclosed-looppolesonthecomplexplane.TheVCLmustbeabletoupdatethemapoftheclosed-looppolesimmediatelyaftertheusermakeschangestotheparametersofthecontrollerorafteranewcontrollerstructureisselected.Obviously,otheranalyticalresultsshouldalsobeupdatedaftersuchchangesoccurinthecontrollerorplantparameters.Anotherkeyconcepttaughttocontrolengineeringstudentsistheprocessofcontrolsynthesis,whichtypicallyconsistsofspecifyinganappropriatecontrolstructure,forinstanceaPIDoralead-lagscheme,aswellasvaluesoftheparametersincludedinthestructure,suchascontrollergainsandtuningtime-constants.Suchspecicationmustmeetclosed-looprequirements,suchaslocationofclosed-looppoles,degreeofovershoot, 17

PAGE 18

settlingtimeofthetime-domainresponse,etc.ArequirementoftheVCLarchitectureistheabilitytoquicklydisplaytheeectofanychangesmadetothecontrollerandtoquantifytheresultingclosed-loopperformance,sincethisallowsthestudentusertocarryoutsynthesiswork.Sincetheassessmentofthecontrollerperformanceinclosed-loopmodeisinfactananalysistask,theVCLshouldbeabletodeliveradesirablecontrolsynthesisenvironmentwheneveritsanalysiscomponentmeetsthearchitecturalspecicationspreviouslydiscussed.ItisofcriticalimportancethattheVCLenablethestudenttocarryoutopen-loopandclosed-loopsimulationsoftime-domainresponses.Numericalsimulationstudiesunderopen-loopmodepermitthecharacterizationoftheplantdynamicsthroughtheobservationofplotsthatdocumenttheresponseproducedbystandardinputsignals,suchasstepandsinusoidalexcitations.Inaddition,theclosed-loopsimulationcapabilitiesallowthestudenttoobservethetime-domaineectofanalyticalfeatures,suchastheeectoftheclosed-looppolesontheplantresponse,aswellastoassessthetime-domainperformanceofalternativecontroldesignchoices.TheVCLshouldthereforeincludesoftwareresourcesthatsolvedierentialequationsusingrobustnumericaltechniques,includingtheabilitytocopewithsituationswheretheequationsaresti.TheVCLmustincludeanumberofsignalgeneratorsthatcaninjectappropriatetypesofinputsattheuser'sdiscretion.Theseelementsshouldbeabletoproducestandardsignalpatterns,includingconstant,step,sinusoidal,ramp,andpulsesignalsofdierentamplitudesandfrequencies.MakingavailablealargepaletteofoptionsincreasestheversatilityoftheVCL,giventhatcertainproblemsrequiretheuseoflesscommonsignals,suchassaw-toothpatterns,andotherspecializedinputwaveforms.Inparticular,thesignalgeneratorsareusedtodeneset-pointsignalsthatarefedtothecontrollerunderclosed-loopcontroloperation,toproduceauser-speciedmanualsignalemanatingfromthecontrollerunderopen-loopoperation,ortoinjectaprocessdisturbancesignalasneededtotestacontroller'sabilitytoperformasaregulator. 18

PAGE 19

AdesirablefeatureinaVCListheabilitytopresenttotheusertheevolutionoftheplantvariablesviatwo-dimensionalorthree-dimensionalgraphicalanimation.Forexample,themotionofaroboticarmmaybeshowninagraphicalformthatgivestheusertheimpressionthathe/sheisobservingthemovementasitwouldoccurinthreedimensions,muchasifthemotionwerepresentedtotheobserverthroughavideo-cameraimage.Asanotherexample,themotionoftheindicationneedleinapressuregaugecanbeshowntoreectchangesingaspressureinsideachemicalreactor.Althoughtheessentialelementsofthecontrollooparetheplantandthecontroller,theVCLshouldhaveenoughexibilitytoincludeothercomponentsthataresometimesthefocusofsignicantpedagogicalinterest,suchasthecaseofsensors,powerconverters,andactuators.TheVCLshouldincludedocumentationthatpresentstotheuserafundamentaldescriptionoftheplantandofthecontroller,includingthedynamicmodelutilizedforanalysis,alongwithacleardescriptionofthemostimportantphysicalparameters.Evenwhentheinterfaceishighlyintuitive,explicitinstructionsarenecessarytoensureeaseofuse.ThisisespeciallytruesincethesystembeingsimulatedorthecontrollertypeusedmaynotbefamiliartobeginnerlearnerswhostandtogainthemosteducationalvaluefromtheVCLtool.Inordertoaccomplishthis,helpdocumentsshouldbeembeddeddirectlyinthevirtuallaboratoryinterface,makingthemconspicuousandeasytoaccess.Thedocumentationshouldbecomplete,thoughconcise,andshouldbepresentedfromaperspectivethatisusefulforcontrolengineeringdesignandanalysis.InclusionintheVCLofallrelevantplantandcontrollerdocumentationaswellasausersguidecontributestomakingtheenvironmentself-contained,andhencemoreeectivefromthepedagogicalpointofview. 2.3RealizationofaVirtualControlLaboratoryTakingintoconsiderationthearchitecturalprinciplesandrequirementsgiveninSection 2.2 ,theauthorshavedevelopedaVCLthatcanbeaccessedbystudentsusingaWebbrowser.Figure 2-1 showsaviewoftheproposedVCLinsideastandard 19

PAGE 20

Web-browserwindow,asitispresentedtoastudent.InthiscasetheplantstudiedisaDCmotorthatisshownasananimatedgraphicsontheupper-leftareatheVCLinterfacewindow.Onthelower-leftrectangleofthewindowtheusercandenedierentphysicalparametersfortheDCmotor,andcanalsoselectacontrollerstructureaswellasspecifyallvaluesofthecontrollerparameters.Finally,theright-halfofthescreenshowsvetabbedwindowsthatcontainvarioustypesofinformation,includingplantandcontrollerdocumentation,aswellastheresultsofanalysisandsimulationstudies. 2.3.1SoftwarePlatformforDevelopmentTheexampleshowninFigure 2-1 isarealizationoftheVirtualControlLabconceptcreatedusingNationalInstruments'LabVIEWsoftware[24].ThissoftwareoersmanypowerfulresourcesfordevelopingandmaintainingaVCL,includingeaseofcreatingstandardinterfacecontrols,andbuilt-inWebpublishingcapability.Inaddition,thesoftwareprovidesasuiteofresourcesusefulforcontrol-engineeringeducation,suchastheLabVIEWSimulationModuleandtheLabVIEWControlDesignToolkit,whichautomatemanyoftheanalysisandsimulationtaskscarriedoutbytheVCL.TheseadvantagesofLabVIEW,combinedwiththeoptionofprogrammingusingagraphicallanguage,makedeveloping,maintaining,andmodifyingtheVCLarelativelysimpleandstraightforwardtask.DevelopmentandimplementationalternativesforconceptsanalogoustotheVLCprototypeproposedherearealsopossibleusingotherprogramminglanguages,asdescribedin[ 23 ];however,inthatapproachadvancedanalysistaskstypicallyrequirecallstoexternalsoftwarepackages. 2.3.2InterfaceDesignBasedonThreeSpecializedPanelsToprovidetheuserwithanorganizedandintuitiveaccess,aninterfaceconsistingofthefollowingthreedistinctareasisproposed:anAnimationPanel,anInteractionPanel,andaNavigationPanel.Figure 2-1 showsanexampleoftheinterface,featuringthethree-panelstructure.TheAnimationPanel(locatedontheupperleftregionoftheinterfacewindow)andtheInteractionPanel(locatedonthelowerleftregion)are 20

PAGE 21

permanentlydisplayed,allowingtheobservationofkeyfeaturesofthesystemandthecontrolleronacontinuousbasis,emulatingaperspectivethattheuserwouldhavewhenrunningtheequivalentexperimentinaphysicallaboratory.TheNavigationPanel,thelargerrectangulararealocatedontherightoftheinterface,containsvetabsthatactivatewindowsthatdisplaydierenttypesofinformation.Wheneverpossible,standarddialogbuttons,inputboxes,andtabpanelsthatresemblethosefoundincommonhome-computersoftwareareused.Thus,usershavetolearnonlythespecicsofthelaboratoryexperimentitself,avoidingtheconfusionthatmaybeintroducedbyesotericinterfacecontrols.TheintuitivenatureoftheVCLinterfaceisalsoenhancedbyassigningeachinputandoutputacolor,andwheneverpossible,labelinginterfaceelementsrelatedtoagivenvariablewiththeappropriatecolor. 2.3.3AnimationandInteractionPanelsTheAnimationandInteractionPanelsarecloselyintegratedwitheachother,anditisthereforeconvenienttodiscussthemjointly.ThepurposeoftheAnimationPanelistogivetheuseravisualrepresentationoftheplant,alongwithindicatorsthatdisplaytwo-dimensionalorthree-dimensionalmotion.ThepurposeoftheInteractionPanelistoallowtheusertodeneplantparameters,andtoadjustcontrolparametersasneeded.ChoicesmadebytheuserintheInteractionPanel(suchasachangeinthecontrollerstatefromManualtoAuto)triggerachangeinwhatisdisplayedintheAnimationPanel.Hence,thetwopanelsarehighlycoupled.Figure 2-2 showsanimplementationoftheAnimationandInteractionPanelsoftheDC-MotorVCL.TheInteractionPanelinFigure 2-2 ashowsasituationwherethecontrollerissetinManualmode(seethearealabeled"ControllerParameters").Hence,thesystemisinanopen-loopconguration.TheAnimationPanelofFigure 2-2 ashowsacontrollerlikedwithanelectricalwiretotheinputofathree-dimensionalrenditionoftheDCmotor.Inturn,themotorhasanelectricalconnectiontotwosensors,namelyvelocityandpositionindicators.TheDCmotorisrepresentedusingageometricalperspective 21

PAGE 22

consistingofastationarycylindricalbody(thestator)connectedtoarotatingdisk(therotor).Anelectricalwireconnectsthemotortotheangularvelocityindicatorrepresentedbyaspeedometer,andalsotoarounddialthatdisplaystheangularposition.ThesetwooutputindicatorsreectthefactthatinthisDCmotorcontrolproblemeithertheangularpositionortheangularvelocity!mayserveastheprocessvariable(PV),namelyasthemeasuredvariablethatiseventuallytobemaintainedatasetpointbythecontroller.SincethecontrollerissettoManualmodeintheInteractionPanel,thesystemisconguredinopen-loopmodeandconsequentlytheAnimationPanelshowsthattheinputwirerepresentingtheprocessvariable(PV)isdisconnectedfromtheindicators.WheninManualmode,thestudentusercanchangeatwilltheoutputofthecontrollerindicatedinthepanelasthemanipulatedvariable(MV),usingeitherthebiasboxintheInteractionPanelorusingasignalgeneratorlocatedintheNavigationPanelthatisdiscussedlaterinthissection.Figure 2-2 bdepictsasituationwheretheuserhasusedtheInteractionPaneltoswitchthecontrollertoAutomode.InthiscasetheInteractionPanelshowsthatthestudenthasspeciedtheangularpositionasthechosenprocessvariable(PV),asindicatedintheboxlabeled"MVtoPVpairing".Thesystemisnowinaclosed-loopcongurationwheretheangularpositionistheprocessvariableusedforfeedbackcontrol.ThestructureoftheloopismadeobvioustotheuserintheAnimationPanel,wherethecontrollerinputwirelabeledPVisconnectedtotheangularpositionindicator.Thewiringschemedenesaclosedloop,henceservingasavisualaidtoenhancethestudent'sawarenessofthefactthatthecontrollerisintheAutomodeofoperation.Notetheappearanceofawhiteneedleontheangularpositionindicator,representingthecontrollersetpoint.Likewise,awhitetriangularmarkerisvisibleonthesideoftheverticalslidebarthatrepresentstheangularposition,alsoindicatingthevalueofthesetpointofthecontroller.Eitherthewhiteneedleorthewhitetriangularmarkercanbemovedtodierentpositionsusingacomputermouse,thusintroducingaset-pointchange.Set 22

PAGE 23

pointchangescanalsobespeciedusingasignalgeneratorlocatedintheSimulationwindowoftheNavigationpanel,asdiscussedlater.TheInteractionPanelshowsthevaluesoftheproportional-integral-derivative(PID)controller,includingagainKc=1,anintegraltimeconstanti=0:8,andaderivativetimeconstantd=1.Immediatelytotheleftoftheboxesthatspecifythevaluesoftheintegralandderivativetimeconstants,theuserndssquarebuttonsdrawnwithareliefperspective,whichcanbepressedviaacomputermouseclicktoactivatetheintegralandderivativeterms.Thegureshowsthattheintegral-actiontermisturnedOFF(whichhastheeectofignoringthenumerically-speciedvalueintheibox),anditalsoshowsthatthederivative-actiontermisturnedON.Inkeepingwiththeobjectiveofmaximizinginteractivity,theparametersshownintheInteractionPanelmaybechangedatanytime,evenwhileasimulationisrunning.Figure 2-2 cshowsanalternativeautomatic-controlcongurationspeciedbytheuserviatheInteractionPanel.Inthiscasetheselectedprocessvariableistheangularvelocity!,asclearlyshownintheAnimationPanelwherethespeedometeriswiredtothecontrollerinputport.Notethatinthiscasethewhiteneedlenowappearsonthespeedometer,identifyingthevalueofthesetpointthattheuserhasdenedfortheangularvelocity.Likewise,awhitetriangularset-pointmarkerappearsontheverticalsliderbardisplaysthevalueoftheangularvelocity.Hence,theAnimationPanelisupdatedinanimmediatefashiontoreectthecongurationdenedintheInteractionpanel. 2.3.4NavigationPanelanditsTabbedWindowsTheNavigationPanelallowstheusertoaccessavarietyofpedagogicalresourcesthroughaseriesofveoftabbedwindowsbearingthefollowingtitles:Information,Plant,Controller,Analysis,andSimulation.First,whentheuserselectstheInformationtab,awindowopensupintheNavigationPaneldisplayinggeneralinformationonhowtoutilizetheVCLenvironment.ThiswindowservesasausersguidethatisembeddedintheVCL,thatgivesthestudentaccessto 23

PAGE 24

informationwithoutleavingtheinterface.Thetextdisplayedcanbescrolleddowntorevealtheentirecontentsofthedocument.TheInformationwindowsuccinctlydescribesthekeyfeaturesoftheAnimation,Interaction,andNavigationpanels.Inaddition,thewindowdenesforthestudentanumberofcontrol-engineeringexperimentsthatcanbecarriedoutwiththeVCL,clearlystatingtheobjectiveofeachexperimentandsuggestingasequenceofstepsthatcouldbefollowedtoaccomplishthestatedgoals.Forexample,oneexperimenthastheobjectiveofcharacterizingtheopen-loopstepresponseoftheDCmotor.ThetextsuggeststotheusertoproceedbyputtingthecontrollerinManualmode,excitingtheplantwithastepchangeinthemanipulatedvariable,andthenidentifyingkeyfeaturesoftheresponseoftheprocessvariableobservedintheSimulationwindowoftheNavigationPanel.Second,selectionofthePlanttabbringsupawindowthatdisplaysalecontaininginformationonthedynamicsoftheDCmotor.TheNavigationPanelofFigure 2-1 showsthatthePlantwindowpresentstotheusertwotransferfunctionsthatdescribetherelationshipbetweentheapplieddrivingpotentialu(themanipulatedvariable)andeachofthetwovariablesthatcanbeofinterest,namelytheangularpositionandangularvelocity!(theprocessvariables).Usingthescrollbarlocatedonthesideofthewindowtheusercandisplaytheremainderofthetext,whichdescribeshowthetransfer-functionsarederivedfromthemechanicalandelectricaldierentialequationsthatdenethedynamicsofthemotor,andalsoshowshowthemodelingequationscanbewritteninastandardstate-spaceform.Third,theControllertabactivatesawindowthatdisplaysalewithinformationaboutthedierentcontrollersthatareavailabletotheuser.InthecaseoftheDC-MotorVCL,theControllerwindowreviewsthekeyprinciplesofPIDcontroltheory,asshowninFigure 2-3 .ThedocumentdescribesthedierencesbetweentheManualandAutomodesofthecontroller,introducestheclassicaltransferfunctionofanidealPIDcontroller,describesapracticalrealizableversionofthePIDlaw,anddiscussestheeectofthe 24

PAGE 25

dierenttuningparametersofthecontroller.Thewindowalsoshowsthenumericalcoecientsofthenumeratoranddenominatorpolynomialsthatdenethetransferfunctionofthecontroller(seetheframeentitled"CurrentController").ThesecoecientsareimmediatelyupdatedwhenthecontrollerparametersaremodiedbytheuserintheInteractionPanel.Inaddition,theControllerwindowshowsamapofthelocationofthepolesandzerosofthecontrollertransferfunction.Fourth,theAnalysistabdisplaysawindowthatcontainstheclassicalanalysistoolssupportedbycontrol-engineeringtheory.AsshowninFigure 2-4 ,thiswindowcontainsatransferfunctionshowingnumericalvaluesforallthecoecientsofthenumeratoranddenominatorpolynomials,aplotthetime-domainresponsetoaunit-stepinputexcitation,amapshowingthelocationofpolesandzerosonthecomplexplane,andaBodeplotthatprovidesfrequencyresponseinformation.WhenthecontrollerisinManualmode,theinformationdisplayedintheAnalysiswindowrefersonlytotheplant.Inthatcasethesystemisinopen-loopmode,andtheDCmotorisdevoidofafeedbackconnectiontothecontroller.Hence,thetransferfunction,stepresponsecurve,pole-zeromap,andtheBodeplotrefertothedynamicsoftheDCmotor.Ontheotherhand,whenthecontrollerissettoAutomode,theinformationdisplayedintheAnalysiswindowreferstotheclosed-looptransferfunctionrelatingthesetpointofthecontrollertotheprocessvariable(eitheror!,asdenedbytheuserintheInteractionPanel).Hence,inthiscasethetransferfunction,step-responsecurve,pole-zeromap,andtheBodeplotdescribethedynamicsoftheclosed-loopestablishedbetweentheDCmotorandthePIDcontroller.Hence,theinformationshownintheAnalysiswindowissynchronizedwiththeuser'schoiceofManualorAutocontrolmodesintheInteractionPanel.Inthatsense,theAnalysiswindowisacontextualwindow,showingdierentanalysisresultsdependingonwhetherthesystemisinopen-looporclosed-loop.AsshowninFigure 2-4 ,theAnalysiswindowclearlydisplaysamessageindicatingwhetherthesystemisinopen-looporclosed-loopmode,henceensuringthattheusercorrectlyinterpretsthecontextualanalyticalresults 25

PAGE 26

presented.Themaximumandminimumvaluesforthehorizontalandverticalaxesofanygraphinthewindowcanbeeasilyadjustedbysimplyclickingonthecorrespondingtickmarkandtypinganewvalue.Thisfeaturegivestheusergreatexibilityinadjustingthescaleofgraphsasneededtorevealfeaturesofinterest.Finally,theSimulationtaboftheDC-MotorVCLbringstothefrontoftheNavigationPanelawindowwithagraphshowingtheresultsofanumericalsimulationthatdescribestheresponseoftheangularpositionandangularvelocity!toauser-selectedinput.Thisisalsoacontextualwindow.Morespecically,whenthecontrollerisinManualmode,thegraphdescribestheopen-loopresponsesoftheplanttoauser-specieddriving-voltageinputu,intheabsenceoffeedbackcontrol.InthiscasetheusercanchangethedrivingvoltagesignalbyselectinganexcitationfromtheSignalGeneratorappearingintheSimulationwindow(seebottomhalfofFigurerefg:sim).ThesignaloptionsavailableintheDC-MotorVCLincludeaconstantorsinusoidalsignal,aswellasasaw-toothandapulsedwaveform.AnalternativescenarioiswhenthecontrollerisswitchedtoAutomodeandthesystemisthereforeconguredinclosed-loopmode.Inthiscasetheusercanchangethesetpointofthecontrollerbyselectinganexcitationwaveformfromthesignalgenerator.Incontrasttothepreviouscase,thevaluesofthedrivingvoltagearenowdictatedbythePIDcontrollaw,reectingthefactthatthecontrollerfeedbackpathisactiveduringclosed-loopoperation.Thenumericalresponseoftheplantoroftheclosedlooptotheselectedexcitationiscomputedusingafourth-orderRunge-Kuttaintegrationalgorithm.Allthesignalsofrelevancearedisplayedinthegraphusinglinesofdierentlinetypesandcolors.Themaximumvalueofthetimeaxiscanbeeasilychangedbyselectingthecorrespondingtickmarkandtypinganewvalue.Inaddition,thesliderbarlocatedbelowthegraphallowstheusertodisplayprevioustracesthatarenolongervisibleincurrentplotscope(cf.Figure5).Thewindowalsohasabuttonlabeled"Run"thatstartsthenumericalsimulationtaskwhenitisdepressed.Thebuttonlabelthenturnsto"Stop",andservestoterminatethecalculationswheneveritis 26

PAGE 27

pressed.Abuttonentitled"ClearChart"allowstheusertoeraseallthetraces.Finally,abuttonentitled"Export"allowstheusertocreateanASCIIlethatcontainsalltheresultsofthesimulationcalculations,aswellasasummaryoftherelevantparametersthatdescribetheplantandthecontroller.AsinthecaseoftheAnalysiswindow,amessageisclearlydisplayedindicatingwhetherthesystemisinopen-looporclosed-loopmode,henceensuringthattheusercancorrectlyinterpretthesimulationresultspresented.ThesimulationwindowallowstheusertomakechangesintheInteractionPanelwhileanumericalsimulationistakingplace.Allchangesareimmediatelyacceptedbythesimulationresources,withoutrequiringtheusertostoparuninprogress.Theauthorshavefoundthattheuseofatabbed-windowformatfortheNavigationPanelprovidesanintuitivemethodfororganizingdierentaspectsoftheVCL.Ontheotherhand,theuseofanexcessivenumberofwindowsappearstobecounterproductive,giventhattheusercanbeconfusedoroverwhelmedbyalargenumberofoptions.Inouropinion,restrictingthenumberoftabstoonlyveappearstobeacompromisethatsuccessfullyresolvesthecompetingneedforasimple,elegantinterface,andtheneedfordisplayinglargeamountsofrelevantanalysisandsimulationresultsaswellasdocumentaryinformation. 2.3.5PublicationandDeploymentontheWorldWideWebLabVIEWhasbuilt-inWebpublishingcapabilitiesthatinprinciplemakethesoftwaredevelopmentsWeb-accessibleviaasimplecommand.ThiscreatesonehypertextinstancethatcanbeaccessedbyonestudentatthetimeusingaWebbrowser.Ontheotherhand,inateachingenvironmentitisnecessarytopermitmultipleuserstohavefullaccesstotheVCLatthesametime.Thiscanbeeasilyaccomplishedbycreatingacommongatewayinterface(CGI)frontendthatuponrequestproducesreplicatesoftheVCLandrunsacopyforeachconnecteduser.Tominimizetheuseofstorageresources,theCGImustalsoremoveanyVCLcopiesthatarenolongerinuse.AneectiveCGIcanbecreatedusingLabVIEW,whichisthealternativeselectedbytheauthors,orcouldbewritten 27

PAGE 28

usingtraditionallanguagessuchasPerl.Anextensivediscussionoftechnicalalternativesispresentedin[ 24 ].InordertomakeavailableonaWebbrowserwindowalltheinteractivefeaturesofDC-MotorVCLcreatedbytheauthors,itisnecessarytoinstalltheLabVIEWRun-TimeEngineintheremotecomputer.Thisengineisavailableasafreedownloadthatinturninstallsanappropriatebrowserplug-inthatisofcriticalimportancetoensureaninteractiveinterface.Fortunately,whenastudentattemptstousetheVCLforthersttimeadialogmessageappearsiftheRun-TimeEngineplug-inisnotinstalled,andtheuserispromptedtodownloadandinstalltherequiredexecutablele.Ontheotherhand,theneedforaplug-insolution,commontomanyinteractivevirtualorremotelaboratories,reducestheexibilityofthelearningenvironmentbecausethestudentsareconstrainedtoutilizingcomputerswheretheyhavetheappropriatepermissiontoinstallabrowserplug-in.TheWeb-basedDC-MotorVCLisdesignedtosupportahighlevelofinteractivitywiththestudentuser.Theusercanchangecontrollerandplantparameters,selectalternativeprocessvariablesforcontrolpurposes,establishaopen-looporclosed-loopmodeofoperation,changetherangesofallthehorizontalandverticalaxisontheanalysisplots,launchsimulationstudies,andchangesetpoints.Inthisfashion,theproposedtoolsatisesakeystructuralrequirement,namely,theattainmentofahighlyinteractiveinterfaceviaaWeb-browserwindow. 2.3.6ModularityOtherSoftwareDesignDetailsLabVIEWroutinesandtheirinterfacesarestoredinlesreferredtoasvirtualinstruments(VIs).Theactualfunctionsthatdenetheplantandcontrollerarestoredinsub-virtual-instruments(subVIs)inLabVIEW.TheDC-MotorVCLisconceivedinahighlymodularfashiondesignedtoenableaninstructortoreusethesoftwaretodesignanotherVCL.TheinstructorwouldproceedtorstchangetheiconsintheAnimationPaneltoreectthephysicsofthenewplant.Second,astate-spacerealizationofthetransferfunctiondescribingtherelationshipbetweentheinputvariableandthe 28

PAGE 29

manipulatedvariablemustbecodedintothehiddenLabVIEWwiringdiagramassociatedwiththeVI.Third,theinputboxesusedtodeneplantparametersintheInteractionPanelVIarewiredtotheircorrespondingentriesinthetransferfunction,sothatanyparameterchangesareimmediatelyreectedinthesimulation.Fourth,updatedlesareembeddedintheInformation,PlantandControllerwindowsoftheNavigationPanelVI,respectivelydescribingthenewVCLexperimentanditsplant.AsubVIisusedtodeneinformationcommontomultipleVIs,suchasvariablenames,forexample,sothattheframeworkcanautomaticallyupdatealltheinterfaceelementsintheNavigationPaneltoreectthecurrentsystemwithoutanyeortonthepartofthedeveloper.ThedeveloperwouldchangetheminimumandmaximumvaluesofthepercentagescaleusedinthegraphoftheAnimationPanel.Othersmallertasksmaybeinvolvedintheprocess,suchasaddingoreliminatingnewplant-parameterboxesintheInteractionPanel,andrearrangingthebuttons,boxes,andothericonstoproduceanaestheticallypleasinginterface.Finally,theresultingVCLcanbepublishedforWebaccess.Inthisfashion,theresultingmodiedVCLanditsoriginalcounterpartwillhaveacommoninterface,eachonecustomizedintermsofanimationandlabelsthatreectthephysicsoftheirrespectiveplants.ThedevelopmenttakesadvantageoftheNILabVIEWSimulationModuletocalculateclosed-looptime-domainresponses.Itisstraightforwardtocongurethesimulationtoallowtheusertoimplementchangestotheplantortothecontrollerasasimulationisinprogress,henceavoidingtheneedtorestartarunwhenevercontrolorprocesschangesaremade.Thisbehaviorisobtainedbyconductingthesimulationoveraseriesofsuccessiveniteperiodsofdurationslargerthantheintegrationstepsizeusedbythesimulator.Datadecimationusedtoreporttheresultsatpre-speciedintervals,andanupdateoftheplantandcontrollerparametersisdoneattheendofeachniteperiodsothatuser-denedchangescanbequicklyrecognizedbythesimulator.WehavefoundthataRunge-Kuttaalgorithmoforderfourwithaxedstepsize[ 25 ]isanadequatesolverforthedierentialequationsthatdescribethedynamicsoftheDCmotor. 29

PAGE 30

2.4PedagogicalScenariosTheVCLcanbeusedinanumberofpedagogicalscenariosconceivedtotakeadvantageofthebenetsofactivelearning,exiblelearning,andlearningviadiscovery.ArstscenarioofinterestistheuseoftheVCLduringatraditionallecturetoillustrateaparticularcontrol-engineeringconcept.Forexample,theinstructorcouldusetheVCLclosed-loopcapabilitiestoshowthestudentshowtheosetcharacteristicofproportional-onlycontrollerscanbereducedbychoosinglargerabsolutevaluesforthecontrollergain.Theinstructorcouldthenshowhowatthepointofultimate-gainthebenetsofreducedosetbegintoerodeduetotheonsetofanundesirablepersistentlyoscillatoryresponse.Thestudentswouldthenobserveanimatedevidenceofhowevenlargervaluesofthecontrollergainleadtoinstability.Otherexamplesofconceptsthatcanbeillustratedistheeectoftimeconstantsontheplanttime-domainresponseandonthefrequency-domainbandwidth,therelationshipbetweenpolelocationsandstability,andtheeectofopen-loopzerosonthetime-domainresponseoftheplant,amongmanyothers.TheinstructorcanchoosetopresenttheVCLdemonstrationeitherbeforeorafterthecontrolengineeringconceptinquestionistreatedinthelecture.PresentingtheVCLdemonstrationbeforethelecturediscussiongivesthestudentsastrongmotivationalincentivetodevelopacuriosityfortheproblem,andhelpsthemtoidentifythekeytechnicalproblemsthatneedtobeaddressedbythetheory.Theensuinglecturediscussionofthetopic,presentedontheblackboardorviaothertraditionaldeliverymethods,suchusingviewgraphsupport,islikelytobemoresuccessfulinkeepingthestudentsfocusedontheproblembecausetheyhavealreadyacquiredrelevantexperienceontheproblemthroughtheVCL.Alternatively,theinstructormaychoosetoconductaVCLdemonstrationafterthetopichasbeenpresentedontheblackboard.InthiscasetheVCLprovidesanopportunityforreinforcingtheknowledgegained,providingvisualexperienceintheformoftwo-dimensionalorthree-dimensionalanimationandsimulated 30

PAGE 31

signalgraphs,thathelpthestudentsmakecognitivelinksbetweenassociatedconceptsthatoftenappearmoreabstractinnaturewhenpresentedontheblackboard.AsecondscenarioofinterestistheuseoftheVCLinthecontextofinformalcooperativelearningexercisesintheclassroom[ 26 ].Usingwell-establishedtechniquestheinstructordirectsthestudentstoorganizethemselvesinsmallworkinggroups,anddenesthegoalsofapre-plannedexercisethatneedstobesolvedcooperativelybyallthestudentsinthegroup[ 26 ]withtheassistanceoftheWeb-accessibleVCL.Examplesofexerciseactivitiesincludetuningcontrollers,identifyingtheorderofaplantfromanopen-loopstepresponse,ndingtheultimategainofaproportional-onlycontroller,etc.Theseactivitiesfallunderthecategoriesofdiscoveryandactivelearning.AthirdscenarioistheuseoftheVCLtosupportformalcooperativeexercises[ 26 ],alsoknownasgroupprojects.Inthiscasethestudentsaregivenalong-termassignmentthatinvolvesseveralintermediatesteps,culminatingwithacomprehensivenalreport.InthiscasetheVCLwouldbeusedtoprovidesupportforthecompletionoftheintermediatesteps,suchasmodelingtheplant,characterizingitsfrequency-domainproperties,comparingandcontrastingthefrequency-domainandtime-domainperformanceofalternativecontrolschemes,andvalidatingthestabilityofproposedcontroldesignintermsofclosed-looppolemaps.Theseactivitiesalsofallunderthecategoryofdiscoveryandactivelearning.AfourthscenarioistheuseoftheVCLasasupporttoolforhomeworkassignments.TheexiblenatureoftheVCLandthecomprehensivenatureofitsanalysisandsimulationresourcescaneectivelysupporttheinstructor'splanforengagingthestudentsinactivelearningwhilecompletinghomeworkassignments.Allaspectsofmodeling,analysis,design,andsimulationcanbeaddressedwiththeaidoftheVCLasasupplementallearningresource.Inthiscase,theVCLisusedtosupportactivelearning.Finally,fromapedagogicalviewpointtheinstructormaynditadvantageoustoexposestudentstonewconceptsinasequentialfashion.Themodularstructureofthe 31

PAGE 32

VCLframeworkallowstheinstructortoenableonlyselectedpartsofthesoftware,preventingthestudentsfromexperimentingprematurelywithadvancedfeatures.Forexample,studentscouldbepresentedwithaVCLversionwherethecontrollerislockedinManualmode,andwheretheSimulationwindowisenabled.Thestudentarethenaskedtoexcitetheplantwithdiagnosticinputsignals,suchasstepsandsinewaveforms,andusetheresultingresponsetoproduceandvalidateaplantmodel.Oncethistaskiscompleted,theinstructorcanreleaseaversionoftheVCLidenticaltothelatterexceptthattheupdatedversionallowsthecontrollertobeswitchedintoAutomode.Thenewexerciseassignedcouldconsistoftuningthecontrollerbyselectingappropriateparametersandassessingresultingtheclosed-loopresponseusingtheSimulationwindow. 2.5ConclusionsThekeyelementsrequiredinanarchitecturaldescriptionofaVCLhavebeenidentiedasaconceptualguidefordevelopingtoolsthatsuccessfullymeettheneedsofexibleandactivelearning.AnexampleVCLspecializedforthecontrolofaDCmotormeetsmostofthearchitecturalrequirements,realizingahighlevelofuserinteractivitywhileremainingaversatilemodularplatformthatcanbemodiedtoaccommodateotherplants.AlthoughtheDC-MotorVCLrepresentsaneectiveandelegantrealizationoftheproposedVCLparadigm,itdoesnotnecessarilysatisfyallthedesirablearchitecturalfeatures.Oneconceivableshortcomingistheneedtoinstallanappropriatebrowserplug-intheircomputers,whichreducesthetool'sabilitytoprovideservicetoallWeb-enabledclientcomputers.This,however,isacommonlimitationofinteractiveWeb-basedsystems,andtheplug-inalternativeselectedintheDC-MotorVCLisagoodcompromisewithinthecontextofthecurrentlyavailabletechnology.IncontrasttothecaseofothervirtuallaboratoriescreatedusingspecializedprogramminglanguagessuchasJava,theDC-MotorVCLpresentstothestudentawidearrayofanalysisandsimulationoptionswithoutmakingexternalcallstoother 32

PAGE 33

supportingpiecesofsoftware.Inthatsense,theVCLisaself-containedtoolbecauseitdoesnotrequirethatneitherthestudent'smachinenortheWeb-servercomputermaintainexternalcontrol-engineeringsupportsoftware.Allthenumericalsimulationcomputations,analysiscalculations,interfacemanagement,andWeb-interactivitysupporttasksarecarriedoutbyasinglesoftwaretool,namelyLabVIEW.Finally,eventhoughdevelopedonaproprietarysoftwareplatform,theproposedDC-MotorVCLoersaveryhighlevelofuser-interactivitywhilepreservingtheabilitytoallowtheinstructortocustomizethebase-casedesigntosuitotherphysicalplants.ThemodularstructureoftheVCLallowsadevelopertocreatenewsimulationsthattakeadvantageofthefeaturesandinterfaceoftheframeworkproposed.Itispossibletomodelawiderangeofplantswithoutsacricinganyoftheinteractivefeaturesofthevirtualcontrollab.ThereusabilityoftheDC-MotorVCLdescribedhereisfairlystraightforwardwhenthenewplantofinteresthasonlyonemanipulatedvariableandeitheroneortwoprocessvariables.Anextensiontothecaseofmultiple-input/multiple-output(MIMO)plantsusingananalogousprocedureiseasilyenvisionedprovidedthattheinstructorhasavailableabase-caseMIMOVCLthatcomplieswiththearchitecturalconstraintsdescribedinSection 2.2 33

PAGE 34

Figure2-1. RealizationofaDCMotorVirtualControlLabaccessedbystudentsusingastandardWebbrowser.Theinterfacepresentsthreedistinctpanels:(1)anAnimationPanel(locatedontheupperleftarea)showingdetailsoftheDCmotorunderclosed-loopcontrol,(2)anInteractionPanel(lowerleft)thatallowstheusertodenetheparametersoftheplantandofthecontroller,and(3)aNavigationPanel(right-halfarea)consistingofaseriesofvetabbedwindowsthatcontaindierenttypesofinformationassuggestedbythetablabels. 34

PAGE 35

Figure2-2. Animationpanelandinteractionpanelshowing(a)thecontrollersetinManualmode,(b)thecontrollersetinAutomodeusingtheangularpositionastheprocessvariableusedasfeedback,(c)thecontrollersetinAutomodeusingtheangularvelocityastheprocessvariableusedasfeedback. 35

PAGE 36

Figure2-3. Navigationpanelshowingthecontrollerwindowselection.AnembeddedPDFledescribesthefeaturesofthecontroller,whiletheframeontheright-columnshowsthecontrollertransferfunctionaswellasthelocationofitspolesandzeros. 36

PAGE 37

Figure2-4. Navigationpanelshowingtheanalysiswindowselection.Theresultsincludetheunit-stepresponse,Bodeplot,pole-zeromap,andthetransferfunctionoftheplantgiventhatthesystemisinopen-loopmode,asindicatedbythemessageacrossthetopofthewindow.EquivalentresultsarepresentedbytheAnalysiswindowwhenthesystemisswitchedtoclosed-loopmode. 37

PAGE 38

Figure2-5. Navigationpanelshowingthesimulationtabselection.Thegraphdisplaysthetracesofalltherelevantvariables,andthesignalgeneratorallowstheintroductionofset-pointchanges. 38

PAGE 39

CHAPTER3DOUBLEINTEGRATORCONTROLDESIGN 3.1Introduction 3.1.1BackgroundMostindustrialapplicationsoperateundersimplefeedbackcontrollerssuchasPID(Proportional-Integral-Derivative)loops.Thesecontrollersareoftenpreferredovermoreadvanceddesignsbecausetheyareinexpensivetoimplementandeasilytuned.TheintegralactioninPIandPIDcontrollersallowsthemtotrackastepchangeinasetpointorrejectastepchangeindisturbancewithoutcausingsteady-stateoset.Foracontrollertobeindustriallyuseful,explicit,easilyimplementableandcomputationallyinexpensiverulesarerequiredfortuningthem.Theserulesoftentaketheformofequationsrelatingtheparametersoftheopenloopsystemtothecontrollerparametersthatmustbespecied.Manyofthemostsuccessfultuningrelationshipsaredevelopedbysimulatingaclosed-loopsystemandseekingtominimizetheerrorbetweenthesetpointandthecontrolvariable[ 27 ][ 28 ].Controllerswithintegralactionarecommonlyusedtocompensatefordisturbanceorset-pointchangesthattaketheformofastepchange.Unlikeapureproportionalcontroller,aproportionalplusintegral(PI)controllercantrackthestepchangeandeventuallyreducetheerrortozero.Similarly,thereisneedforcontrollersthatwilltrackramptypeinputs.APIcontrollerproducessteady-stateosetforarampsignalsimilarlytothewayaproportionalcontrollerdoesforastepchange.Acontrollerwithtwointegratorsisrequiredtoreducethiserrortozero.Thisphenomenoniswelldocumentedbutfewdetailedstudiesexist[ 29 ][ 30 ].ArelevantstudybyBelangerandLuybenproposesaformofaproportional-integral-doubleintegral(PI2)controller[ 31 ].Alvarez-Ramirezetal.furtherexaminedthiscontrollerandcoinedthenomenclaturePI2[ 32 ].BelangerandLuybentakeananalyticalapproachtothetuningofPI2controllers,andrelatetheprescribedtuningparameterstotheultimategainandperiodoftheplant. 39

PAGE 40

Theyassumethattheplantmodelconsistsofonlyanintegratorplusdead-time,whichisapproximatelyastandardrst-orderplusdead-timemodelonlyforlargetimeconstants.Thissimpliestheproblemtoapointwhereananalyticsolutiondescribingtheoptimaltuningisfeasible.Furthermore,theseauthorsrestrictthetwointegraltime-constantsofthePI2controllertohavevalueswhichcauserepeatedrootsinthecontrollertransferfunction,minimizinganyadverseimpactontheclosed-loopstability.Theseassumptionsallowtheparameterstobechosenasafunctionoftheultimategainandperiodtoachievedesiredcharacteristicssuchasaspeciedclosed-loopdampingratio.Thisstudyisbasedonanumericaloptimizationapproachratherthananalgebraicone,andusesamoregeneralerrormetricthandampingratio.Theeectoftherestrictionrelatingthetwocontrollerintegraltimeconstantscanbeexaminedsinceoptimaltuningswithandwithouttherestrictioncanbecalculated.Additionally,fastersystemscanbecontrolledusingthisapproachsincetruerstorderresponsesareallowedbythechosenmodelform,incontrasttothemorelimitedtypeofresponsesavailablewhenthemodelisrestrictedtoconsistofonlyoneintegrator.Thischapterproposestuningrelationshipsthat,likeexistingPIDtuningschemes,involvetheparametersofarst-orderplusdead-time(FODT)plant.SimulationsofaclosedloopsystemcombiningthisFODTtransferfunctionwithoneforaPI2controlleryielderrormetricswhichcanbeoptimizedtodevelopanappropriatetuningrelationship.TheremainderofthisSectionproposesthreepossibleschemesforimplementingaPI2controllerandcomparesthemtotheschemeproposedbyBelangerandLuyben.Amethodfortuningeachoftheseschemesviatheoptimizationoftheintegralofthetime-weightedabsoluteerrorisdescribedinSection 3.2 .Section 3.3 presentstheresultingoptimaltuningrelationshipswhileSection 3.4 analyzestheseresultsandcomparesthemtotuningprescriptionspresentedintheliterature. 40

PAGE 41

3.1.2ProblemStatementAPI2controllercouldberealizedthroughvariousstructureswherethecommonfeatureisthedoubleintegrator.TransferfunctionsforthreePI2aregiveninthefollowingschemes: Scheme1G(s)=Kc+Ki11 s+Ki21 s2 (3{1) Scheme2G(s)=Kc1+1 i1s1+1 i2s (3{2) Scheme3G(s)=Kc1+1 i1s2 (3{3) Scheme1isequivalenttotheformusedbyBelangerandLuyben,butiswrittenintermsofthreeseparategainsratherthantimeconstants[ 31 ].Thisschemeisthemostgeneralofthethree;however,fromanimplementationstandpointitwouldbeusefultocongureaPI2controllerusingtwostandardPIcontrollersinseries.Scheme2showssuchanapproach.SincethegainsofthetwoPIcontrollersaremultiplicativethegainofthesecondcontrollerintheseriescanbetakenasunitywithoutlossofgenerality,thusonlyonegainisusedinScheme2.ThereishoweveralossofgeneralityinScheme2relativetotheformofScheme1.AnyScheme-2controllercanbeeasilyrepresentedasaScheme-1controller,butScheme-1controllerswhereK2i1<4KcKi2donothaveaScheme-2representation.ThisinequalityconstraintarisesfromthefactorizationofScheme1andthedesiretoavoidcomplexvaluesfortheintegraltimeconstantsinScheme2.ItisalsoofinteresttoconsiderthespecialcaseofScheme2wherethetwointegraltimeconstantsarerestrictedtohavethesamevalue,asinScheme3.ThisisanalogoustotherestrictionintimeconstantsproposedbyBelangerandLuybenandalsosimpliessubstantiallythetuningprocedure.IfScheme3yieldssimilarperformancetoScheme2,thenitwouldbeamoreattractiveoptiontoadoptduetothesimplicityofimplementationandtuning.Allthreeschemesareconsideredinthisstudy. 41

PAGE 42

3.2ApproachForthepurposesofthisstudythesePI2controlschemesarearrangedwithaplantinastandardfeedbackloopasshowninFigure 3-1 whereristhedesiredsetpoint,uistheinputprescribedbythecontrollerandyistheplantoutput.Theperformanceofacontrollercanbemeasuredinavarietyofways.Acommonapproachistouseoneofseveralmetricsoftheerrorofthesystemovertime.Inthisstudyweadopttheintegralofthetime-weightedabsoluteerror(ITAE)asdenedbytheexpression ITAE=Z10tjejdt(3{4)wheree=ryistheclosedloopfeedbackerror.[ 28 ].Inthisstudy,thetransferfunctionGcisaPI2controlschemeandGpistherst-orderplusdeadtimeplanttransferfunction Gp(s)=K s+1es(3{5)TheITAEisattractiveasaperformancemeasurementbecauseitisweightedlightlyatearlytimeswhereerrorisinevitablesincethesystemcannotrespondinstantly,andweightedmoreheavilyatlatertimes,whereerrorcouldbeindicativeofinstabilityorsteady-stateoset.Sincethepresenceofanadditionalintegratortendstodestabilizetheclosed-loopsystem,areasonablyconservativemetricsuchastheITAEisareasonablechoiceforthedoubleintegratorproblemandpreferredtootherstandardmetricssuchastheintegraloftheaverageerrorandtheintegralofthesquareoftheerror.TheITAEapproachisfurtherattractivesinceoptimizationofthismetrichasbeenshowntobeaneectivemethodfordevelopingtuningrulesfortraditionalPIDcontrollers.SimulationscanbeconductedtocalculatetheITAEforasetofplantandcontrollerparametersandthecontrollerparameterscanbeoptimizedforavarietyofplantparametersetstoyieldtuningrelationships.AnumericalsimulationsystemisestablishedusingMATLAB.Thesimulationincludestherst-orderplusdead-timeprocess,asinEquation( 3{5 ),controlledbyaPI2 42

PAGE 43

controllersubjectedtoarampsetpointinput.Thethreeparametersoftheprocess{gain,timeconstant,anddelay{andtheparametersofthecontrollercanallbevaried.TheITAE( 3{4 ),formallyaninniteintegral,istruncatedto tf=15max(;)(3{6)Thisnaltimeislongerthanthetimeinwhichonewouldexpectanyeectivecontrollertoreducetheerrortonearlyzero.ThusthetruncatedITAEissucienttoidentifypoorlyperformingcontrollersandprovidesagoodapproximationofthetrueITAEvalueforoptimizedcontrollers.Theprocessofsettingthemodelparametersandoptimizingthecontrollerparametersisrepeatedforavarietyofmodelparameters.AsimplexalgorithmischosenfortheoptimizationofparametersbecauseofitsabilitytoconvergeonaccuratevalueinreasonabletimeandbecauseitdoesnotrequiregradientswhicharechallengingtodetermineforthetruncatedITAE.Theresultsofthisalgorithmwereconrmedforselectedvaluesoftheparametersbyperforminganexhaustivenumericalsearch.ThedatafromtheoptimizationisnondimentionalizedandplottedversustheprocessparametersinafashionsimilartothatadoptedbyexistingtuningcorrelationsforPIDcontrollers.Thegeneralshapeoftheresultingcurvesuggestsmethodsforttingalinearapproximationtothelogarithmofthedata.Thisisaccomplishedviaaleast-squaresapproach.Resultsfromoptimizationsdoneonallthefunctionalformsofthecontroller{namelySchemes1,2and3{areobtainedandexaminedindependentlyofeachother. 3.3OptimalTuningRelationshipsTuningrelationshipsforeachofthePI2schemesareindependentlydevelopedusingthesimulationroutinediscussedabove.Therelationshipsapparentfromtheresultingdataarewellsuitedtobeingrepresentedasexponentialfunctionsofnondimentionalizedparameters.InafashionsimilartoexistingPItuningrelationships,theparametersofthecontrollerarenondimentionalizedusingtheplantgainandtimeconstant,asappropriate 43

PAGE 44

Table3-1. Tuningrelationshipsforallthreeschemes SchemeGcPI2TransferFunctionTuningPrescription 1G(s)=Kc+Ki11 s+Ki21 s2KKc=1:1616 0:6880if 1KKc=0:8133 0:1894if <1KKi1=0:2892 1:79422KKi2=1:2358 1:04952G(s)=Kc1+1 i1s1+1 i2sKKc=1:0083 0:7854if 1KKc=0:8480 0:1600if <1 i1=0:4880 0:5501 i2=0:3805 0:81503G(s)=Kc1+1 i1s2KKc=1:0760 0:7868if 1KKc=0:8318 0:0964if <1 i=0:4317 0:6820 [ 28 ][ 30 ].TheresultingrelationshipsaregiveninTable 3-1 .Thesameresultsarealsoexaminedgraphically.Forthepurposesofclarity,onlyaverageddataandtherelationshipsdescribedinTable 3-1 areshownintheplots.Figure 3-2 showstheoptimizationresultsforScheme1.InafashionanalogoustothatusedforPIDtuning,theoptimalcontrollerparametersarenondimentionalizedusingtheplantparametersandplottedonalog-logscale.Allthreeparametersareplottedagainsttheratiooftheplant'sdeadtimeandlagtimeconstant.Forthetwointegralgains,thisyieldsplotsthatcanbereasonablyapproximatedbyalinearleast-squarest.ThisisconvenientforuseinactualtuningsinceitallowsasimpletuningruleequationsuchasthoseusedforPIcontrollers[ 28 ].TheproportionalgainplotinFigure 3-2 ashowsacurvethatdoesnotlenditselftodescriptionviaasimplemathematicalform.Sincethisrepresentationisextremelyusefulfortuning,thegraphisbrokenintotwopartseachofwhichissubjectedtoseparatelinearleastsquaresttingprocedures.Theselinearts,representedbythedashedtracesinFigure 3-2 ,aregeneratedbytheequationsgiveninlineoneofTable 3-1 .Figure 3-3 showsplotssimilartothoseinFigure 3-2 butcorrespondingtotheoptimaltuningforScheme2.Itshouldbenotedthatinadditiontosuggestingameansof 44

PAGE 45

physicallyimplementingthecontroller,structuringthePI2controllerastwoseparatePIcontrollerssimpliesthenondimentionalizationtask.Ratherthanrequiringacombinationofseveralconstants,theintegralresetparameterscanbenondimentionalizedsimplybyusingtheplantlagtime-constant.Againitisusefultosplittheproportionalgaincurveintotwosegmentstoreasonablymodelthedatawithalineart.Forsimplicityinapplyingthetuningrules,thesamesplitpoint, =1,isused.Figure 3-4 showstheoptimaltuningsfortheScheme-3PI2controller.Onlyoneplotispresentedfortheintegraltimeconstant,sincethisparameteristhesameforbothintegraltermsofScheme3.Thenondimentionalizationprocedureandthesplittingofthegaincurveintotwosectionsforregressionpurposesareidenticaltothoseperformedonthepreviousgures. 3.4AnalysisandDiscussion 3.4.1ComparisonofthePI2SchemesConsideredClosed-looprampresponsesaresimulatedtoexaminetheeectsofthecontrollersprescribedbytheaboverelationships.Fourplantmodelsareused:(i)aplantwithsimilartimeconstantanddelay,(ii)aplantwithalongdelay,(iii)aplantwithashortdelay,and(iv)andaplantwithalargedelayandalargetimeconstant.Therecommendedtuningforeachofthecontrollerformsisused,andtheresultsareplottedinFigure 3-5 .Sinceallthreeschemesperformsimilarlywell,thepreferredschemecanbechosenbasedonpracticalimplementationconcernsratherthanperformance.Scheme3ispreferredintheabsenceofdierenceinperformancebecauseitcanbeimplementedusingstandardPIcontrollersandhastwoparametersthatrequiretuningratherthanthree. 3.4.2ComparisonwithLiteratureTimeresponsesarealsocalculatedbasedonthecontrollerschemeandtuningproposedbyBelangerandLuyben[ 31 ].BelangerandLuyben'scontrolschemeassumesthattheplantcanberepresentedasanintegratorplusadeadtime.Thusaplantwithalargetimeconstantwithrespecttothedeadtimewouldsatisfytheassumption,whilea 45

PAGE 46

plantwithasmallertimeconstantwouldnot.Thus,theBelanger-Luybentuningswouldbeexpectedtobelesseectiveforthelattercase.Figure 3-6 ashowsascenariowhereandareofsimilarmagnitude.Inthiscasebothcontrollerssuccessfullyachievethezeroosetrequirementatsucientlylongtime.Thisisnotsurprising,asitcanbeshownthateitherformeventuallyeliminatesosetprovidedthecontrollerproducesastableclosedloop.HoweverthePI2controllerproposedbyScheme3showssubstantiallylesstransientbehaviorandwouldbedescribedasabetterperformingoptionbymostmetrics.Toprovideafaircomparisonitisnecessarytoexaminethecasewheretheplanttimeconstantissolargethattherstorderlagcanbeapproximatedbyanintegrator,asthisisthecaseforwhichBelangerandLuyben'stuningsaredesigned.Thus,ascenariowithalargelagtimeconstantandsmallergainsanddeadtimesisexaminedinFigure 3-6 b.Thetwocontrollersproducesimilarresponses.Eithercontrollercouldbedescribedassuperiordependingonyourperformancemetricofchoice,butbothcontrollersperformadequately.Finally,theFigure 3-6 cconsidersascenariointheotherextreme,wherealargedead-timeisusedwithasmallerlag.ThisisbothachallengingcontrolproblemandwelloutsidetherealmforwhichBelangerandLuyben'stuningcriteriawasdesigned.HeretheScheme3controllershowsalongtransientbehaviorthatisnotidealbutprobablyunavoidablegiventhelargedead-time.TheBelanger-Luybentuningprescription,however,showsunstablebehavior,makingitunsuitableforthisplant. 3.4.3ValidationoftheOptimizationResultsSeveralapproachesweretakentoverifythevalidityoftheresults.AlloptimaltuningvaluesproducedbythenumericaloptimizationdatapointsareexaminedviaNyquisttechniquestoverifythestabilityoftheclosedloop.Furthermorecontourplotsconstructedforselectedvaluesofthe ratiotoexaminetheconvexityoftheITAEasafunctionofthecontrollerparametersprovideasupplementalvalidationtool.Figure 3-7 showstheITAEasafunctionoftheScheme-3parameters,Kcandiforvariousvaluesoftheplant 46

PAGE 47

parameters.Specically,theplotsshowaKvalueof1,avalueof10andvaluesof2.5,10and250.SimilargraphsforvaryingKvaluesarenotshownsincetheseserveonlytoscaletheresults,onlychangingtheratioofdelaytotimeconstantalterstheshapeofthecontour.Itshouldbenotedthatthevaluespredictedbythecorrelationsintable 3-1 areneartheoptimalvaluesformoderatetoratios,andarealsowellawayfromtheregionswithsteepgradientsthatcharacteristicallyindicateanapproachtothestabilityboundary.Whiletheseplotsdonotconstituteaproofofconvexityofthesystemsinquestion,theydoservetolendcredibilitytotheaccuracyoftheminimaidentiedbythenumericaloptimizationprocedure. 3.5ConclusionsAPI2double-integralcontrollerimplementedeitherasageneralthreetermcontrollerorasthemorespecializedbuteasilyimplementabletwo-PI-controllers-in-seriesschemeisshowntoreducesteadystateerrortozeroforstepandramptypesetpointsandtuningcorrelationsaredevelopedforrstorderplusdeadtimesystems.ThesecorrelationsprescribecontrollerparametersthatminimizetheITAEasanexponentialfunctionofnondimentionalizedsystemparameters.Ramp-responsesimulationsshowthattheperformanceofthethereproposedcontrolschemesarecomparable,soScheme3isthepreferredoptionduetoitssimplicityofimplementationandtuning.ThesetuningsarealsoshowntobecompetitivewithandinsomecasessuperiortotheanalyticallydevelopedtuningrelationsproposedbyBelangerandLuybenforsimilarPI2controllers[ 31 ].StabilityofthesystemsrecommendedbythecorrelationhasbeenestablishedviaNyquistmethodsbutstudyingtheirrobustnesswithrespecttobotherrorsinthecorrelationsandintheprocessparameterswouldprovetobeinterestingfuturework. 47

PAGE 48

Figure3-1. Closedloopcontrolstructurefeaturingaplant(Gp),aPI2controller(Gc)andthesetpoint(r),output(y),input(u)andfeedbackerror(e)signals. Figure3-2. Scheme1optimalITAEtuningrelationship. 48

PAGE 49

Figure3-3. Scheme2optimalITAEtuningrelationship. Figure3-4. Scheme3optimalITAEtuningrelationship. 49

PAGE 50

Figure3-5. TimeresponsesofthethreePI2schemesforvariousvaluesofplantparameters.TuningparametersareasprescribedbyTable 3-1 50

PAGE 51

Figure3-6. ComparisonofLuyben'sPI2controllerwithscheme2proposedhere. 51

PAGE 52

Figure3-7. Contourplotsforonetimeconstantcalculations. 52

PAGE 53

CHAPTER4OFFSETFREEMODELPREDICTIVECONTROL 4.1IntroductionModelpredictivecontrollers(MPC)areinwidespreaduseinindustrytoadjustthemanipulatedvariables(alsoknownasinputs)ofsystemsofinterestwiththegoalofensuringgoodperformanceofasetofcontrolledvariables(alsoknownasoutputs).TheclassofsystemssusceptibletoMPCcontrolisverylarge,anditincludeschemicalreactors,distillationcolumns,hydrocarboncrackers,entirereneries,aswellascementmillsandovens,robots,automobiles,aircraft,trains,amphibiousvehicles,autonomousvehicles,gasturbines,andjetengines,amongothers.AdditionalsystemsofinterestforwhichMPCcandeliverappropriateautomaticmanipulationstoensuresuitablefunctionalperformanceincludebiologicalenvironments,encompassing,forexample,bioreactorsdesignedtoproduceinsulin,andhumanorganssuchasthepancreasanditsinsulinandglucagonsproductionprocesses,amongmanyothers.Thesystemsofinterestalsoincludeabstracthumanconstructsthatinvolveinputsandoutputsthatchangewithtime,suchasthestockmarket,nancialmodels,manufacturingproductionmodels,schedulingnetworksforproductionandtransportation,forexample.Thesesystemsofinterestareoftenreferredtoastheplant.ThepopularityofMPCcontrollersstemsfromtheirabilitytodelivergoodperformancewhendeployedoncomplexsystemsfeaturinglargenumbersofinputsandoutputs.Furthermore,theMPCcontrollerscansatisfyconstraintsimposedbytheuser,suchasensuringthatinputvaluesdonotdeviatefromarangespeciedbymaximumminimumbounds.TheMPCtechnologyhasestablisheditselfastheprimarymeansofcontrollingindustrialsystemswithmultipleinputsandmultipleoutputs.Inspiteofitshigh-valuecapabilities,itiswellknownthatspecialmeasuresmustbetakentoensurethattheMPCdesignmaketheoutputsachievetheirspeciedset-pointvaluesatsteadystatewithoutappreciableerror.Theerrorobservedafteralltransient 53

PAGE 54

responsessettleintoapproximatelyconstantpatterns,knownassteady-stateoset,isoftenmagniedbythepresenceofdisturbancesaectingtheplant.Acomprehensivediscussionofdesignmeasuresproposedforobtainingoset-freeperformanceatsteadystateformodelpredictivecontrollersisgiveninMuskeandBadgwell[ 33 ]andinPannocchiaandRawlings[ 34 ].ThedesignmeasuresadvocatedinthereferencesmentionedabovearechallengingtodeployinasystematicfashionbecausetheyrequireincludinginthemodelusedfordesigningtheMPCcontrolleradynamicrepresentationofthedisturbancesthatisofteninconsistentwiththestructuralformoftheactualdisturbancesthatactontheplant.Hence,theproposeddisturbancerepresentationsarectitiousmodelsadoptedbecauseincasesofinteresttheybringaboutthebenecialconsequenceofeliminatingsteady-state.Unfortunately,dierentctitiousrepresentationscancausedierentperformancequalitiesduringthetransientperiodthatprecedestheonsetofsteadystate.Therearenosystematicguidelinesonhowtospecifyanoptimaldisturbancestructure.Theseapproachesarethereforenotsucientlysystematicbecausethedesignprocesscallsforsignicantinvestmentsofengineeringeorttodiscriminateamongmanypossiblerepresentationsofdisturbancedynamics.Inaddition,thetechniquesinthesereferencesinvolvetheinclusionofadditionaldesignvariablesknownasstateandinputtargetsthatareinvolvedinthesolutionofasupplementarynumericaloptimizationproblemthatmustbesolvedonlineateveryinstantthatthecontrollerneedstomakeanadjustment.TheprocessofoptimizationincreasesthenumericalcostofdeployingtheMPCalgorithm,andintroducesadditionaldesigncomplicationsbecausespecialcostmatricesmustbespeciedwithoutthebenetofestablishedguidelines.Furthermore,nosystematicguidelinesaregiventoassistinthespecicationofeectivevaluesfortheinputtargetvariable,animpedimenttoensuringadequatetransientperformancewithoutinvestingasignicantamountofaddedengineeringeort.Itisoftenobservedthatthebehaviorofthemodelpredictive 54

PAGE 55

controllersdesignedaccordingtotheguidelinesgiveninthesereferencesbehaveinanunintuitivefashion,andthusthemethodologylackstherobustnessdesirableinamission-criticalorhigh-performancecontrolsystem.Performancedegradationintheformofunacceptablesteady-stateosetscanalsobeobservedwhentheoperatingconditionsoftheplantchange,whichmaycallforasignicantinvestmentinadditionalengineeringresourcestoredesigntheMPCstructuretoeectivelyaddressneedsofnewenvironmentalconditions.Thischapterpresentsmodelpredictivecontrolalgorithmsthatdonotsuerfromtheshortcomingsidentiedabove.Themodelpredictivecontrollersdiscloseddeliveroset-freeperformance,andcanbedeployedinasystematicfashion.Furthermore,thenumericalcomputationalburdenrequiredmaybesignicantlylowerthanthatofalternativeapproachesdocumentedintheliterature. 4.2ProblemStatementThischapterpresentsfourmethodsfordesigningandimplementingModelPredictiveControllers(MPC)thatdeliverresponsesthatarefreeofsteady-stateoseterrors.TheMPCmethodsdisclosedareeectiveforeliminatingoset,andhenceensureperfectset-pointtrackingatsteadystate,evenwhentheplantbeingmanipulatedfortheMPCcontrollerissubjecttothepresenceofunmeasuredinputdisturbancesandoutputdisturbancesthatadoptconstantvaluesatsteadystate.AllMPCmethodsdescribedinthischaptereliminatetheneedforincludinginthetypicalMPCperformanceindexanumberofcomputationallyexpensiveandunintuitiveconcepts,includingthesteady-statetargetsrequiredbypreviousMPCimplementations.AllMPCmethodsuseaperformanceindexthatincludesaset-pointerrortrackingcost.Figure 4-1 isablockdiagramillustratingasystemwithaModelPredictiveController(MPC).Itincludesandaplantandcontroller.Theplanttakesoneormoreinputs,u,andproducesoneormoreoutputs,y.TheoutputandadesiredvalueknownasthesetpointarefedtotheMPCController.AteverysamplingintervalMPCControllerdeterminesthe 55

PAGE 56

inputbymakinguseofaModelSystem,anEstimationSystem,aPredictionSystem,andanOptimizationSystem.AtypicalcontrolperformancespecicationisthattheinputprescribedbytheMPCControllerproducesasequenceofoutputvaluesthatareclosetoaspeciedsequenceofset-pointvalues.Thesteady-stateoset,oftenreferredtosimplyasoset,ess,isdenedasthevalueadoptedbythefeedbackerrorafterasucientlylongtimetoallowalltransientchangesintheoutputtodisappearandallowtheoutputtoadoptaconstantpattern.Mathematically,theosetisrepresentedasalimit,namelytheerrorastimetendstoinnity,whereasinpracticetheconceptofaninnitelylongtimeindexisreplacedbyasucientlylongbutnitetimeindex.Theosetisthereforeameasureofhowmuchtheoutputdiersfromthespeciedset-pointvalueafterarelativelylongperiodofoperation.TheMPCcontroldesignmethodologydiscussedhereensuresthattheresultingcontrollersattainzerooset,henceforcingtheoutputtomatchthesetpointatsteadystateevenunderconditionswherethereisachangeinthesetpointorwherePlant105issubjecttothepresenceofdisturbancesthatarenotmeasured.Thus,thedesirablefeaturedeliveredbytheMPCdesignsdisclosedhereistheattainmentoftheconditionTheModelSystemofanMPCControllerisamathematicalalgorithmthatdescribestheoperationoftheplant.Avarietyoflinearandnon-linearmodelscanbeused,butthefocusofthischapterisonthelinearMIMOstatespaceplantmodel x(k)=Ax(k1)+Bu(k1)+d(k) (4{1) y(k)=Cx(k)+p(k) (4{2) wherexisavectorofstatesoftheplant,uistheinputfromthecontroller,yistheoutputfromtheplanttothecontrolleranddandparestateandoutputdisturbancesrespectively.Thesizesofthesevectorsaregivenby: x2Rn;y2Rm;u2Rp;d2Rn;p2Rm(4{3) 56

PAGE 57

ThePredictionSystemdescribestheanticipatedfuturevaluesofthevariablesassociatedwiththeModelSystemandthePlant.AtsamplingintervalkthePredictionSystemcanproducenumericalestimatesofthefuturevaluesofoutputsandstates,andofcurrentandfuturevaluesoftheinputandinput-incrementvectors.Thesepredictionscanberepresentedcompactlyviatheaugmentedvectors x=266666664x(k+1)x(k+2)...x(k+Np)377777775y=266666664y(k+1)y(k+2)...y(k+Np)377777775u=266666664u(k)u(k+1)...u(k+Nc)377777775(4{4)where u=266666664u(k)u(k+1)...u(k+Nc)377777775(4{5) u(k)=u(k)u(k1)(4{6)TheindicesNpandNcareknownasthepredictionhorizonandcontrolhorizonrespectively.Therefore,theMPCcontrolleradjustsatotalofNcpmanipulatedvariables,andthevectorycontainsatotalofNpmrows.Furthermore,therstprowsofucontainthecurrentinputvector.ThePredictionSystemconsideredinthischapteroperatesbyiterativelysolvingthemodelequations( 4{1 )and( 4{2 )forfuturesamplingintervals.Thisprocesscanberepresentedbytheaugmentedpredictorintheequations x=Aax(k)+Bau+p (4{7) y=Cax+d (4{8) 57

PAGE 58

where Aa=266666664AA2...ANp377777775 (4{9) Ba=2666666666666666666664B0:::00ABB:::00...............ANc1BANc2B:::ABBANcBANc1B:::A2BP1i=0AiBANc+1BANcB:::A3BP2i=0AiB...............ANp1BANp2B:::ANpNc+1BPNpNci=0AiB3777777777777777777775 (4{10) Ca=266666664C0:::00C:::0............0C:::C377777775 (4{11) andwhere d=266666664d(k)d(k+1)...d(k+Np)377777775p=266666664p(k)p(k+1)...p(k+Np)377777775(4{12)TheEstimationSystemofanMPCControllerisusedtocomputeexactorapproximatevaluesofthestatevectorusingavailablemeasurementsandinformationprovidedbytheModelSystem.Morespecically,theEstimationSystemgeneratesasequenceofestimatedstatevalues^xthatcorrespondexactlyorapproximatelytothesequenceofstatevaluesxproducedbytheplant.Exactvaluesofdisturbancesmaynotbeavailable.Thus,the 58

PAGE 59

notationof^dand^pisintroduced.Similarto^x,thesevectorscontainvaluesthatmaybeexactorestimated.Sinceunlikeprioroset-freeMPCimplementations,thosedescribedheredonotrequiredisturbanceestimation,anotherpossiblealternativeistoset^d=^p=0.Whenthe^dand^pvectorsarenotmeasuredoridenticallyzero,thesequenceofestimatedstatescanbecalculatedusinganEstimationSystemdesignedusingschemesforstateestimationthatarewellknownintheliterature.Forexampletheestimator: ^x(k)=A^x(k1)+Bu(k)+Lx[yC^x] (4{13) ^d(k)=^d(k1)+Ld[yC^x] (4{14) ^p(k)=^p(k1)+Lp[yC^x] (4{15) (4{16) whereLx,Ld,andLparetheestimatorgains,iswelldocumentedintheMPCliterature.Detailsregardingestimation,includingoneparticularalternativeestimatorarediscussedinsection( 4.4 ).TheOptimizationSystemofanMPCControllerndstheoptimalvalueoftheinputvectoruortheincrementalinputvectoru,usedasinputstothePlant.Typicallythisisdonebyminimizingaperformanceindex.Thisminimizationisoftenperformedunderrestrictionsdenedinaconstraintset.Sincethegoalofthischapterisosetelimination,whichcannotbeachievedwhenconstraintsareactiveatsteadystate,unconstrainedminimizationisthefocusofthisdiscussion.AnMPCControllerusesasetofMPCParametersthataredenedbytheuserofthecontrolsystem.TwocentralMPCParametersarethepredictionhorizonNpandthecontrolhorizonNp,whichcanbespeciedtoMPCController110asxedvaluesorasrulesthatdescribetimevaryinghorizons.OtherMPCParametersincludethevaluesofcostmatrices,constraints,andheuristicinformation,suchasthevarianceofmeasurementanddisturbancenoise. 59

PAGE 60

ThefollowingsectionexaminesfourdierentpredictivecontrolmethodsthateliminateosetinthePlant.MethodIusesanintegralstatevectorintheModelSystem,MethodIIusesaninput-velocitycost,MethodIIIcombinesbothanintegralstatevectorandaninputvelocity,andnallyMethodIVusesarstintegralstatevectorandasecondintegralstatevector. 4.3Oset-FreeMPCMethodsProposedTheconceptsofintegralstatesandvelocityweightingcanbecombinedtoyieldseveraldierentMPCdesignmethods.Perhapsthesimplestapproachintheadditionofexplicitintegralstatestotheprediction. 4.3.1MethodI:IntegralStatesInMethodItheintegralstateequation z(k)=z(k1)+r(k1)y(k1)(4{17)isaddedtotheplantmodeldescribedbyequations( 4{1 )and( 4{2 ).Thisplantmodelisusedtocreatethepredictionsystem x(k+j)=Ax(k+j1)+Bu(k+j1) (4{18) y(k+j)=Cx(k+j) (4{19) z(k+j)=z(k+j1)+r(k+j1)y(k+j1) (4{20) Deningtheaugmentedvectorzas z=266666664z(k+1)z(k+2)...z(k+Np)377777775(4{21) 60

PAGE 61

resultsinapredictorthatcanbewrittenintheform y=CI^x(k)+DIu+FI^d+EI^p (4{22) z=MI[z(k)+r(k)y(k)]+NI(ry) (4{23) where CI=266666664CACA2...CANp377777775 (4{24) DI=2666666666666666666666666664CB0:::00CABCB:::00...............CANc1BCANc2B:::CABBCANcBCANc1B:::CA2B1Xi=0CAiBCANc+1BCANcB:::CA3B2Xi=0CAiB...............CANp1BCANp2B:::CANpNc+1BNpNcXi=0CAiB3777777777777777777777777775 (4{25) 61

PAGE 62

FI=2666666666664C1Xi=0CAi...Np1Xi=0CAi3777777777775 (4{26) EI=266666664II...I377777775 (4{27) MI=266666664II...I377777775 (4{28) NI=26666666400:::0I0:::0............II:::I377777775 (4{29) TheOptimizationSystemminimizestheperformanceindex J=(ry)TQa(ry)+uTRau+zTSaz(4{30)whichcontainstermsforset-pointtracking,inputactionandintegralstates,where r=266666664r(k+1)r(k+2)...r(k+Np)377777775(4{31) 62

PAGE 63

Inthesimplestcongurationtheweightsaredenedas Qa=diag(Q;Q;:::;Q;Q)) (4{32) Ra=diag(R;R;:::;R;R(NpNc)) (4{33) Sa=diag(S;S;:::;S;S) (4{34) whereQ,R,andTareweightingmatricesthatdenetherelativecostsassociatedwitheachoutput,inputandintegral,respectively.TheminimumofthisperformanceindexoccursatrJu=0.Solvingtheminimizationoftheperformanceindexforuyieldsthecontrollaw u=(KI+KzINI)[rCI^x(k)]FI^d(k)EI^p(k)]+KzIMI[z(k)+r(k)y(k)] (4{35) KI=DTINTISaNI+QaDI+Ra1DTIQa (4{36) KzI=DTINTISaNI+QaDI+Ra1DTINTISa (4{37) (4{38) AteachsamplingintervaltheMPCcontrollerperformsthiscalculationandimplementstherstprowsofuasu(k).Iftheplantiscontrollableandtheclosedloopisstable,thisMPCcontrollerissucienttoeliminateosetfordisturbancesorchangesinsetpointthathaveaconstantnalvalue.Anotheralternativeforaddressingthepossibilityofdisturbanceintheuseofincrementalequations.Iftheequations( 4{1 )and( 4{2 )areshiftedbackwardsintime 63

PAGE 64

byoneintervalandsubtractedfromtheirunshiftedforms,theresultis ^x(k+j)=^x(k+j1)+A^x(k+j1)A^x(k+j2) (4{39) +Bu(k+j1)+^d(k+j)^d(k+j1)^y(k+j)=^y(k+j1)+C^x(k+j)C^x(k+j1) (4{40) +^p(k+j)^p(k+j1)Sincethecasewherethedisturbancesreachconstantvaluesintimeisofinterest,theassumptions ^p(k+j)=^p(k+j1) (4{41) ^d(k+j)=^d(k+j1) (4{42) (4{43) arerelevantandreduce( 4{40 )and( 4{41 )to ^x(k+j)=^x(k+j1)+A^x(k+j1)A^x(k+j2)+Bu(k+j1) (4{44) ^y(k+j)=^y(k+j1)+C^x(k+j)C^x(k+j1) (4{45) Anapparentalternativemethodofformulatingtheincrementaloutputequationwouldbetousetheequationforthecurrenttimeinterval,k,ratherthanthek+j1interval,yielding ^y(k+j)=^y(k1)+C^x(k+j)C^x(k1)(4{46)Howeverbothofthesemethodsyieldtheaugmentedpredictionsysteminequations y=C0II^x(k)CII^x(k1)+DII0u+EIy(k) (4{47) z=MI[z(k)+r(k)y(k)]+NI(ry) (4{48) 64

PAGE 65

where C0II=266666666666664C1Xi=0AiC2Xi=0Ai...CNpXi=0Ai377777777777775 (4{49) CII=266666666666664C1Xi=0AiI!C2Xi=0AiI!...CNpXi=0AiI!377777777777775 (4{50) DII=2666666666664CB0:::00C1Xi=0AiCB:::0............CNp1Xi=0AiCNp2Xi=0Ai:::CNpNc+1Xi=0AiCNpNcXi=0Ai3777777777775 (4{51) 0=266666664I00:::0II0:::0...............000:::I377777775 (4{52) 1=266666664I0...0377777775 (4{53) (4{54) 65

PAGE 66

Thisresultsinthecontrollaw u=(KI+KzINI)[rCII(^x(k)^x(k1))+DII1u]+KzIMI[z(k)+r(k)y(k)] (4{55) (4{56) where KI=h(DII0)TNTISaNI+Qa(DII0)+Rai1(DII0)TQa (4{57) KzI=h(DII0)TNTISaNI+Qa(DII0)+Rai1(DII0)TNTISa (4{58) (4{59) andwhere 0= (4{60) 1= (4{61) Theseformscanalsobecombined,usinganincrementalstateequationwithastandardoutputequationorviceversa.TheestimationandcontrolequationsforthesevariantsandothersthatarediscussedbelowarelistedinTable 4-1 4.3.2MethodII:Input-VelocityCostAnMPCcontrollercanalsobeconguredtoperformapredictivecontrolmethodthatutilizesaninput-velocitycost.Sinceinputvelocitiesarethegoalofthemethod,itisanaturalchoicetouseanincrementalmodelliketheonedescribedinequations( 4{45 )and( 4{45 )sincetheyalreadydescribetheplantintermsoftheinputvelocityu.TheconstructionoftheEstimationandPredictionsystemsproceedsasinMethodIyieldingtheaugmentedpredictionequation y=CII[^x(k)^x(k1)]+DIIu+EIy(k)(4{62) 66

PAGE 67

Theoptimizationsystemthenminimizestheperformancefunction J=(ry)TQa(ry)+uTTau(4{63)whichincludesthekeyelementofMethodII,apenaltyontherateofchangeofinputs.Minimizingthisequationproducesthecontrollaw u=KII[CII[^x(k)^x(k1)]+EIy(k)](4{64)where KII=DTIIQaDII+Ta1DTIIQa(4{65)ThisapproachisfundamentallysimilartothatdiscussedforMethodI.Thissimilaritymightleadonetobelievethatthevelocity-weightingmethodcouldbeimplementedbasedonnon-incrementalplantmodelsusingthe0matrixdenedin( 4{60 ).However,thesimulationstudypresentedinthenextsectionshowsthatanincrementalstateequationisrequiredforvelocity-weightingtocorrectlyeliminateoset.Theuseofanon-incrementaloutputequationisstillaviablestrategy,whichisoset-freeinsomecases. 4.3.3MethodIII:IntegralStatesandVelocityControlAnMPCcontrollercanalsoperformacombinationofMethodsIandII.TheModel,Estimation,andPredictionSystemsproceedasdescribedforMethodII.Ifanincrementalmodelisused,theresultingpredictorisgivenby y=CII[^x(k)^x(k1)]+DIIu+EIy(k) (4{66) z=MI[z(k)+r(k)y(k)]+NI(ry) (4{67) Theoptimizationsystemusestheperformanceindex J=(ry)TQa(ry)+uTTau+zTSaz(4{68) 67

PAGE 68

whichincludetermsforbothinputvelocityandintegralstates.Theminimizationofthisperformanceindexyieldsthecontrollaw u=(KI+KzINI)rC0II^x(k)+CII^x(k1)EIy(k)+KzIMI[z(k)+r(k)y(k)] (4{69) (4{70) where KII=DTIINTISaNI+QaDII+Ra1DTIIQa (4{71) KzII=DTIINTISaNI+QaDII+Ra1DTIINTISa (4{72) (4{73) Theseequationsarebasedontheincrementalmodelequations( 4{45 )and( 4{45 ).Ifnon-incrementalstateequationsareused,thevelocitycontrolmechanismfailstoaddanintegrator,buttheintegratingstatescontinuetoperformnormally.Inthefollowingsectionthisisdemonstratedviatheapplicationofarampinset-point.Ifbothintegratingmethodsareeective,thecontrollerisabletoeliminateosetwithrespecttoaramp.Ifonlyoneintegratoriseective,thecontrolleriscapableofoset-freetrackingforstepset-pointchangesbutnotforramps.Table 4-1 detailswhichformulationsofMethodIIItrackrampsandthuscontaintwointegrators. 4.3.4MethodIV:DoubleIntegralStatesAnalternativemethodofincludingtwointegratorsintheMPCcontrolleristwoincludetwosetsofexplicitintegralstates.InthiscasetheModelSystemisaugmentedto 68

PAGE 69

theformgiven x(k+j)=Ax(k+j1)+Bu(k+j1) (4{74) z(k+j)=z(k+j1)+r(k+j1)y(k+j1) (4{75) w(k+j)=w(k+j1)+z(k+j1) (4{76) y(k+j)=Cx(k+j) (4{77) (4{78) Thisleadstothepredictor y=CI^x(k)+DIu+FI^d+EI^p (4{79) z=MI[z(k)+r(k)y(k)]+NI(ry) (4{80) w=MI[w(k)+z(k)]+NIw (4{81) (4{82) Theperformanceindex J=(ry)TQa(ry)+uTRau+zTSaz+wTWaw(4{83)includestermsthatweightthedoubleintegralstates.Theminimizationofthisperformanceindexyieldsthecontrollaw u=KIV+KzIVNI+KwN2I[rCI^x(k)]FI^d(k)EI^p(k)] (4{84) +(KzIV+Kw)MI[z(k)+r(k)y(k)]+Kw[w(k)+z(k)] (4{85) 69

PAGE 70

where KIV=hDTIN2ITWaN2I+NTISaNI+QaDI+Rai1DTIQa (4{86) KzIV=hDTIN2ITWaN2I+NTISaNI+QaDI+Rai1DTINTISa (4{87) Kw=hDTIN2ITWaN2I+NTISaNI+QaDI+Rai1DTIN2ITWa (4{88) (4{89) AsinMethodI,standardorincrementalmodelscanbeusedtodescribetheplantwithoutanimpactontheintegralsorthesteadystatebehaviorofthesystem. 4.4EstimatorsSeveraloptionsfortheestimationsystemareavailable.Ifx,dorparemeasurable,thentheycanbeuseddirectly.Ifdorpisnotmeasurable,zerovaluescanbeusedfor^dor^p.Ifthisisnotdesirable,orifunmeasuredstatesareunavoidablethenanestimatorisrequired.Themostlcommonestimatorsaretheonesshowninequations( 4{14 4{16 ).Thisestimatorworkseectivelyforallmethods,buttheincrementalmodelsdiscussedabovesuggestthealternativeestimatorformulation: ^x(k)=(A+I)^x(k)A^x(k1)+Bu(k1) (4{90) +Lx[y(k)y(k1)C^x(k)+C^x(k1)] (4{91) ^d(k)=^d(k1)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]^p(k)=^p(k1)+Lp[y(k)y(k1)C^x(k)+C^x(k1)] (4{92) (4{93) Ifdesired,theincrementalmodelstateestimatorcanbereducedtothestandardformviathesubstitution v(k)=264^x(k)^x(k1)375 (4{94) (4{95) 70

PAGE 71

thisyieldsthesystem v(k+1)=0B@264A+IAI0375KvCC1CAv(k)+264B0375u(k)+Kv(y(k)y(k1)) (4{96) allowingthegainKvtobedesignedbytheusualtechniques,suchaspoleplacement. 4.5SimulationStudyThevariousmethodsforoseteliminationareexaminedvianumericalsimulationstudy.Eachisimplementedforseveraldierentplantmodels.Theresultsregardingsteadystatebehaviorarelistedintable 4-1 .Therstcolumndescribesthemethodused.Thesecondandthirdcolumnsshowwhetherstandardorincrementalmodelsareusedforthestateandoutputequationsrespectively.Thefourthcolumnlistswhetherornotstateanddisturbanceestimatorsareincluded.Whentheyarenotincluded,thesimulationsareperformedundertheassumptionthatxisdirectlymeasurableand^dand^parebothzero.Thefthandsixthcolumnsshowthenalpredictorandcontrollerequations.Theremainingcolumnsshowthesteadystatebehavior.Inthetrackingcolumnan"oset"entrymeansthatthecontrollercannottrackastepchangeinsetpointwithoutoset,a"step"entrymeansthecontrollercantrackastepchangewithoutoset,andanentryof"ramp"meansthecontrollercantrackstepsorrampsinsetpointwithoutsteadystateoset.Thedandpcolumnslistifthecontrolcongurationcanrejectconstantnal-valuedisturbancesofthetypesdescribedaboveasdandp.Thefocusofthischapterhasissteadystatebehavior,butforcompletenesstransientbehaviormustbeconsideredaswell.Tothisendtheeectofthecontrollersonaplantconsistingofamodelofastirredtankreactorisexamined.PannocchiaandRawlings[ 34 ] 71

PAGE 72

proposethereactormodel x(k+1)=2666640:25113:3681037:05610411:060:32962:545001377775x(k) (4{97) +2666645:4261031:5301051:2970:121806:592102377775u(k)+d(k) (4{98) y(k+1)=266664100010001377775+p(k+1) (4{99) (4{100) wherethestatesxaretheconcentrationofreactant,temperature,andliquidlevelinthereactor,theinputsuarethecoolanttemperatureandtheowrateofmaterialoutofthereactor.FortheMPCdesignthehorizonsarechosentobeNp=12andNc=3.Theeectsofbothset-pointchangesanddisturbancesonavarietyofcontrollersisconsidered.Stepchangesinsetpointareappliedtobothofthecontrolledoutputsandthedisturbance d(k)=1:7621057:7841026:592102T(4{101)isintroducedattime,t=175;priortothatthedisturbanceiszero.Thereactormodelisplacedinaclosedloopandsubjectedtochangesinsetpointanddisturbances.Figure 4.6 showstheresponseofthisplanttoacontrollerformulatedaccordingtoMethodIasdescribedinrow1ofTable 4-1 .Oseteliminationinbothoutputsissuccessfullyachieved.Usingtheweightingmatrices Q=266664100000001377775;R=2640:5000:5375;S=264100001375(4{102) 72

PAGE 73

thedynamicperformanceshownisarchived,ifspecicperformancecriteriaaredesiredtheweightscanbeadjustedtoarchivedthese.AsintheexamplesusedbyPannocchiaandRawlingsthesecondoutputisnotcontrolled.Tothisendtheweightsonbothitanditsintegralaresettozero.Thoughstateanddisturbanceestimationarenotrequiredforthemethodslistedhere,itcanstillbeworthwhiletodemonstratehowtoimplementanobserver.ThusthenextsimulationisperformedusingMethodIIwithanincrementalmodelestimatorasdescribedinrow23ofTable 4-1 .Theweightingmatrices Q=266664100000001377775;T=2640:5000:5375 (4{103) (4{104) areusedaswellastheobservergainfromequation( 4{105 ). Lx=2666640:50000:50000:5377775 (4{105) TheresultsareshowninFigure 4.6 .Againtheosetiscompletelyeliminatedandthedynamicperformancecanbefurtheradjustedviatuningshouldausersodesire. 73

PAGE 74

ControllersdesignedaccordingtomethodsIIIandIValsohavetheabilitytotrackramps.Todemonstratethis,considertheplantmodel x(k+1)=2666666640:500000:600000:500000:6377777775x(k) (4{106) +2666666640:5000:40:25000:6377777775u(k)+d(k) (4{107) y(k+1)=26411000011375+p(k+1) (4{108) (4{109) Figure 4.6 showsthisplantusedwithamethodIIIcontrollerformulatedaccordingtorow34ofTable 4-1 .Thiscontrollerusestheweights Q=2640:5000:5375;T=2640:5000:5375;S=2640:5000:3375(4{110)andthehorizonsNp=5andNc=4.Figure 4.6 ademonstratestheresponsetoastepinsetpointandindisturbance.Figure 4.6 bshowstheresponsetoarampinsetpoint.Ineithercasethesteady-stateosetiseliminated.Figure 4.6 showsasimilarsetofresponsesfortheMethodIVcontrollerdescribedonrow39ofTable 4-1 .Thiscontrolleristunedusingtheweights Q=2641001375;R=2640:5000:5375;S=2641001375;W=2640:5000:5375(4{111) 74

PAGE 75

ItisapparentfromFigure 4.6 thattheMethodIVcontrollersuccessfullyeliminatesosetfrombothstepandrampset-pointchangesaswellasstepchangesindisturbance. 4.6ConclusionFourdierentmethodsforoseteliminationinpredictivecontrolaresuggested.Simulationresultsdemonstratethatallarecapableofeliminatingsteadystateosetfromstepchangesininputorconstantnal-valuedisturbances.ThespecicformswhichareeectiveareoutlineinTable 4-1 whichcanbesummarizedbythefollowingobservations.Explicitintegralstates,asusedinmethodsI,IIIandIV,introduceanintegratorregardlessofthemodelformused.Input-velocityweightingcontrol,asusedinmethodsIIandIIIintroducesanintegratorinthetrackingbehavioronlywhenpairedwithanincrementalstateequation,andensuresosetfreeperformanceinthedisturbancerejectionproblemonlywhenpairedwithanincrementaloutputequation. 75

PAGE 76

Table4-1. EquationsandsimulationresultsforvariousMPCmethods MethodStateOutputEstimatorTrackingDisturbanceEquations 1IStandardStandardNoneStepBoth^y=CI^x(k)+DIuz=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTI+kTzNI[rCI^x(k)]+kTzMI[z(k)+r(k)^y(k)] 2IStandardStandardStandardStepBoth^y=CI^x(k)+DIu+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTI+kTzNIhrCI^x(k)FI^d(k)+EI^p(k)i+kTzMI[z(k)+r(k)^y(k)] 3IStandardStandardIncrementalStepBoth^y=CI^x(k)+DIu+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTI+kTzNIhrCI^x(k)FI^d(k)+EI^p(k)i+kTzMI[z(k)+r(k)^y(k)] 76

PAGE 77

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 4IIncrementalStandardNoneStepBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTI+kTzNIrC0II^x(k)+CII^x(k1)+DII1u(k1)+kTzMI[z(k)+r(k)^y(k)] 5IIncrementalStandardStandardStepBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTI+kTzNIrC0II^x(k)+CII^x(k1)+DII1u(k1)EI^p(k)+kTzMI[z(k)+r(k)^y(k)] 6IIncrementalStandardIncrementalStepBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTI+kTzNIrC0II^x(k)+CII^x(k1)+DII1u(k1)EI^p(k)+kTzMI[z(k)+r(k)^y(k)] 77

PAGE 78

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 7IStandardIncrementalNoneStepBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTI+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 8IStandardIncrementalStandardStepBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTI+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 9IStandardIncrementalIncrementalStepBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]u=kTI+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 78

PAGE 79

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 10IIncrementalIncrementalNoneStepBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTI+kTzNI[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzMI[z(k)+r(k)^y(k)] 11IIncrementalIncrementalStandardStepBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTI+kTzNI[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzMI[z(k)+r(k)^y(k)] 12IIncrementalIncrementalIncrementalStepBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]u=kTI+kTzNI[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzMI[z(k)+r(k)^y(k)] 13IIStandardStandardNoneOsetNone^y=CI^x(k)+DI10u+DI2u(k1)u=kTII+kTzNI[rCII^x(k)] 79

PAGE 80

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 14IIStandardStandardStandardOsetNone^y=CI^x(k)+DI10u+DI2u(k1)+FI^d(k)+EI^p(k)^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIhrCII^x(k)FII^d(k)+EII^p(k)i 15IIStandardStandardIncrementalOsetNone^y=CI^x(k)+DI10u+DI2u(k1)+FI^d(k)+EI^p(k)^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIhrCI^x(k)FI^d(k)+EI^p(k)i 16IIIncrementalStandardNoneStepdOnly^y=C0I^x(k)CI^x(k1)DI1u(k1)+DI0uu=kTII+kTzNIrC0II^x(k)+CII^x(k1) 17IIIncrementalStandardStandardStepBoth^y=C0II^x(k)CII^x(k1)+DII0u+EI^p(k)^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIrC0II^x(k)+CII^x(k1)EI^p(k) 80

PAGE 81

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 18IIIncrementalStandardIncrementalStepBoth^y=C0II^x(k)CII^x(k1)+DII0u+EI^p(k)^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIrC0II^x(k)+CII^x(k1)EI^p(k) 19IIStandardIncrementalNoneOsetNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 20IIStandardIncrementalStandardOsetNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i 21IIStandardIncrementalIncrementalOsetNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i 22IIIncrementalIncrementalNoneStepBoth^y=CII(^x(k)^x(k1))+DII0uu=kTII+kTzNI[rCII(^x(k)+^x(k1))] 81

PAGE 82

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 23IIIncrementalIncrementalStandardStepBoth^y=CII(^x(k)^x(k1))+DII0u^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNI[rCII(^x(k)+^x(k1))] 24IIIncrementalIncrementalIncrementalStepBoth^y=CII(^x(k)^x(k1))+DII0u^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNI[rCII(^x(k)+^x(k1))] 25IIIStandardStandardNoneStepNone^y=CI^x(k)+DI10u+DI2u(k1)z=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTII+kTzNI[rCI^x(k)]+kTzMI[z(k)+r(k)^y(k)] 26IIIStandardStandardStandardStepNone^y=CI^x(k)+DI10u+DI2u(k1)+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIhrCI^x(k)FI^d(k)+EI^p(k)i+kTzMI[z(k)+r(k)^y(k)] 82

PAGE 83

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 27IIIStandardStandardIncrementalStepNone^y=CI^x(k)+DI10u+DI2u(k1)+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIhrCI^x(k)FI^d(k)+EI^p(k)i+kTzMI[z(k)+r(k)^y(k)] 28IIIIncrementalStandardNoneRampBoth^y=C0II^x(k)CII^x(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTII+kTzNIrC0II^x(k)+CII^x(k1)+kTzMI[z(k)+r(k)^y(k)] 29IIIIncrementalStandardStandardRampBoth^y=C0II^x(k)CII^x(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIrC0II^x(k)+CII^x(k1)EI^p(k)+kTzMI[z(k)+r(k)^y(k)] 83

PAGE 84

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 30IIIIncrementalStandardIncrementalRampBoth^y=C0II^x(k)CII^x(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIrC0II^x(k)+CII^x(k1)EI^p(k)+kTzMI[z(k)+r(k)^y(k)] 31IIIStandardIncrementalNoneStepNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 32IIIStandardIncrementalStandardStepNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 84

PAGE 85

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 33IIIStandardIncrementalIncrementalStepNone^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNIhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzMI[z(k)+r(k)^y(k)] 34IIIIncrementalIncrementalNoneRampBoth^y=CII(^x(k)^x(k1))+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]u=kTII+kTzNI[rCII(^x(k)+^x(k1))]+kTzMI[z(k)+r(k)^y(k)] 35IIIIncrementalIncrementalStandardRampBoth^y=CII(^x(k)^x(k1))+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTII+kTzNI[rCII(^x(k)+^x(k1))]+kTzMI[z(k)+r(k)^y(k)] 85

PAGE 86

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 36IIIIncrementalIncrementalIncrementalRampBoth^y=CII(^x(k)^x(k1))+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]u=kTII+kTzNI[rCII(^x(k)+^x(k1))]+kTzMI[z(k)+r(k)^y(k)] 37IVStandardStandardNoneRampBoth^y=CI^x(k)+DIuz=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIzu=kTIV+kTzIVNI+kTwN2I[rCI^x(k)]+kTzIVMI[z(k)+r(k)^y(k)] 38IVStandardStandardStandardRampBoth^y=CI^x(k)+DIu+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTIV+kTzIVNI+kTwN2IhrCI^x(k)FI^d(k)+EI^p(k)i+kTzIVMI[z(k)+r(k)^y(k)] 86

PAGE 87

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 39IVStandardStandardIncrementalRampBoth^y=CI^x(k)+DIu+FI^d(k)+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTIV+kTzIVNI+kTwN2IhrCI^x(k)FI^d(k)+EI^p(k)i+kTzIVMI[z(k)+r(k)^y(k)] 40IVIncrementalStandardNoneRampBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIzu=kTIV+kTzIVNI+kTwN2IrC0II^x(k)+CII^x(k1)+DII1u(k1)+kTzIVMI[z(k)+r(k)^y(k)] 87

PAGE 88

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 41IVIncrementalStandardStandardRampBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTIV+kTzIVNI+kTwN2IrC0II^x(k)+CII^x(k1)+DII1u(k1)EI^p(k)+kTzIVMI[z(k)+r(k)^y(k)] 42IVIncrementalStandardIncrementalRampBoth^y=C0II^x(k)CII^x(k1)DII1u(k1)+DII0u+EI^p(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^p(k+1)=^p(k)+Lp[y(k)y(k1)C^x(k)+C^x(k1)]u=kTIV+kTzIVNI+kTwN2IrC0II^x(k)+CII^x(k1)+DII1u(k1)EI^p(k)+kTzIVMI[z(k)+r(k)^y(k)] 88

PAGE 89

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 43IVStandardIncrementalNoneRampBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIzu=kTIV+kTzIVNI+kTwN2IhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzIVMI[z(k)+r(k)^y(k)] 44IVStandardIncrementalStandardRampBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTIV+kTzIVNI+kTwN2IhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzIVMI[z(k)+r(k)^y(k)] 89

PAGE 90

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 45IVStandardIncrementalIncrementalRampBoth^y=CI^x(k)+DIu+EIy(k)EICIx(k)+FI^d(k)z=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]^d(k+1)=^d(k)+Ld[y(k)y(k1)C^x(k)+C^x(k1)]u=kTIV+kTzIVNI+kTwN2IhrCI^x(k)EIy(k)+EICIx(k)FI^d(k)i+kTzIVMI[z(k)+r(k)^y(k)] 46IVIncrementalIncrementalNoneRampBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIzu=kTIV+kTzIVNI+kTwN2I[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzIVMI[z(k)+r(k)^y(k)] 90

PAGE 91

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations 47IVIncrementalIncrementalStandardRampBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=A^x(k)+Bu(k)+^d(k)+Lx[y(k)C^x(k)^p(k)]^d(k+1)=^d(k)+Ld[y(k)C^x(k)^p(k)]^p(k+1)=^p(k)+Lp[y(k)C^x(k)^p(k)]u=kTIV+kTzIVNI+kTwN2I[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzIVMI[z(k)+r(k)^y(k)] 48IVIncrementalIncrementalIncrementalRampBoth^y=CII(^x(k)^x(k1))DII1u(k1)+DII0uz=MI[z(k)+r(k)^y(k)]+NI[r^y]w=MI[w(k)+z(k)]+NIz^x(k+1)=(A+I)^x(k)A^x(k1)+Bu(k1)+Lx[y(k)y(k1)C^x(k)+C^x(k1)]u=kTIV+kTzIVNI+kTwN2I[rCII(^x(k)+^x(k1))+DII1u(k1)]+kTzIVMI[z(k)+r(k)^y(k)] 91

PAGE 92

Table 4-1 Continued MethodStateOutputEstimatorTrackingDisturbanceEquations CI=266666664CACA2...CANp377777775DI=2666666666666666666666666664CB0:::00CABCB:::00...............CANc1BCANc2B:::CABBCANcBCANc1B:::CA2B1Xi=0CAiBCANc+1BCANcB:::CA3B2Xi=0CAiB...............CANp1BCANp2B:::CANpNc+1BNpNcXi=0CAiB3777777777777777777777777775FI=2666666666664C1Xi=0CAi...Np1Xi=0CAi3777777777775EI=266666664II...I377777775 C0II=266666666666664C1Xi=0AiC2Xi=0Ai...CNpXi=0Ai377777777777775CII=266666666666664C1Xi=0AiI!C2Xi=0AiI!...C0@NpXi=0AiI1A377777777777775DII=2666666666664CB0:::00C1Xi=0AiCB:::0............CNp1Xi=0AiCNp2Xi=0Ai:::CNpNc+1Xi=0AiCNpNcXi=0Ai3777777777775 MI=266666664II...I377777775NI=26666666400:::0I0:::0............II:::I3777777750=266666664I00:::0II0:::0...............000:::I3777777751=266666664I0...0377777775 92

PAGE 93

Figure4-1. AblockdiagramillustratingthevariouselementsofanMPCsystem. 93

PAGE 94

Figure4-2. TheresultsofaMethodIMPCdesignappliedtoasimulatedchemicalreactorplant. 94

PAGE 95

Figure4-3. TheresultsofaMethodIIMPCdesignappliedtoasimulatedchemicalreactorplant. 95

PAGE 96

Figure4-4. TheresultsofaMethodIIIMPCdesignappliedtoasimulatedplant. 96

PAGE 97

Figure4-5. TheresultsofaMethodIVMPCdesignappliedtoasimulatedplant. 97

PAGE 98

CHAPTER5COLLECTIVEMOTIONUSINGFLATNESS-BASEDCONTROLMETHODS 5.1IntroductionSwarmintelligencehasbeenagrowingareaofresearchinterestduetoitsmultiplepotentialapplications,suchasvictimsearch-and-rescueandmilitaryoperations.Theincreasinglymorepowerfulcomputingandcommunicationscapabilitiesnowavailableformobilerobotshasalsocontributedtothesurgeininterestsonthistopic.Thispaperaddressestwofundamentalproblemsinswarmcontrol.Therstisthedesignofaockingstrategy,whichseekstoensurethatmultiplevehiclesfollowaprespeciedtrajectorywhilemaintainingapresetformation.Thesecondisthedesignofaformation-recoveryandcollision-avoidancestrategy,whichconsistsofasetoftacticsdeployedtoavoidcollisionsandtorestorethepresetformationwhenoneormorevehiclesdeviatefromtheirassignedpositions.Whetherarotatingornon-rotatingformationisused,thecontrollerencourageseachvehicleintheplatoontoconvergetoitscorrectpositionintheformation,butitdoesnotprovideanyguaranteecollisionsbetweenmembersoftheplatoonwillbeavoided.Mobilerobotsoperatinginrealenvironmentsarelikelytoencounterobstacleanduneventerrain,whichmaycausedeviationsfromtheirdesiredpaths.Thereisalsopotentialforequipmentmalfunctions,whichcouldcausearobottotravelinanundesireddirection,spinorstopaltogether.Inordertobeusefulaplatoonmustberobustagainsttheseproblems,sorecoverystrategiesthatallowvehiclestoavoidanyplatoon-matesthatdeviatefromtheirdesignedtrajectoriesareessential.Theseavoidancealgorithmsarealsousefulwhenvehiclesattempttoassumeformationfromanarbitraryinitialcondition.Inordertobesuccessfularecoverystrategymustbeabletomaintaincohesionintheplatoonwhileavoidinganout-of-controlvehicleandtoreformthedesireformationafterthedangerhaspassed. 98

PAGE 99

Thispaperpresentssolutionstobothoftheseproblemsforthecaseofnonholonomictwo-wheeledmobilerobots. 5.1.1LiteratureReviewMostofthesolutionsproposedforockingcontroldealwiththeformationrecoveryandcollisionavoidanceproblemsinanindirectmanner.Forinstance,themethodologyofElkaimandKelbley[ 35 ]isbasedonpotentialfunctions,ascommonlyusedinmobileroboticsforobstacleavoidanceproblems.Theneteectistheadditionofvirtualforcesthatdetermineanappropriatedirectionoftravelsincethevehicleisattractedtoitsgoal,inthiscaseacirclearoundavirtualleader,whilerepulsedbyobstaclesandothervehicles.Ifthevirtualleadermovessucientlyslowly,theplatooncanavoidcollisionandtravelbetweenobstacleswhilemaintainingitsoverallformation.Thedrawbacksofthistechniquearethatthevehiclesdonotuseacontrollawthatguaranteesconvergencetothedesiredposition.Instead,theseschemescomputeonlyanoptimaldirectionoftravel,withouttakingintoaccountthedynamicsofthevehicles.Toensurethatobstaclescanbeactuallyavoided,aconservativeapproachcanbeadoptedbydeningasafezonearoundeachvehicle.Theradiusofthezoneisdenedsothatitislargeenoughforthevehiclestostopwithoutactuallytouchinganyobstacles.AnapproachofthisnatureisexploredbyJusthandKrishnaprasad[ 36 ],butenhancedbytheincorporationofavailableinsightaboutthevehicledynamics.Morespecically,eachvehiclehasacontrollawthatincludesonecomponentthatmakesthemheadinthesamedirectionastheirneighbors,andasecondcomponentthatmakesthemsteerawayfromtheirneighborsiftheygettooclosetoanothervehicle,oralternatively,tosteertowardseachotheriftheyaretoofarapart.Atargetseparationdistancebetweenthevehiclesisestablishedviaauser-denedparameter. 5.1.2ProposedDesignThispaperextendsthecontrolstrategyproposedforasingletwo-wheeledmobilerobotbyBuccieriet.al.toaddressaswarmofsuchvehicles[ 37 ]. 99

PAGE 100

Thecollisionavoidancetechniquepresentedinthispaperisinspiredontheideasofreferences[ 35 ]and[ 36 ]discussedabove,sincetheapproachproposedinvolvestheexplicitinclusionofpotentialfunctionstocreatevirtualattractionandrepulsionforces.However,insteadofadoptingaglobalperspective,collisionavoidanceandockingcontrolaretreatedseparatelybymeansofswitchingcontrollersthatareactivatedbasedonthemeritsofparticularsituations.Furthermore,smoothswitchingfunctionsareintroducedtoaddresspotentialinstabilityproblemsassociatedwithswitchingstrategies[ 38 ][ 39 ]andtoaugmentthesizeofthedomainofstabilizingparameters.Thepaperisorganizedasfollows.Section 5.2 introducesthedynamicsofthevehicle,arestatementoftheargumentsthattheunderlyingnonlineardynamicsareat,andareviewoftheatness-basedcontrollerforasinglevehicleoriginallyproposedby[ 37 ].Section 5.3 introducestheextensionsofthesingle-vehiclecontroldesigntothecaseofacollectivemotionprobleminvolvingmultiplevehicles.Section 5.4 presentstheresultsofcomprehensivesimulationstudiestoillustratetheperformanceoftheproposedcontrolstrategy. 5.2Flatness-BasedControllerforaSingleVehicleThissectiondiscussesthesingle-vehicledynamicsconsidered,andrestatesthecontrollerdesign[ 37 ].Thenonholonomicwheeledmobilerobotconsideredinthisstudycanberepresentedbythekinematicmodel _x1=v1cosx3_x2=v1sinx3 (5{1) _x3=v2wheretheinputsv1andv2respectivelyrepresentthescalarlinearandangularvelocitiesoftherobot,thestatesx1andx2arerespectivelydenotethehorizontalandverticalCartesiancoordinatesoftherobot'sposition,andstatex3istheheadingangle,denedas 100

PAGE 101

theanglebetweenthelinearvelocityvectorandthepositivehorizontalCartesiansemiaxis(Figure 5-1 ).Buccierietal.[ 37 ]showedthatthenonlineardynamics( 5{1 )areat.Tobeconsideredat,agenericdynamicsystem_x=f(x;v),wherex2
PAGE 102

andcontinuousstatetrajectoriesxref1forx1andxref2forx2,isgivenbythecontrollaws v1=q _21+_22 (5{4) v2=2_1+1_2 12+22kp[x3arctan2(1;2)] (5{5) thatrespectivelydenethelinearandangularvelocitiesofthevehicle,wherethecontrolstatesevolveaccordingtothelineardynamicalsystem _1=k1(x1xref1)+k2(1_xref1)+xref1 (5{6) _2=k3(x2xref2)+k4(2_xref2)+xref2 (5{7) wherexref1,_xref1,and_xref1arerespectivelythereferenceposition,velocity,andaccelerationofthevehicle,andwheretheproportionalfeedbackgainkpandtheconstantski,i=1;2;3;4,andaretunableparameters.Notethatthecontrolstates1and2in( 5{6 )-( 5{7 )areinterpretedastheelementsofthetargettrajectory-velocityvector=[12]T2<2intheatstate-space.Theproportionalfeedbacktermwithadjustablegainkpisaddedtotheangularvelocitycontrollaw( 5{5 )toforcetheheadinganglex3toconvergetoitsspeciedvalue.Thisfeaturehasbeenshowntoensureasymptoticconvergencetothereferencetrajectory[ 37 ].Notethatforthespecialcaseofregulation,wherexref1=xref2=0foralltimes,thecontroller'sstate-spaceequations( 5{6 )-( 5{7 )reducetothesimplerlineardynamics _1=k1x1+k21_2=k3x2+k42(5{8) 5.3CollectiveMotionIthasbeenshownvianumericalandexperimentalstudiesthatthecontrolstrategy( 5{4 )-( 5{7 )yieldssubstantiallybettertrackingperformancethanclassicaldynamicfeedbackforthecaseofasinglerobot[ 37 ].Thissectionpresentsanextensionofthe 102

PAGE 103

atness-basedsingle-vehiclecontrollerdiscussedinthesection 5.3 tothecaseofcollectivemotioninvolvingaplatooncomprisingNidenticalvehiclesdeningtheplatoondynamics: _x1i=v1icosx3i_x2i=v1isinx3i (5{9) _x3i=v2ii=1;2;:::;NTheapproachconsistsofassigningtoeachrobotintheformationaatness-basedtrackingcontroller,modiedtoaddresstheadditionalconstraintsimposedbytherequirementofpreservingformation.Robustnessagainstobstacles,orthefailureofanagentintheplatoonarediscussedinanensuingsection.Thereferencetrajectoryofeachvehicleinaplatoonisdenedrelativetothatofaleader.Oneofthephysicalvehiclesintheswarmisdesignatedastheleader,oralternativelyavirtualleadermaybeused.Inthelattercaseeachvehicletrajectoryisspeciedrelativetoadesiredtrajectoryfortheplatoonratherthanthemeasuredpositionofanyvehicle.Thisdesiredtrajectorymaynotnecessarilycorrespondtothedesiredtrajectoryofanyparticularvehicleintheplatoon.Inthisdiscussionweassumethattheplatoonhasanassignedphysicalleader. 5.3.1NotationandDenitionsLettheplatoonbecomprisedofNidenticalvehiclesofthekinddescribedinsection 5.3 .ThehorizontalandverticalCartesiancoordinatesandtheheadingangleforeachindividualrobotiintheplatoonarerespectivelydenotedasx1i,x2i,andx3i,i=1;2;:::;N,andforthespecialcaseoftheplatoonleaderas(identiedbythesubscripti=L)asx1L,x2L,andx3L.Analogously,thecoordinatesforthereferencetrajectoryforeachrobotaredenotedasxref1iandxref2i,andthecoordinatesofthereferencetrajectoryfortherealorvirtualleaderoftheplatoonaredenotedasxref1Landxref2L. 103

PAGE 104

Aformationisdenedasthesetofpositiontrajectoriesspeciedfortheentireplatoon xref:=f(xref1i;xref2i);i=1;2;:::;Ng(5{10)wherexref1iandxref2iarecontinuoustwice-dierentiablefunctionsoftime.Theformationthenuniquelydenesthesetofvelocitiesandaccelerations _xref:=f(_xref1i;_xref2i);i=1;2;:::;Ng(5{11)and xref:=f(xref1i;xref2i);i=1;2;:::;Ng(5{12)respectively. 5.3.2ControlDesignforTrajectoryTrackingThecontrollaw v1i=q _21i+_22i (5{13) v2i=2i_1i+1_2i 1i2+2i2kp[x3arctan2(1i;2i)] (5{14) _1i=k1(x1ixref1i)+k2(1i_xref1i)+xref1i (5{15) _2i=k3(x2xref2i)+k4(2i_xref2i)+xref2i (5{16) resultswhenthescheme( 5{4 )-( 5{7 )isappliedtoeachindividualrobot.Thestateequations( 5{15 )-( 5{16 )mustbespecializedforeachformationunderconsideration,substitutingthecorrespondingreferencetrajectoryappearingontherighthandside. 5.3.2.1FormationSchemesTwoformationsareconsidered,namelyanonrotationalandarotationalscheme.Inbothcasesthereferencetrajectoryforeachrobotisdenedintermsofanosetfromthe 104

PAGE 105

referencetrajectoryoftheleader.Inanonrotationalformationtwoconstantosets1iand2ifromaxedCartesian-coordinatesframeareusedtodenethereferenceNotethat1L=2L=0. xref1i:=xref1L+1ixref2i:=xref2L+2ii=1;2;:::;N(5{17)asillustratedinFigure 5-2 .Inthealternativecaseofarotationalformation,twotime-dependentrotationaloperators1iand2iareusedtodeneequations xref1i=xref1L+1i (5{18) xref2i=xref2L+2i (5{19) 1i=1icosx3L2isinx3L (5{20) 2i=2icosx3L+1isinx3L (5{21) (5{22) where1iand2iaretwoconstantosetsdenedinarelativeframeofreferencecenteredontheleadervehicle,andorientedsothatthepositivevertical-semiaxisofthemovingframeisorientedalongthelinearvelocityvector[_x1L_x2L]T,asillustratedinFigure 5-3 .Obviously,1L=2L=0.Therelativepositionsofthevehiclesstayinvariantinthenonrotatingformation,regardlessofthepositionandorientationoftheleaderoftheplatoon.Thistypeofformationissimpletoimplementandimposesalowcomputationalburden;however,becausethesizeoftheentireformationdoesnotchange,itmaybeanundesirablechoiceincaseswheretheplatoonmustmaneuverinsmallspaces.Incontrast,inarotatingformationeachrobothasconstantosets1and2withrespecttoamoving 105

PAGE 106

Cartesian-coordinatesframeofreferencecenteredontheleader,andhencehavenon-constantosets1and2withrespecttoaxedCartesian-coordinatesframe. 5.3.2.2LeaderControllerBeforedevelopingthegoverningcontrolequationsfortheleaderoftheplatoon,itisusefultorstestablishtheleader'sangularacceleration x3L=2tanx3L (1+tan2x3L)2(5{23)ThisexpressionisreadilyobtainedafterrecognizingfromFigure( 5-1 ) x3L=arctan2(_x1L;_x2L)(5{24)takingthesecondderivativewithrespecttotimetoobtain x3L=2(_x2L=_x1L)[1+(_x2L=_x1L)2]2(5{25)andnallyinvokingtherelationship _x2L=_x1L=tanx3L(5{26)thatisimpliedinthedynamics( 5{9 ).Furthermore,since( 5{9 )alsoimpliesthat x3L=_v3L(5{27)from( 5{23 )itfollowsthat _v2L=2tanx3L (1+tan2x3L)2(5{28)Forthecaseoftheplatoonleader,wherei=L,thereferencetrajectories( 5{17 )and( 5{19 )reducetothesimpleform xref1i:=xref1Lxref2i:=xref2L(5{29)becausetheosetsandrotationaloperatorsareidenticallyzero. 106

PAGE 107

Then,theatness-basedcontrollerfortheleaderoftheplatoonresultswhenequations( 5{13 )-( 5{16 )arespecializedtothecasewherei=L,andwherethereferencetrajectoriesofthevehiclearegivenby( 5{29 ),yielding: v1L=q _21L+_22i (5{30) v2L=2i_1L+1_2i 1L2+2i2kp[x3arctan2(1L;2i)] (5{31) _1L=k1(x1Lxref1L)+k2(1L_xref1L)+xref1L (5{32) _2L=k3(x2xref2L)+k4(2L_xref2L)+xref2L (5{33) _v2L=2tanx3L (1+tan2x3L)2 (5{34) Notethattheangular-velocityderivative_v2Lgiveninthestateequation( 5{34 )isnotneededforthepurposesofcontrollingthepositionoftheleaderoftheplatoon.Itplaysanimportantrole,however,inthecontrolalgorithmfortheothervehiclesinthecaseofrotationalformations.Notealsothatthecontrolequationsfortheleaderareidenticalforbothformationsunderconsideration,giventhatbydenition,theosetsandhencetherotationaloperators,areidenticallyzeroforthecasei=L.Thecontroldesignfornonleadervehiclesispresentedintheensuingsections,specializedforeachformationofinterest. 5.3.2.3NonleaderController:NonrotationalFormationInanonrotationalformationthetrajectoriesofeachvehiclearegivenby( 5{17 ),whichfeaturesconstantosets.Hence,thevelocityandaccelerationtrajectoriesarethesameasthoseoftheleader.Then,thecontroldynamics( 5{13 )-( 5{16 )foreachrobotin 107

PAGE 108

theplatoonisexpressedasfollows: v1i=q _21i+_22i (5{35) v2i=2i_1i+1_2i 1i2+2i2kp[x3arctan2(1i;2i)] (5{36) _1i=k1(x1ixref1L1i) (5{37) +k2(1i_xref1L)+xref1L (5{38) _2i=k3(x2ixref2L2i) (5{39) +k4(2i_xref2L)+xref2L (5{40) (5{41) Thecaseofrotationalformationsissomewhatmorecomplicated,andthedetailsofthecontroldesignproposedaregivenbelow. 5.3.2.4NonleaderController:RotationalFormationThereferencevelocityandaccelerationtrajectoriesforeachvehicleinarotationalformationaredeterminedtakingtheappropriaterst-andsecond-ordertimederivativesof( 5{19 ).Thisyieldstheexpressions: _xref1i=_xref1L+_1i (5{42) _xref2i=_xref2L+_2i (5{43) xref1i=xref1L+1i (5{44) xref2i=xref2L+2i (5{45) Then,specializingthecontroldynamics( 5{14 )-( 5{16 )toanindividualmemberioftheplatooninarotationalformation,whileobeyingthesetrajectoryderivatives,leadstothe 108

PAGE 109

controlequations v1i=q _21i+_22i (5{46) v2i=2i_1i+1_2i 1i2+2i2kp[x3arctan2(1i;2i)] (5{47) (5{48) where _1i=k1(x1ixref1L1i)+k2(1i_xref1L_1i)+xref1L+1i (5{49) _2i=k3(x2ixref2L2i)+k4(2i_xref2L_2i)+xref2L+2i (5{50) andwheretherotationaloperatorsandtheirrstandsecondderivativesaredenedas 1i=1icosx3L2isinx3L (5{51) 2i=2icosx3L+1isinx3L (5{52) _1i=v2L2i (5{53) _2i=v2L1i (5{54) 1i=_v2L2iv2Lx3L1i (5{55) 2i=_v2L1iv2Lx3L2i (5{56) Equation( 5{55 )resultsafterdierentiating( 5{53 )andusingtherelationships( 5{23 )and( 5{28 );analogouslyforequation( 5{56 ).Notethat_v2L,thederivativeoftheangularvelocityoftheleadergivenin( 5{34 ),isutilizedin( 5{55 )and( 5{56 )tocalculatethesecond-derivativesoftherotationaloperators.Forcodingpurposes,itisusefultoremarkthatthecontrolalgorithmforrotationalformations( 5{49 )-( 5{56 )presentedinthissectionreducestothecaseofcontrolfora 109

PAGE 110

nonrotationalformation( 5{35 )-( 5{37 )byincorporatingthefollowingredenitionofvariablesin( 5{51 )-( 5{56 ): 1i:=1i (5{57) 2i:=2i (5{58) _1i:=0 (5{59) _2i:=0 (5{60) 1i:=0 (5{61) 2i:=0 (5{62) Hence,inthissense,therotational-formationcontrolalgorithmcanbeconsideredasageneralcasefordesignencompassingbothoftheformationsconsideredinthescopeofthiswork. 5.3.2.5NumericalSimplicationsTheusermaynditdesirabletoreducethecomputationalburdenassociatedwiththecalculationofthetangentfunctionin( 5{34 .Oneviableoptionistosubstitutetheexactexpressionfor_v2Lgivenin( 5{34 )withthenumericaldierentiationapproximation ^_v2L=v2L(t)v2L(t) (5{63)wheretdenotesthecontinuoustimevariable,and>0isasmalltimeinterval,andwherethevariablev2L(t)representsatime-delayedversionofv2L(t).Suchanapproximationmayrequiretheadoptionofstandardtechniquesneededtorobustifythenumericalestimationofderivatives,includingappropriatesignallteringoperationstoavoidthepropagationofnoise-inducederrors. 110

PAGE 111

Anothersimplicationthatleadstoareductionincomputationalcostisthegrossapproximation xref1i:=xref1Lxref2i:=xref2L(5{64)thatisobtainedafterignoringtheosetsin( 5{19 )forthepurposeofcalculationofthesecondderivativesoftherotationaloperators.Theresultisamodiedalgorithmreadilyobtainedbysettingi1=i2=0in( 5{46 )-( 5{56 ),andignoringequation( 5{34 )becausethevariable_v2Lisnolongerneeded.Hence,thisapproachalsoavoidsapotentiallyexpensivetangent-functioncalculation.Unfortunatelythisapproximationleadstosteady-stateerrorsinposition.Therefore,theapproximation( 5{64 )isonlyacceptablewhenitcanbeestablishedapriorithattheunavoidablesteady-stateerrorsinpositionareacceptablefortheformation.Whenthereferencevelocities_xref1Land_xref2Landaccelerationsxref1Landxref2Lfortheleaderarenotexplicitlygiveninanalyticalform,theycanbeestimatedon-lineusingthenite-dierenceapproximations ^_x1L=x1L(t)x1L(t) (5{65) ^_x2L=x2L(t)x2L(t) (5{66) ^x1L=x1L(t)2x1L(t)+x1L(t2) 2 (5{67) ^x2L=x2L(t)2x2L(t)+x2L(t2) 2 (5{68) where>0isasmalltimeinterval,andwheretheargumenttandt2representtime-delayedsignals.Numericaldierentiationformulasofhigherordercanbealternativelyusedforimprovedprecision. 5.3.3ControlDesignforCollisionAvoidanceandRecoveryTheliteraturereportscollisionavoidanceschemesforplatoonsofrobotsimplementedusingsimulatedrepulsionpotentialsestablishedbetweentwovehiclesastheycomein 111

PAGE 112

proximitywitheachother[ 35 ].Inthispaperwealsoadoptapotential-basedcollisionavoidancealgorithm,butinaddition,weintroduceaswitchfunctionijthatturnsonthecollision-avoidanceschemewhenasafe-distancethresholdisviolated.Morespecically,eachvehicleisassignedacollision-correctionzoneintheformofatoruscenteredaboutthevehicle,andcharacterizedbyanouterradiusR>0andaninnerradiusr>>><>>>>:0ifdij>R1dijr RrifrdijR1ifdij
PAGE 113

orthemoredesirable,thoughcomputationallymoredemanding,smoothswitchingmap ij=1 2+1 2tanh(R+r2dij) Rr (5{74) Thenonsmoothswitchingpolicy( 5{72 )isaneectivealternative,butmaycauseabruptchangesinvehicletrajectories,alongwithlargecontrolactionsthatcouldtosignalsaturationsshouldthemaximalvalueofinputsbeexceeded.Suchadverseeectsareavoidedusing( 5{72 ).Thecollisionavoidancestrategycaneasilybeextendedtoavoidcollisionswithstationaryormovingobstacles(otherthanplatoonvehicles),bytothepotentialfunctionthecoordinatesofeachoneoftheseobjectsalongwiththeirdistancestoeachvehicle.Theswitchingfunctionijtendstoavalueofzeroforeachvehicleifthedistancesdijforallvehicle-vehicleandvehicle-obstaclepairsaregreaterthantheradiusR.Inthatcasethecollision-avoidancepolicyisswitchedo.Thecollision-avoidancemechanismisturnedonassoonasavehicleentersthetoruswithouterradiusRandinnerradiusr.Theswitchingfunctionprogressivelyactivatesthecollision-avoidancemechanismasthevehiclesgetclosertoadistancer,startingfromthethreshold-violationdistanceR. 5.4SimulationStudiesSimulationstocomparerecoverystrategiesareperformedusingaplatoonofthreevehicles(i.e.,N=3)inarotatingformation.Theleader(identiedbythesubindexi=L)isassignedacircularreferencetrajectory.Thevehicles'initialpositionsandheadingsarearbitraryandnotconsistentwiththeformation.Inallcasesconsideredtheinitialpositionofthevehiclesasdenedatt=0arelistedinTable 5-1 alongwiththeformationosetsandcontrollerparameters.Theleader'sreferencetrajectoryisthecircledenedby xset1L=A[cos!t)+sin(!t)] (5{75) xset2L=A[cos!t)+sin(!t)] (5{76) 113

PAGE 114

Table5-1. Parametersusedinthesimulationstudies x1L(0)=3 1L=0 x2L(0)=2 2L=0 x3L(0)=0 12=0:5 x12(0)=2 22=0:5 x22(0)=2 13=0:5 x32(0)=0 23=0:5 x13(0)=3:5 x23(0)=2:5 x33(0)=0 1i(0)=2;i=L;2;3 2i(0)=1;i=L;2;3 =1 r=0:3 k1=100 R=0:5 k2=20 A=1 k3=100 !=2 k4=20 kp=2 Figure 5-6 ascenariowithnorecoverystrategyemployed.Collisionsoccurasthevehiclesattempttorestoretheirformation.Onesuchcollisionisdenotedbythepointatwhichthecurveshowingthedistancebetweenthemdropsbelowthemarkedthreshold.Figure 5-7 showstheresultsofasimulationwherethenonsmoothcollisionrecoverypolicy( 5{72 )isimplemented..Herethecollisionsareavoided,butattheexpenseofafairlyenergeticcontrolaction.Notethatdramaticchangesinthevaluesofthecontrolinputsareobservedwhenthecollisionavoidancepartofthecontrollawisactivated.Sincethesecontrolinputsarerobotvelocities,therobotsmotorsmuchbecapableofproducingthelargeaccelerationsrequiredforthissimulationtoberealistic.Thismaynotbepracticalformanyapplications,Figure 5-8 showstheimprovementsincontrol-energyrealizedwhenthesmoothrecoverypolicy( 5{74 )isimplemented.Inthisscenariocollisionsareagainavoided.Whilemorecontrolactionisneededthaninthecasewherenocollisionavoidanceisimplemented,theundesirablylargechangesinvelocitycausedbythenonsmoothrecoverystrategyareprevented. 114

PAGE 115

5.5ConclusionsSeveralsolutions,allbasedontheone-vehicleatness-basedcontroller,wereproposedfortheockingproblemandcanbeusedfordierentapplications,dependingonthesizeandtheavailablecomputingpoweroftheplatoon.Thecollisionavoidanceandtherecoveryproblemswereaddressedatoncebyintroducingaswitchingstrategybetweentheockingcontrolmodeandapotential-basedcollisionavoidancemode.Asmootherswitchingtechniquewasalsotestedtoreducethelikelihoodofunwantedoscillationsbetweenthetwomodes.Theresultingcontrollerhasbeensimulated,showedgoodresults,andcanthusbeusedforcollectivemotioncontrol. 115

PAGE 116

Figure5-1. Thecoordinatesystemofthewheeledmobilerobot.x1andx2aretheCartesianpositionoftherobot,x3istheheadingangleoftherobotandthecontrolinputsv1andv2arethelinearandangularvelocitiesrespectively. Figure5-2. Anon-rotatingformation.1and2aretheosetsoftheshadedvehiclefromthewhitevehicle,theleader,inthelabframe. 116

PAGE 117

Figure5-3. Arotatingformation.1and2aretheosetsoftheshadedrobotfromthewhiterobot,theleader,intheleadersframe.Theseresultintheosets1and2inthelabframe. Figure5-4. Arobotusingcollisionavoidance.TherobotreactstoanotherrobotapproachingtowithinadistanceRbychangingitscurrentreferencefromthebolddashedvector,tothesolidone,representingthesumofthetrackingandavoidancevectors. 117

PAGE 118

Figure5-5. TherobotreactstoanotherrobotapproachingtoadistancebetweenRandrbypartiallyactivatingcollisionavoidance,resultinginthereferencetrajectoryrepresentedbythebolddashedvector,representingthesumofthetrackingandavoidancevectors. Figure5-6. PlotAshowsthetrajectoriesofaswarmofthreerobotswithnorecoverystrategy.Thesolidlinerepresentstheleader,whichisattemptingtotrackacirculartrajectory.Thedottedlinesarethepathsoftwoothervehiclestrackingtheirpositionsintheformation.PlotBshowsthedistancedbetweeneachvehicleandtheleaderasafunctionoftime.Thepointatwhichthedistancedropsbelow0.3indicatesavehiclecollidingwiththeleader.PlotsCandDshowthecontrolactionsv1andv2respectivelyneededforeachvehicle. 118

PAGE 119

Figure5-7. PlotAshowsthetrajectoriesofaswarmofthreerobotswithadiscontinuousrecoverystrategy.Thetrajectoriesaresimilartothoseshowningure 5-6 exceptwheretherobotsacttoavoideachother.PlotBshowsthedistancebetweeneachvehicleandtheleader.NotethatthecloseapproachseeninFigure 5-6 isavoided.Thehorizontallineisaddedtoshowthedistancesratwhichcollisionavoidancebegins.PlotCshowstherequiredcontrolactionforthesecondvehicle.Bothinputschangedramaticallywhenthecollisionavoidancemodeisactivated. 119

PAGE 120

Figure5-8. PlotAshowsthetrajectoriesofaswarmofthreerobotswithacontinuousrecoverystrategy.Thetrajectoriesaresimilartothoseshownforthediscontinuousstrategy.PlotBshowsthedistancedbetweeneachvehicleindicatingthatnocollisionsoccur.ThetwohorizontallinesshowthedistanceRatwhichcollisionavoidancebeginsandthedistanceratwhichthecollisionavoidancelawreachesitsfulleect.PlotCshowstherequiredcontrolactionforthesecondvehicle.Bothcontrolactionslackthedramaticspikesseenabove. 120

PAGE 121

CHAPTER6CONCLUSIONThevirtualcontrollabframeworkoutlinedhereovercomesthelimitedinteractivityandmodularityofmanyothervirtuallabsallowingittobeusedtodevelopsimulationtoolsforawidevarietyofpedagogicalscenarios.Thisshouldmakeitevenmorevaluableforprovidingastudentwiththelearningvalueofalaboratorywithoutphysicalaccesstolabequipment.Somepotentialscenariosinwhichitcouldbeparticularlysuccessfularealsoexamined,howeverexibilityisakeytraitofthevirtualcontrollab,soitshouldbeeasyforaninstructortoapplyittoscenarioswhichwehavenotspecicallyconsidered.API2typedoubleintegralcontrollerimplementedeitherasaspecializedthreetermcontrollerorastwoPIcontrollersinseriesreducessteadystateerrortozeroforstepandramptypeinputsandtuningcorrelationshavebeendevelopedforrstorderplusdeadtimesystems.ThesecorrelationsgivethecontrollerparametersthatminimizetheITAEasanexponentialfunctionofthenondimentionalizedsystemparameters.StabilityofthesystemsrecommendedbythecorrelationhasbeenestablishedviaNyquistmethods.Modelpredictivecontrollersmodiedwithintegralstatesorvelocitycontroleectivelyeliminateosetwithrespecttostepchangesinsetpointorconstantvaluedisturbances.Controllerscontainingbothintegralstatesandvelocitycontrolormultiple-integralstatescansuccessfullyeliminateosetwithrespecttorampchangesinsetpoint.Theatness-basedcontrollerpreviouslyimplementedforasinglewheeledmobilerobot.Aswitchingtechniqueallowsforockingcontrolandcollisionavoidancetobeimplementedsimultaneously.Simulationofaswarmofrobotsoperatingundersuchacontrollershowspromisingresults,especiallywhenusedwithsmoothswitching. 121

PAGE 122

APPENDIXAALGEBRAICBACKGROUNDFORRAMPTRACKINGCONTROLLERS A.1ComparisonofPI2ControllerswithBelangerandLuyben'sDesignIntheirworkBelangerandLuybenuseaPI2controlleroftheform Gc=Kc1+1 i1s+1 i2s2(A{1)whilethecontrollerreferredtoasScheme2inchapter 3 isoftheform Gc=Kc1+1 i1s1+1 i2s2=Kc1+i1+i2 i1i2s+1 i1i2s2(A{2)Despitesupercialdierencesthesetwocontrollersarequitesimilar.ThiscanbeestablishedbyequatingthetwotransferfunctionsandlabelingBelangerandLuyben'stimeconstantswithaprime,yielding Kc1+1 i10s+1 i20s2=Kc1+i1+i2 i1i2s+1 i1i2s2(A{3)Dividingbythegainandequatinglikepowersofsyieldstherelationships i10=i1i2 i1+i2(A{4)and i20=i1i2(A{5)Luybenarguesthattherelationship i20=4i102(A{6)shouldbeusedtoreducetheimpactonthestabilityandthenumberofparametersneedingtuning.Substitutingtheequivalentgroupingsofthetimeconstantsusedhereyields i1i2=4i1i2 i1+i22 (A{7) (i1+i2)2=4i1i2 (A{8) 0=2i12i1i2+2i2 (A{9) i10=1 2i1=1 2i2 (A{10) i20=2i1 (A{11) ThusacontrollerofthetypeputforthbyBelangerandLuybencanbewrittenforanyschemetwoorschemethreecontrollerdevelopedfromthetuningcorrelationswrittenhere 122

PAGE 123

andanycontrollerthatfollowsLuyben'stuningrulescanbewrittenasaschemethreecontrollersincetheirtimeconstantrelationshipisanalogoustothetimeconstantsoftwoPIcontrollersbeingthesame. A.2ConstraintsRelatingtheDierentPI2schemesTheconstraintK2i1>4KcKi2describesthesubsetofscheme1controllerswhichcanalsoberepresentedbyscheme2controllerswithrealparameters.Thisrelationshipcanbefoundbyequatinglikepowersandsolvingtheresultingquadratics.Thetransferfunctionsfortheschemeoneandtwocontrollersare G(s)=Kc+Ki11 s+Ki21 s2(A{12)and G(s)=Kc1+1 i1s1+1 i2s(A{13)respectively.Equatinglikepowersyields Kc=Kc (A{14) Ki1=Kc1 i1+1 i2 (A{15) Ki2=Kc1 i1i2 (A{16) 1 i1=Ki2i2 Kc (A{17) Ki1=KcKi2i2 Kc+1 i2 (A{18) Ki1i2=Ki22i2+Kc (A{19) 0=Ki22i2Ki1i2+Kc (A{20) i2=Ki1+p K2i14Ki2Kc 2Ki2 (A{21) Thisequationhasrealsolutionsonlyfor K2i14KcKi2>0 (A{22) K2i1>4KcKi2 (A{23) A.3ProofofOsetEliminationforSeveralFormsofPI2TrackingRampsOneofthegoalsofthecontrollerssetforthhereistoachievezerosteadystateoset,sotheclosedlooperrorasdescribedinsection2mustgotozeroatlongtimes. lims!0se(s)=0(A{24) 123

PAGE 124

Theerrorcanbeexpressedasfollowsforarampsetpointinput. e(s)=Gre(s)r(s) (A{25) e(s)=1 1+Gp(s)Gc(s)1 s2 (A{26) Therstorderplantisrepresentedbythetransferfunction Gp(s)=K s+1exps(A{27)Whilethereareseveralschemesforthecontrollerasdiscussedinsectiontwo. Gc1(s)=Kc+Ki11 s+Ki21 s2 (A{28) Gc2(s)=Kc1+1 i1s1+1 i2s (A{29) Gc3(s)=Kc1+1 i1s2 (A{30) Ingeneralthesecanbewritten Gcj(s)=Kc s2Nj (A{31) e(s)=1 1+KKcNj (s+1)s2exps1 s2 (A{32) (A{33) Thezero-osetrequirementisthusequivalentto lims!01 s+KKcNj (s+1)sexps=0 (A{34) lims!0s(s+1) Nj=0 (A{35) Thisisclearlytrueinthecasewhen lims!0Nj6=0(A{36) 124

PAGE 125

itmayalsobetruewithNjdoesapproachzero,butthatanalysisisnotneededsinceitcanbeshown lims!0N1=lims!0s2+Ki1 Kcs+Ki2 Kc=Ki2 Kc (A{37) lims!0N2=lims!0s2+i1+i2 i1i2s+1 i1i2=1 i1i2 (A{38) lims!0N2=lims!0s2+2 i1s+1 2i1=1 2i1 (A{39) (A{40) ThusNjapproachesanonzeroconstantvalueforeachcaseandthezero-osetconditionismet. 125

PAGE 126

APPENDIXBEXAMPLEMATLABCODE B.1DoubleIntegralControllerTuningThefollowingcodecalculatedtheoptimaltuningvaluesusedtoproducethecurvesinchapter 3 whichareinturnthebasisfortherecommendedtuningrelationships. B.1.1TuneCorMain.m %MaintuningcorrelationgenerationroutingforPI2controllersglobaltf;globalform;globalKc;globaltaui1;globaltaui2;globalKi;globalKi2;globalr;globaly;globalt;form=3;sp=0;kpoints=20;tpoints=20;rpoints=50;KA=logspace(-1,1,kpoints);tauA=logspace(0,2,tpoints);ratioA=logspace(log10(.1),log10(50),rpoints);flags=zeros(rpoints,kpoints,tpoints);err=zeros(rpoints,kpoints,tpoints);if(form==1)KcOpt=zeros(rpoints,kpoints,tpoints);Taui1Opt=zeros(rpoints,kpoints,tpoints);Taui2Opt=zeros(rpoints,kpoints,tpoints);clearKiOpt;clearKi2Opt;elseif(form==2)KcOpt=zeros(rpoints,kpoints,tpoints);Taui1Opt=zeros(rpoints,kpoints,tpoints);clearTaui2Opt; 126

PAGE 127

clearKiOpt;clearKi2Opt;elseKcOpt=zeros(rpoints,kpoints,tpoints);KiOpt=zeros(rpoints,kpoints,tpoints);Ki2Opt=zeros(rpoints,kpoints,tpoints);cleartaui1Opt;cleartaui2Opt;endoptions=optimset('fminsearch');options=optimset(options,'MaxFunEvals',1000,'TolFun',1e-2,'Display','off');fid=fopen(['tc'datestr(date,29)'f'num2str(form)'.txt'],'a');%fori=1:rpointsfori=43:rpointsdisp(['Processing'num2str(ratioA(i))]);forj=10:kpointsK=KA(j);fork=1:tpointstau=tauA(k);theta=ratioA(i)*tau;tf=max(theta,tau)*15;%Guessedbasedonpreviousruns(improvesspeedgreatly)if(form==2)ifratioA(i)<1Kc0=(.990*ratioA(i).^-.835)/K;elseifratioA(i)<10Kc0=(0.822*ratioA(i).^-.0975)/K;elseKc0=0.6/K;endtaui0=tau/(.443*ratioA(i).^-.672);x0=[Kc0taui0taui0];elseif(form==1)ifratioA(i)<1Kc0=(.995*ratioA(i).^-.826)/K;elseifratioA(i)<10Kc0=(.837*ratioA(i).^-.1152)/K;elseKc0=0.65/K;endx0=[Kc0tau/(.505*ratioA(i).^-.558)...tau/(.390*ratioA(i).^-.789)]; 127

PAGE 128

elseifratioA(i)<1Kc0=(1.076*ratioA(i).^-.7868)/K;elseifratioA(i)<10Kc0=(0.8318*ratioA(i).^-.0964)/K;elseKc0=0.6/K;endtaui0=tau/(.443*ratioA(i).^-.672);x0=[Kc0Kc0*2*(.4317*(ratioA(i)).^-.6820)/tau...Kc0*(.4317*(ratioA(i)).^-.6820)^2/tau^2];end[minx,min,flag]=fminsearch(@itae,x0,options);%Simplexfprintf(fid,'%f\t%f\t%f\t%f\t%f\t%f\n\r',K,tau,theta,minx(1),minx(2),minx(3));endendendfclose(fid); B.1.2itae.m functione=itae(x)%GeneratesITAEcostfunction%requiresKc,taui1,taui1,Ki,Ki2,t,y,r,tfandformalreadybesetglobalKc;globaltaui1;globaltaui2;globalKi;globalKi2;globalform;globalt;globaly;globalr;globaltf;Kc=x(1);if(form~=3)taui1=x(2);Ki=1;Ki2=1;else 128

PAGE 129

Ki=x(2);Ki2=x(3);taui1=1;taui2=1;endif(form==1)taui2=x(3);elsetaui2=1;endtry%Simulatesim('tunecor',tf);%CalculateErrore=0;fork=1:size(y)-1e=e+abs(r(k)-y(k))*t(k)*(t(k+1)-t(k));endcatchwarning('SimulationError');e=1E100;endif(sum(abs(r(length(t)-5:length(t))-y(length(t)-5:length(t)))..../r(length(t)-5:length(t)))>.05)e=e+1E20;end B.2OsetFreeMPCThefollowingisthecodeforproducinggure 4.6 .Similarscriptstoexample1.mcanbecreatedtosimulateotherplantandcontrollercombinations. B.2.1Example1.m %GenerateCSTRexamplewithMethodIcontrollerclear;closeall;plant=2;%WeightsQ=[1001];S=[1001];%numberofelementsequaltothenumberofrowsofCm 129

PAGE 130

T=[500.5];%numberofelementsequaltothenumberofrowsofCmR=[0.50.5];%numberofelementsequaltothenumberofcolumnsofBm%NowdefinethepredictionNpandthecontrolNchorizon.Np=10;Nc=3;%ObserverLx=0.2*eye(3);Ld=0.1*eye(3);Lp=0.03*eye(3);method=1;stateEqn=0;outputEqn=0;est=0;[rdp]=MPCGenerateSP(Np,Nc,[000],[0.200.2],[000],...[-1.762e-57.784e-26.592e-2],[000],[-1.762e-506.592e-2]);MPCMain;subplot(311);holdon;stairs(1:ttotal,r(1:ttotal,1),'k-');stairs(1:ttotal,y(1:ttotal,1),'k--');h=legend(['r_1';'y_1']);set(h,'FontName','Times','FontAngle','italic','FontWeight','Bold');axis([0400-0.10.4]);subplot(312);holdon;stairs(1:ttotal,r(1:ttotal,3),'k-');stairs(1:ttotal,y(1:ttotal,3),'k--');h=legend(['r_3';'y_3']);set(h,'FontName','Times','FontAngle','italic','FontWeight','Bold');axis([0400-0.10.4]);subplot(313);holdon;stairs(1:ttotal,u(1:ttotal,1),'k');stairs(1:ttotal,u(1:ttotal,2),'k--');h=legend(['u_1';'u_2']);set(h,'FontName','Times','FontAngle','italic','FontWeight','Bold');axis([0400-2010]);xlabel('t');set(gcf,'PaperPosition',[0075]) 130

PAGE 131

B.2.2MPCMain.m %MainOperatingCodeforOffsetFreeMPC%Callaftermodel,method,weightsandsetpointhavebeeninitialized%===INITIALIZATION===%ifmethod==0||method==2S=zeros(size(S));endifmethod~=4T=zeros(size(T));end[ApBpCpAmBmCm]=MPCLoadPlant(plant);[QRST]=MPCAugmentWeights(Q,R,S,T,Np,Nc);[delta0delta1delta2]=MPCDeltas(Bm,Np,Nc);if(stateEqn==1)[AIIaAIIadBIICIICII0DIIEIIMINI]=MPCMatsII(Am,Bm,Cm,Np,Nc);[KyKzKw]=MPCGains(1,method,DII,NI,delta0,Q,R,S,T);else[AIBICICI0DIEIFIMINI]=MPCMatsI(Am,Bm,Cm,Np,Nc);[KyKzKw]=MPCGains(0,method,DI,NI,delta0,Q,R,S,T);end%ErrorCheckif(method>4)error('Methodnotcurrentlyavailable');endif(est>2||est<0)error('Unknownestimatortype');endif(outputEqn>3||outputEqn<0)error('Unknownouputequationtype');endif(stateEqn>2||stateEqn<0)error('Unknownstateequationtype');end%===CALCULATIONLOOP==%ykm1=zeros(size(Cm,1),1); 131

PAGE 132

ykm2=ykm1;xkm1=zeros(size(Am,1),1);ukm1=zeros(size(Bm,2),1);dukm1=ukm1;dkm1=zeros(size(Am,1),1);xhatkm1=xkm1;xhatkm2=xkm1;dhatkm1=dkm1;phatkm1=ykm1;zk=ykm1;wk=ykm1;rkm1=r(1,:);y=[];x=[];u=[];xhat=[];dhat=[];phat=[];ttotal=size(r,1)-Np;fork=1:ttotal%Plantxk=Ap*xkm1+Bp*ukm1+dkm1;yk=Cp*xk+p(k,:)';%ConstructrvectorfromsetpointdatarAug=reshape(r(k+1:k+Np,:)',[],1);%Estimatorif(est==0)xhatk=xk;dhatk=zeros(size(xk));phatk=zeros(size(yk));elseif(est==1)xhatk=Am*xhatkm1+Bm*ukm1+dhatkm1+Lx*(ykm1-Cm*xhatkm1-phatkm1);dhatk=dhatkm1+Ld*(ykm1-Cm*xhatkm1-phatkm1);phatk=phatkm1+Lp*(ykm1-Cm*xhatkm1-phatkm1);elseif(est==2)%incrementalpredictorxhatk=(Am+eye(size(Am)))*xhatkm1-Am*xhatkm2+Bm*dukm1...+Lx*(ykm1-ykm2-Cm*xhatkm1+Cm*xhatkm2);dhatk=dhatkm1+Ld*(ykm1-Cm*xhatkm1-phatkm1); 132

PAGE 133

phatk=phatkm1+Lp*(ykm1-Cm*xhatkm1-phatkm1);endif(stateEqn==0)if(outputEqn==0)yhat=CI*xhatk;elseif(outputEqn==1)yhat=CI0*xhatk+EI*yk;elseif(outputEqn==2)yhat=CI0*xhatk+EI*yk;elseif(outputEqn==3)yhat=CI*xhatk+EI*phatk;endif(method==2||method==3)yhat=yhat+DI*delta2*ukm1;endelseif(stateEqn==1)if(outputEqn==0)yhat=CII0*xhatk+CII*xhatkm1;elseif(outputEqn==1)yhat=CII0*(xhatk-xhatkm1)+EII*yk;elseif(outputEqn==2)yhat=CII0*(xhatk-xhatkm1)+EII*yk;elseif(outputEqn==3)yhat=CII0*xhatk+CII*xhatkm1+EII*phatk;endif~(method==2||method==3)yhat=yhat-DII*delta1*ukm1;endelseif(stateEqn==2)if(outputEqn==0)yhat=CI*xhatk+FI*dhatk;elseif(outputEqn==1)yhat=CI0*xhatk+FI*dhatk+EI*yk;elseif(outputEqn==2)yhat=CI0*xhatk+FI*dhatk+EI*yk;elseif(outputEqn==3)yhat=CI*xhatk+FI*dhatk+EI*phatk;endif(method==2||method==3)yhat=yhat+DI*delta2*ukm1;endend%IntegratingState(ifneeded) 133

PAGE 134

z=MI*(zk+r(k,:)'-yk)+NI*(rAug-yhat);zkp1=zk+r(k,:)'-yk;w=MI*(wk+zk)+NI*z;wkp1=wk+zk;%ControlLawuk=Ky*(rAug-yhat)+Kz*z+Kw*w;uk=uk(1:length(ukm1));%Onlyimplementnextuif(method==2||method==3)%Forvelocityweightedmethodscalculated"u(k)"isreallydeltau(k)dukm1=uk;uk=uk+ukm1;elsedukm1=uk-ukm1;endx=[x;xk'];y=[y;yk'];u=[u;uk'];xhat=[xhat;xhatk'];dhat=[dhat;dhatk'];phat=[phat;phatk'];xkm1=xk;ykm2=ykm1;ykm1=yk;ukm1=uk;dkm1=d(k,:)';xhatkm2=xhatkm1;xhatkm1=xhatk;dhatkm1=dhatk;phatkm1=phatk;zk=zkp1;wk=wkp1;end B.2.3MPCLoadPlant.m function[ApBpCpAmBmCm]=MPCloadPlant(p)%Loadmodelandrelateddataforoneofseveralpredefinedplants%Usage[ApBpCpAmBmCm]=MPCloadPlant(p)switchpcase1%system:twointerconectedtanks,fromCoughanowr 134

PAGE 135

%D.ProcessSystemsAnalysisandControl.pg459Am=[-32;4-5];Bm=[10;02];Cm=[10;01];Ap=Am;Bp=Bm;Cp=Cm;Bd=0.7*[11]';Dp=0.7*[11]';case2%TankReactormodelfortheprobleminRawlingsAICHEpaper.Am=[0.2511-3.368e-3-7.056e-4;11.060.3296-2.545;001];Bm=[-5.426e-31.530e-5;1.2970.1218;0-6.592e-2];%Cm=[100;001];Cm=eye(3);Ap=Am;Bp=Bm;Cp=Cm;Bd=[-1.762e-57.784e-26.592e-2]';Dp=[000]';case3%Ill-conditioneddistilationmodelfromRawlingsphi=exp(-5/75);Am=[phi0;0phi];Bm=phi*[0.878-0.864;1.082-1.096];Cm=eye(3);Ap=Am;Bp=phi*[0.878-0.88;1.1-1.096];Cp=Cm;Bd=[11]';Dp=[00]';case4%SimpleintegratingmodelfromMuskeandBadgewellcase5%PlantAfromRawlings'93Am=[4/3-2/3;10];Bm=[1;0];Cm=[-2/31]; 135

PAGE 136

Ap=Am;Bp=Bm;Cp=Cm;case6%PlantCfromRawlings'93Am=diag([0.50.60.50.6]);Bm=[0.50;00.4;0.250;00.6];Cm=[1100;0011];Ap=Am;Bp=Bm;Cp=Cm;case7%LargedistillationmodelfrormMuskeandBadgewell%Note:only8stateversionimplemented.Full28statemodelnot%availableAm=[10000000;01000000;000.799-0.00172-0.4060.000867-0.0457-0.0158;0000.6450.4440.5350.1590.139;00000.1200.00229-0.128-0.0431;000000.03620.0127-0.296;0000000.003860.00220;00000000.000372];Bm=[-7.57.507.5-7.5;-7.57.5-7.500;0.0713-0.0729000.374;-0.00697-0.0757000.179;0.0131-0.0148000.00567;-0.03500.0392000.00265;0.00323-0.0037300-0.00633;-0.001700.00201000.00435];Cm=[001.86-0.3960.318-0.001360.001390.000702;002.280.354-0.0741-0.00396-0.00102-0.000229;010.00102-0.000119-0.0004980-0.000203-0.000131;-10-0.00119-0.0001060.00046000.000139-0.000170];Ap=Am;Bp=Bm/.75; 136

PAGE 137

Cp=Cm;Bd=[11111111]';Dp=[0000]';end B.2.4MPCGenerateSP.m function[rdp]=MPCGenerateSP(Np,Nc,r0,rf,d0,df,p0,pf)%Generatestepsetpointandstepdisturbancesforuseinsimulations%Usage:[rdp]=MPCGenerateSP(Np,Nc,r0,rf,d0,df,p0,pf)tfinal=Np*(32+5*length(r0));r=[];d=[];p=[];fork=1:tfinal+Nprk=r0;fori=1:length(r0)if(k>Np*(2+5*(i-1)))rk(i)=rf(i);endendif(k>Np*(2+5*length(r0)))dk=df;elsedk=d0;endif(k>Np*(17+5*length(r0)))pk=pf;elsepk=p0;endr=[r;rk];d=[d;dk];p=[p;pk];endend B.2.5MPCGenerateRampSP.m function[rdp]=MPCGenerateRampSP(Np,Nc,r0,rSlope,d0,df,p0,pf)%Generaterampsetpointandstepdisturbancesforuseinsimulations%Usage:[rdp]=MPCGenerateRampSP(Np,Nc,r0,rSlope,d0,df,p0,pf)tfinal=Np*(32+5*length(r0));r=[]; 137

PAGE 138

d=[];p=[];fork=1:tfinal+Nprk=r0;fori=1:length(r0)if(k>Np*(32+5*(i-1)))rk(i)=rSlope(i)*(k-Np*(32+5*(i-1)));endendif(k>Np*2)dk=df;elsedk=d0;endif(k>Np*17)pk=pf;elsepk=p0;endr=[r;rk];d=[d;dk];p=[p;pk];endend B.2.6MPCDeltas.m function[delta0delta1delta2]=MPCDeltas(B,Np,Nc)%Generatesthedeltamatricesrequiredforrelating%standardandincrementaloutputequations%Usage:[delta0delta1delta2]=MPCDeltas(B,Np,Nc)%deltau=delta0*u+delta1*u(k-1)%u(k)=delta0^-1*deltau(k)+delta2*u(k-1)nu=size(B,2);delta0=[];delta1=eye(nu);delta2=[];fori=1:Ncif(i~=1)delta1=[delta1;zeros(nu)];enddelta2=[delta2;eye(nu)];endfori=1:Nc 138

PAGE 139

rd=[];forj=1:Ncif(i==j)rd=[rdeye(nu)];elseifi==(j+1)rd=[rd-eye(nu)];elserd=[rdzeros(nu)];endenddelta0=[delta0;rd];endend B.2.7MPCMatsI.m function[AIBICICI0DIEIFIMINI]=MPCMatsI(A,B,C,Np,Nc)%Generateaugmentedmatricesfornon-incrementalstatemodels%Usage:[AIBICICI0DIEIFIMINI]=MPCMatsI(A,B,C,Np,Nc)nx=size(A,1);nu=size(B,2);ny=size(C,1);AI=[];BI=[];CI=[];CI0=[];DI=[];EI=[];FI=[];MI=[];NI=[];TF=0;TB=eye(nx);fori=1:NpAI=[AI;A^i];CI=[CI;C*A^i];CI0=[CI0;C*(A^i-eye(nx))];EI=[EI;eye(ny)];TF=TF+A^(i-1);FI=[FI;C*TF];MI=[MI;eye(ny)];rB=[];rD=[]; 139

PAGE 140

rN=[];forj=1:Ncifj
PAGE 141

CII=[];DII=[];EII=[];MII=[];NII=[];TA=eye(nx);fori=1:NpTA=TA+A^i;AIIa=[AIIa;TA];AIIad=[AIIad;eye(nx)-TA];CII0=[CII0;C*TA];CII=[CII;C*(eye(nx)-TA)];EII=[EII;eye(ny)];MII=[MII;eye(ny)];TB=B;rB=[];rD=[];rN=[];ifi>Nc+1forj=Nc+1:i-1TB=TB+A^(j-Nc)*B;endendforj=1:Nc%Buildrowsbackwarssosummationsworkoutif(j
PAGE 142

BII=[BII;rB];DII=[DII;rD];NII=[NII;rN];endend B.2.9MPCGains.m function[KyKzKw]=MPCGains(stateEqn,method,D,N,delta0,Q,R,S,T)%Determinegains%Usage:[KyKzKw]=MPCGains(stateEqn,method,D,N,delta0,Q,R,S,T)%Setgradientofywithrespecttou(usefulforcalculatinggains)ifmethod==2||method==3if(stateEqn==1)grad=D;elsegrad=D*inv(delta0);endelseif(stateEqn==1)grad=D*delta0;elsegrad=D;endendKd=grad'*(Q+N'*S*N+N'*N'*T*N*N)*grad+R;Ky=inv(Kd)*grad'*Q;Kz=inv(Kd)*grad'*N'*S;Kw=inv(Kd)*grad'*N'*N'*T;end 142

PAGE 143

REFERENCES [1] T.F.JungeandC.Schmid,\Web-basedremoteexperimentationusingalaboratory-scaleopticaltracker,"inProceedingsoftheAmericanControlConfer-ence,vol.4,ChicagoIllinois,June2000,pp.2951{2954. [2] D.Gillet,C.Salzmann,R.Longchamp,andD.Bonvin,\Telepresence:Anopportunitytodeveloppracticalexperimentationinautomaticcontroleducation,"inEuropeanControlsConference,BrusselsBelgium,July1997. [3] D.Gillet,F.Georoy,K.Zeramdini,A.V.Nguyen,Y.Rekik,andY.Piguet,\Thecockpit:Aneectivemetaphorforweb-basedexperimentationinengineeringeducation,"InternationalJournalofEngineeringEducation,vol.19,no.3,pp.389{397,2002. [4] D.GilletandG.Fakas,\eMersion:Anewparadigmforweb-basedtraininginengineeringeducation,"inInternationalConferenceonEngineeringEducation,OsloNorway,July2002. [5] A.BhandariandM.H.Shor,\Accesstoaninstructionalcontrollaboratoryexperimentthroughtheworldwideweb,"inProceedingsoftheAmericanControlConference,Philadelphia,June1998. [6] H.Latchman,C.Salzmann,S.Thotapilly,andH.Bouzekri,\Hybridasynchronousandsynchronouslearningnetworksindistanceeducation,"inInternationalConfer-enceonEngineeringEducation,RiodeJaneiro,Brazil,1998. [7] B.ArmstrongandR.Perez,\Controlslaboratoryprogramwithanaccentondiscoverylearning,"IEEEControlSystemsMagazine,February,pp.1{20,2001. [8] D.R.YangandJ.Lee,\Javacontrolmodule,"http://dot.che.gatech.edu/information/research/issicl/che4400/javamodule.html,GeorgiaInstituteofTechnology,Atlanta,GA. [9] W.MessnerandD.Tilbury,\Controltutorialsformatlab,"http://www.engin.umich.edu/group/ctm/,UniversityofMichigan,AnnArbor,MI. [10] J.W.OverstreetandA.Tzes,\Internet-basedclient/servervirtualinstrumentsdesignsforrealtimeremote-accesscontrolengineeringlaboratory,"inProceedingsoftheAmericanControlConference,vol.2,SanDiego,1999,pp.1472{1476. [11] C.Schmid,\ThevirtualcontrollabVCLabforeducationontheweb,"inProceedingsoftheAmericanControlConference,vol.2,Philadelphia,1998,pp.1314{1318. [12] O.D.CrisalleandH.A.Latchman,\Virtualcontrollaboratoryformultidisciplinaryengineeringeducation,"1993,nSFAwardNo.DUE-9352523,proposalfundedbytheNationalSciencefoundationundertheInstrumentationandLaboratoryImprovementProgram. 143

PAGE 144

144 [13] D.Gillet,C.Salzmann,H.Latchman,andO.Crisalle,\Recentadvancesinremoteexperimentation,"inProceedingsoftheAmericanControlConference,vol.4,ChicagoIllinois,June2000,pp.2955{2956. [14] X.Vilalta,D.Gillet,andC.Salzmann,\Contributiontothedenitionofbestpracticesfortheimplementationofremoteexperimentationsolutions,"inIFACWorkshoponInternetBasedControlEducationIBCE'01,Madrid,Spain,December2001. [15] B.Kuo,AutomaticControlSystems,1995. [16] W.Luyben,ProcessModelingSimulationandControl,1990. [17] K.Ogata,ModernControlEngineering,2002. [18] B.OgunnaikeandW.H.Ray,ProcessDynamics,Modeling,andControl,1994. [19] G.Franklin,J.D.Powell,andA.Emami-Naeini,FeedbackControlofDynamicSystems,4thed.PrenticeHall,2001. [20] D.Seborg,T.Edgar,andD.Mellichamp,ProcessDynamicsandControl,2nded.Wiley,2004. [21] N.Nise,ControlSystemsEngineering,4thed.Wiley,2004. [22] R.Stefani,B.Shahian,C.Savant,andG.Hostetter,DesignofFeedbackControlSystems,4thed.OxfordUniversityPress,2002. [23] S.J,S.Dormido,R.Pastor,andF.Esquembre,\Interactivelearningofcontrolconceptsusingeasyjavasimulations,"inIFACWorkshoponInternetBasedControlEducationIBCE04,Grenoble,France,2004. [24] C.C.Ko,B.M.Chen,andJ.Chen,CreatingWeb-BasedLaboratories.Springer-Verlag,2004. [25] S.C.ChapraandR.P.Canale,NumericalMethodsforEngineers.McGraw-Hill,1990. [26] P.C.WankatandF.S.Oreovicz,TeachingEngineering.McGraw-Hill,1993. [27] A.Lopez,P.Murrill,andC.Smith,\Controllertuningrelationshipsbasedinintegralperformancecriteria,"InstrumentationTechnology,vol.14,1967. [28] C.SmithandA.Corripio,PrinciplesandPracticeofAutomationControl,1997. [29] J.D.P.G.FranklinandA.Emami-Naeini,FeedbackControlofDynamicSystems,2001. [30] W.Brogan,ModernControlTheory,1991.

PAGE 145

145 [31] P.BelangerandW.Luyben,\Designoflow-frequencycompensatorforimprovementofplantwideregulatoryperformance,"Ind.Eng.Chem.Res.,vol.36,p.5359. [32] R.Monroy-Loperena,I.Cervantes,A.Morales,andJ.Alvarez-Ramirez,\Robustnessandparametrizationoftheproportionalplusdouble-integralcompensator,"Ind.Eng.Chem.Res.,vol.38,p.2013. [33] K.R.MuskeandT.Badgwell,\Disturbancemodelingforoset-freelinearmodelpredictivecontrol,"JournalofProcessControl,vol.12,p.617. [34] G.PannocchiaandJ.Rawlings,\Disturbancemodelsforoset-freemodel-predictivecontrol,"AIChEJournal,vol.49,no.2,p.426. [35] G.ElkaimandR.Kelbley",\Alightweightformationcontrolmethodologyforaswarmofnon-holonomicvehicles,"IEEEAerospaceConference,2006. [36] E.JusthandP.Krishnaprasad",\Equilibriaandsteeringlawsforplanarformations,"SystemsandControlLetters,vol.52,p.25. [37] D.Buccieri,D.Perritaz,P.Mullhaupt,Z.Jiang,andD.Bonvin,\Velocityschedulingcontrollerforanonholonomicmobilerobot:Theoreticalandexperimentalresults,"IEEEAerospaceConference,2006. [38] J.Daafouz,P.Riedinger,andC.Iung,\Stabilityanalysisandcontrolsynthesisforswitchedsystems:Aswitchedlyapunovfunctionapproach,"IEEEJournalofAutomaticControl,vol.47,no.11,2002. [39] D.Liberzon,Equilibriaandsteeringlawsforplanarformations,2003.

PAGE 146

BIOGRAPHICALSKETCH ChristopherScottPeekwasbornin1980inMidlothian,Virginia.HegrewupintheRichmondareagraduatingfromChestereldCountyMathematicsandScienceHighSchoolatCloverHillin1999.HeearnedhisB.S.inchemicalengineeringattheUniversityofVirginia.InAugust2003hejoinedtheUniversityofFlorida,wherehelaterbecameamemberoftheProcessControlResearchGroupundertheguidanceofDr.OscarD.Crisalle.Asamemberofthegroup,heinvestigatedresearchtopicsonpredictivecontrolfromboththeoreticalandappliedperspectives. 146