AnIntroductiontoData-Driven ControlSystems
AliKhaki-Sedigh K.N.ToosiUniversityofTechnology
Iran
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Contents
Preface xi
Acknowledgements xv ListofAcronyms xvii
1Introduction 1
1.1Model-BasedControlSystemDesignApproach 1
1.1.1TheEarlyDevelopments 1
1.1.2Model-basedControlSystemStatusQuo 2
1.1.3ChallengesofModelsinControlSystemsDesign 3
1.1.4AdaptiveandRobustControlMethodologies 5
1.2Data-drivenControlSystemDesignApproach 5
1.2.1TheDesignerChoice:Model-basedorData-drivenControl? 7
1.2.2TechnicalRemarksontheData-DrivenControlMethodologies 9
1.3Data-DrivenControlSchemes 10
1.3.1UnfalsifiedAdaptiveControl 10
1.3.1.1UnfalsifiedControl:SelectedApplications 11
1.3.2VirtualReferenceFeedbackTuning 12
1.3.2.1VRFT:SelectedApplications 14
1.3.3SimultaneousPerturbationStochasticApproximation 15
1.3.3.1SPSA:SelectedApplications 16
1.3.4TheWillems’FundamentalLemma 16
1.3.4.1FundamentalLemma:SelectedApplications 18
1.3.5Data-DrivenControlSystemDesignBasedonKoopmanTheory 18
1.3.5.1Koopman-basedDesign:SelectedApplications 21
1.3.6Model-freeAdaptiveControl 23
1.3.6.1MFAC:SelectedApplications 24
1.4OutlineoftheBook 25 References 29
2PhilosophicalPerspectivesoftheParadigmShiftinControl SystemsDesignandtheRe-EmergenceofData-Driven Control 35
2.1Introduction 35
2.2BackgroundMaterials 36
2.2.1ScientificTheory 36
2.2.2ScientificRevolutionsandParadigmShifts 37
2.2.3RevolutionsinControlSystemsDesignfromKuhn’sPerspective 39
2.2.4PhilosophicalIssuesinControlEngineeringandControlSystems Design 41
2.2.5AGeneralSystemClassification 43
2.3ParadigmShiftsinControlSystemsDesign 44
2.3.1Pre-historyandPrimitiveControl 44
2.3.2Pre-classicalControlParadigm 44
2.3.3GeneralSystemTheoryandthePhilosophicalFoundationsof Model-BasedControl 45
2.3.4Model-BasedDesignParadigm 46
2.3.4.1PhilosophicalDiscussionsonModelPrevalenceinFeedback Control 46
2.3.5ClassicalControlDesign 49
2.3.6ModernControlDesign 50
2.4UncertaintyCombatParadigm 54
2.4.1UncertaintyandPerformanceProblem 54
2.4.2UncertaintyCombat:theRobustControlApproach 56
2.4.3UncertaintyCombat:theAdaptiveControlApproach 57
2.4.4UncertaintyCombat:theSoftComputing-basedControlApproach 59
2.5TheParadigmShiftTowardsData-drivenControlMethodologies 61
2.5.1UnfalsifiedPhilosophyinControlSystemsDesign 64
2.6Conclusions 68 References 69
3UnfalsifiedAdaptiveSwitchingSupervisoryControl 73
3.1Introduction 73
3.2APhilosophicalPerspective 75
3.3PrinciplesoftheUnfalsifiedAdaptiveSwitchingControl 77
3.3.1BasicConceptsandDefinitionsintheUASCMethodology 78
3.3.2TheMainResults 79
3.4TheNon-MinimumPhaseController 87
3.5TheDALPhenomena 88
3.6PerformanceImprovementTechniques 91
3.6.1FilteredCostFunction 91
3.6.2ThresholdHysteresisAlgorithm 92
3.6.3Scale-IndependentHysteresisAlgorithm 93
3.7IncreasingCostLevelAlgorithmsinUASC 95
3.7.1IncreasingCostLevelAlgorithm 97
3.7.2LinearIncreasingCostLevelAlgorithm 98
3.8Time-varyingSystemsintheUASC 101
3.9Conclusion 104
Problems 106
References 108
4Multi-ModelUnfalsifiedAdaptiveSwitchingSupervisory Control 111
4.1Introduction 111
4.2TheMulti-ModelAdaptiveControl 113
4.3PrinciplesoftheMulti-ModelUnfalsifiedAdaptiveSwitching Control 116
4.4PerformanceEnhancementTechniquesintheMMUASC 126
4.4.1DifferentMMUASCCostFunctions 126
4.4.2AdaptiveWindowintheMMUASC 127
4.5Input-constrainedMulti-ModelUnfalsifiedSwitchingControl Design 129
4.5.1Multi-ModelUnfalsifiedConstrainedAnti-WindupControl 130
4.5.2TheFeasibilityProblem 135
4.5.3QuadraticInverseOptimalControl 138
4.5.4Multi-ModelUnfalsifiedConstrainedGeneralisedPredictive Control 141
4.5.5VirtualReferenceSignalintheMMUCGPCScheme 143
4.5.6SwitchingAlgorithmintheMMUCGPC 144
4.6Conclusion 147 Problems 148 References 151
5Data-DrivenControlSystemDesignBasedontheVirtual ReferenceFeedbackTuningApproach 155
5.1Introduction 155
5.2TheBasicVRFTMethodology 156
5.2.1FilterDesign 159
5.3TheMeasurementNoiseEffect 163
5.3.1TheInstrumentalVariableSelection 164
5.4TheNon-MinimumPhasePlantsChallengeintheVRFTDesign Approach 166
5.5ExtensionsoftheVRTFMethodologytoMultivariablePlants 171
5.6OptimalReferenceModelSelectionintheVRFTMethodology 177
5.6.1TheParticleSwarmOptimisationScheme 180
5.7Closed-loopStabilityoftheVRFT-BasedData-DrivenControl Systems 183
5.7.1AnIdentification-BasedApproach 183
5.7.2AnUnfalsification-BasedApproach 184
5.8Conclusions 187
Problems 188 References 190
6TheSimultaneousPerturbationStochastic Approximation-BasedData-DrivenControlDesign 193
6.1Introduction 193
6.2TheEssentialsoftheSPSAAlgorithm 195
6.2.1TheMainTheoreticalResultoftheSPSAAlgorithm 198
6.3Data-DrivenControlDesignBasedontheSPSAAlgorithm 201
6.3.1ThePIDControl 202
6.3.2TheMPCApproach 202
6.4ACaseStudy:Data-DrivenControlofUnder-actuatedSystems 205
6.4.1TheLiquidSloshExample 205
6.4.2TheBallandBeamExample 210
6.5Conclusions 212 Problems 213 References 215
7Data-drivenControlSystemDesignBasedonthe FundamentalLemma 217
7.1Introduction 217
7.2TheFundamentalLemma 218
7.3SystemRepresentationandIdentificationofLTISystems 222
7.3.1EquivalentData-drivenRepresentationsofLTISystems 222
7.3.2Data-drivenState-spaceIdentification 224
7.4Data-drivenState-feedbackStabilisation 225
7.5RobustData-drivenState-feedbackStabilisation 228
7.6Data-drivenPredictiveControl 233
7.6.1TheData-enabledPredictiveControl(DeePC) 235
7.6.1.1Input–OutputDataCollection 235
7.6.1.2StateEstimationandTrajectoryPrediction 235
7.6.1.3TheDeePCAlgorithm 237
7.6.2LTISystemswithMeasurementNoise 239
7.6.3Data-drivenPredictiveControlforNonlinearSystems 241
7.7Conclusion 247 Problems 247 References 250
8KoopmanTheoryandData-drivenControlSystemDesignof NonlinearSystems 253
8.1Introduction 253
8.2FundamentalsofKoopmanTheoryforData-drivenControlSystem Design 254
8.2.1BasicConceptsandDefinitions 254
8.2.2Finite-dimensionalKoopmanLinearModelApproximation 258
8.2.3ApproximatingtheKoopmanLinearModelfromMeasuredData:The DMDApproach 259
8.2.4SystemStateVectorResponsewithDMD 262
8.2.5ApproximatingtheKoopmanLinearModelfromMeasuredData:The EDMDApproach 265
8.3Koopman-basedData-drivenControlofNonlinearSystems 269
8.3.1Koopman-WillemsLemmaforNonlinearSystems 270
8.3.2Data-drivenKoopmanPredictiveControl 272
8.3.3RobustStabilityAnalysisoftheData-drivenKoopmanPredictive Control 275
8.3.4RobustData-drivenKoopmanPredictiveControl 277
8.4ACaseStudy:Data-drivenKoopmanPredictiveControlofthe ACUREXParabolicSolarCollectorField 281
8.4.1Data-drivenKoopmanPredictiveControloftheACUREXSolar CollectorField 285
8.5Conclusion 288 Problems 288 References 290
9Model-freeAdaptiveControlDesign 293
9.1Introduction 293
9.2TheDynamicLinearisationMethodologies 295
9.2.1TheCompactFormDynamicLinearisation 296
9.2.2ThePartialFormDynamicLinearisation 297
9.2.3TheFullFormDynamicLinearisation 300
9.3ExtensionsoftheDynamicLinearisationMethodologiesto MultivariablePlants 302
9.3.1CFDLDataModelforNonlinearMultivariablePlants 303
9.3.2PFDLDataModelforNonlinearMultivariablePlants 303
x Contents
9.3.3FFDLDataModelforNonlinearMultivariablePlants 304
9.4DesignofModel-freeAdaptiveControlSystemsforUnknown NonlinearPlants 304
9.4.1Model-freeAdaptiveControlBasedontheCFDLDataModel 305
9.4.2Model-freeAdaptiveControlBasedonthePFDLDataModel 308
9.4.3Model-freeAdaptiveControlBasedontheFFDLDataModel 310
9.5ExtensionsoftheModel-freeAdaptiveControlMethodologiesto MultivariablePlants 314
9.5.1MFACDesignBasedontheCFDLDataModelforNonlinear MultivariablePlants 314
9.5.2MFACDesignBasedonthePFDLDataModelforNonlinear MultivariablePlants 318
9.5.3MFACDesignBasedontheFFDLDataModelforNonlinear MultivariablePlants 320
9.6ACombinedMFAC–SPSAData-drivenControlStrategy 330
9.7Conclusions 337 Problems 338 References 339
Appendix 341
ANorms 341
BLyapunovEquation 343
CIncrementalStability 343
DSwitchingandtheDwell-time 344
EInverseMoments 346
FLeastSquaresEstimation 349
GLinearMatrixInequalities 351
HLinearFractionalTransformations 353 References 355
Index 357
Preface
Model-basedcontrolsystems.Model-basedcontrolsystemanalysisanddesign approacheshavebeenthedominantparadigmincontrolsystemeducationand thecornerstoneofcontrolsystemdesignfordecades.Thesemethodologies relyonaccuratemathematicalmodelsandassumptionstoachievethedesired systembehaviour.Intheearlydecadesofthelastcentury,despitethetremendous interestinmodel-basedcontrolapproaches,manyPIDcontrollersintheindustry weredesignedbasedonthedata-driventechniqueofZiegler–NicholsPID parametertuning,whichisconsideredthefirstdata-drivencontrolapproach. Later,theadvancedadaptiveandrobustmodel-basedcontroltechniquesevolved tocombattheuncertaintychallengeintheestablishedmodel-basedtechniques. Theseadvancedcontroltechniquessuccessfullycontrolledmanyreal-worldand industrialplants.Yet,bothstrategiesrequiremathematicalmodelsandprior plantassumptionsmandatedbythetheory.
Data-drivencontrolmethodologies.Thelimitationsanduncertaintiesassociatedwithmodelsandassumptions,ontheonehand,andtheemergenceof progressivelycomplexsystems,ontheotherhand,havesparkedaparadigm shifttowardsdata-drivencontrolmethodologies.Theexponentiallyincreasing numberofresearchpapersinthisfieldandthegrowingnumberofcourses offeredinuniversitiesworldwideonthesubjectclearlyshowthistrend.Thenew data-drivencontrolsystemdesignparadigmhasre-emergedtocircumvent thenecessityofderivingofflineoronlineplantmodels.Manyplantsregularly generateandstorehugeamountsofoperatingdataatspecificinstantsoftime. Suchdataencompassesalltherelevantplantinformationrequiredforcontrol, estimation,performanceassessment,decision-makingandfaultdiagnosis. Thisdataavailabilityhasfacilitatedthedesignofdata-drivencontrolsystems.
Intendedaudience.Thisbookisanintroductiontodata-drivencontrolsystems andattemptstoprovideanoverviewofthemainstreamdesignapproachesin thefield.Theselectedapproachesmaybecalledwithcautiontheconventional approaches,notincludingtheapproachesbasedonsoftcomputingtechniques.
Auniquechapterisdevotedtophilosophical–historicalissuesregardingthe emergenceofdata-drivencontrolsystemsasthefuturedominantcontroldesign paradigm.Thischapterwillbeparticularlyappealingtoreadersinterestedin gaininginsightsintothephilosophicalandhistoricalaspectsofcontrolsystem designmethodologies.Conceptsfromthephilosophyofscienceandhistorical discussionsarepresentedtoshowtheinevitableprevalenceofdata-driven techniquesinthefaceofemergingcomplexadaptivesystems.Thisbookcancover agraduatecourseondata-drivencontrolandcanalsobeusedbyanystudent orresearcherwhowishestostartworkinginthefieldofdata-drivencontrol systems.Thisbookwillpresenttheprimarymaterial,andthereadercanperceive ageneraloverviewofthedevelopingdata-drivencontroltheory.Thebook presentationavoidsdetailedmathematicalrelationsandderivationsthatare availableinthecitedtechnicalpapersoneachsubject.However,algorithmsfor easyimplementationofthemethodswithnumericalandsimulationexamples areprovided.Thesoftwarecodesareavailableuponrequestfromtheauthor. Data-drivencontrolisalsoahotresearchtopic;manyfinal-yearundergraduate andpostgraduatestudentsareinterestedinstartingaresearchprojectinits differentareas.Theavailablereadingsourcesarethetechnicalpapersandthe limitednumberofresearchmonographsandbooksonthesubject.However,the technicalpapersareveryspecialisedandinvolvedeepmathematicalderivations. Thelimitednumberofpublishedmonographsandbooksalsospecialiseinspecific subjectareasanddonotprovideageneralintroductionandoverviewofdifferent methodologiesforafirst-timereaderindata-drivencontrol.Theselectedtopics inthisbookcanbeindividuallytaughtinmanydifferentcoursesonadvanced controltheory.Also,foraninterestedresearcherinanyofthecoveredfields,it wouldbebeneficialtolearnaboutthebasicsofotheralternativemethodologies toplanaresearchprogramme.
Prerequisites.Thebookisdesignedforgraduate-levelcoursesandresearchers specialisingincontrolsystemsacrossvariousengineeringdisciplines.Thebook assumesthatthereaderpossessesasolidunderstandingoffeedbackcontrol systemsaswellasfamiliaritywiththeprinciplesofdiscrete-timecontrolsystems andoptimisationproblems.Moreover,abasicunderstandingofsystemidentification,adaptivecontrolandrobustcontrolcanenhancethereader’scomprehension andappreciationofdata-drivencontrolmethodologies.
Overviewofthebook.Thebookisorganisedasfollows.Chapter1introduces boththemodel-basedanddata-drivencontrolsystemdesignapproaches. Itdiscussestheearlydevelopmentsandthecurrentstatusquoofmodel-based controlsystems,aswellasthechallengestheyface.Thechapteralsoexplores adaptiveandrobustcontrolmethodologiesasameanstoovercomesomeofthese
Preface
challenges.Subsequently,thedata-drivencontrolsystemdesignapproachis presented,andthetechnicalaspectsofdifferentdata-drivencontrolschemesare discussed.
Chapter2takesaphilosophicalperspectivetoanalysetheparadigmshiftsin controlsystemdesign.Itpresentsscientifictheory,revolutionsandparadigm shifts,drawingparallelstotheevolutionofcontrolsystemdesignmethodologies. Thehistoricaldevelopmentofcontrolsystemsdesignparadigmsandtheir philosophicalfoundationsisintroduced,andageneralclassificationofcontrol systemsisgiven.Thechapterconcludeswithanexplorationoftheparadigm shiftstowardsdata-drivencontrolmethodologies,withafocusontheinfluence oftheunfalsifiedphilosophy.
Chapters3and4presentdata-drivenadaptiveswitchingsupervisorycontrol andmulti-modeladaptiveswitchingsupervisorycontrol,respectively.ThephilosophicalbackboneofthepresentedmethodologiesisPopper’sfalsification theory,whichisintroducedinadata-drivencontrolcontextbySafonov.Itis showninChapter3thattheunfalsifiedadaptiveswitchingsupervisorycontrol caneffectivelycontrolunknownplantswithguaranteedclosed-loopstability undertheminimumassumptionoftheexistenceofastabilisingcontroller. Althoughseveralclosed-looptransientimprovementtechniquesarepresentedin Chapter3,themulti-modelunfalsifiedadaptiveswitchingcontrolisintroduced inChapter4toensureasuperiorclosed-looptransientperformance.Itisshown thatperformanceimprovementisachievedbyutilisingamodelsettoselect theappropriatecontrollerbasedonthefalsifyingtheory.Theadaptivememory concept,input-constraineddesignproblemsandquadraticinverseoptimal controlnotionarealsodiscussed.
Chapter5presentsthevirtualreferencefeedbacktuningapproach.Itisshown thatbyformulatingthecontrollertuningproblemasacontrollerparameter identificationproblem,adata-basedcontrollerdesignmethodologyisderived. Inthisapproach,avirtualreferencesignalisintroduced,anditisassumedthat thecontrollerstructureisknownapriori.Afterintroducingthebasicconcepts andmethodology,theproblemsofappropriatefilterdesign,measurementnoise, non-minimumphasezerochallenges,closed-loopstabilityandextensionsto multivariableplantsareaddressedinthischapter.
ThesimultaneousperturbationstochasticapproximationoptimisationtechniqueisintroducedandutilisedinChapter6forthedesignofdata-driven controlsystems.Itisshownthatthiscircumventsthenecessityofananalytical closed-formsolutiontothecontroloptimisationproblemsthatrequirealmost exactmathematicalmodelsofthetrueplant.Theessentialsofthetechnique arepresentedforplantswithunknownexactmathematicalmodels.Then,after
selectingacontrollerwithafixedknownstructurebutunknownparameters,by minimisingacostfunction,thecontrollerparametersarederived.Thepresented data-drivencontrolmethodologyisthenappliedtounknown,under-actuated systemsasapracticalcasestudy.
Chapter7presentsaclassofdata-drivencontrollersbasedonWillem’sFundamentalLemma.Itisinitiallyshownthatpersistentlyexcitingdatacanbeused torepresenttheinput–outputbehaviourofalinearsystemwithouttheneedto identifythelinearsystem’smatrices.Thederivedso-calledequivalentdata-based representationsofalineartimeinvariant(LTI)systemaresubsequentlyutilised todesigndata-drivenstate-feedbackstabilisersandpredictivecontrollerscalled Data-enabledPredictiveControl,orDeePCforshort.Resultswithmeasurement noiseandnonlinearsystemsarealsogiveninthischapter.
Chapter8presentsdata-drivencontrollersbasedonKoopman’stheoryandthe FundamentalLemmapresentedinChapter7.ThefundamentalsofKoopman’s theoryarebrieflyreviewedfordata-drivencontrol.Itisshownthatnonlinear dynamicalsystemsarepresentedbyhigherdimensionallinearapproximations. Themainnotionsof lifting or embedding andtheeffectivetoolsof(extended) dynamicmodedecompositionsareintroducedandadata-drivenKoopman-based predictivecontrolschemeispresentedbyincorporatingWillem’sFundamental LemmaofChapter7.Arobuststabilityanalysisisprovided,andtheresultsare finallyappliedtotheACUREXsolarcollectorfield.
Themodel-freeadaptivecontroldesignisadata-drivencontroldesignapproach basedondynamiclinearisationmethodologiesandispresentedinChapter9. Thethreemaindynamiclinearisationsdiscussedinthischapterareshownto capturethesystem’sbehaviourbyinvestigatingtheoutputvariationsresulting frominputsignals.Thesedatamodelsareutilisedforcontrollerdesignandtheir virtualnaturemakestheminappropriateforothersystemanalysispurposes.Also, inChapter9,thevirtualdatamodelresultsandtheircorrespondingmodel-free adaptivecontrollersareextendedtomultivariableplants.
Somepreliminaryconceptsthatareusefulforthechaptersarepresentedin theAppendix.Thechaptersareaccompaniedbyproblemsetsthatprovidereaderswiththeopportunitytoreinforcetheirunderstandingandapplytheconcepts discussed.Asolutionmanualisalsoprovidedforinstructorsteachingaclasson data-drivencontrolusingthisbookbycontactingtheauthor.
October2023
AliKhaki-Sedigh DepartmentofElectricalEngineering K.N.ToosiUniversityofTechnology
Acknowledgements
Thepreparationofthisbookhasgreatlybenefitedfromtheinvaluablecontributionsandsupportofnumerouspostgraduatestudentsandcolleagueswho generouslydedicatedtheirtimetoshareexpertise,valuablesuggestionsand correctionsthroughouttheprocess.Iextendmysinceregratitudetothefollowing individuals,whosesignificanteffortshavegreatlyenrichedthecontentofthis book:
● AmirehsanKarbasizadehfromtheDepartmentofPhilosophyattheUniversity ofIsfahan,forhisinsightfuldiscussionsonthephilosophyofscienceandhis invaluablecommentsthatnotablyenhancedChapter2.
● MojtabaNooiManzarfromFacultyofElectricalandComputerEngineering, ShahidBeheshtiUniversity,forhiscontributionstotheinitialdraftofChapters 3and4,aswellascorrectionsandthesimulationresultsforthesechapters.
● MohammadMoghadasi,MehranSoleymaniandMaedehAlimohamadi, mymaster’sstudentsintheAdvancedControlLaboratory,fortheirdiligent proofreadingofChapter3.
● BahmanSadeghiandMaedehAlimohamadi,mymaster’sstudentsinthe AdvancedControlLaboratory,fortheirvaluablecontributionstoChapter4.
● MohammadJeddiandFatemehHematiKheirabadi,mymaster’sstudentsinthe AdvancedControlLaboratory,fortheirinsightfulcontributionstoChapter5.
● SepidehNasrollahi,myPhDstudentintheAdvancedControlLaboratory,for hercontributionstoChapter6,aswellashervaluablecontributionstoother chaptersandthecreationofnumerousfiguresthroughoutthebook.
● TaherehGholaminejad,myPhDstudentintheAdvancedControlLaboratory, forhersignificantcontributionstoChapters7and8.
● SaraIman,PhDstudentfromIranUniversityofScienceandTechnology,forher meticulousproofreadingandusefulcommentsonChapters7and8.
● AliRezaei,mymaster’sstudentintheAdvancedControlLaboratory,forhis valuablecontributionstoChapter9andtheefforthededicatedtosimulations throughoutthebook.
Finally,Iwouldliketoexpressmysincereappreciationtotheanonymous reviewerswhoprovidedinvaluablefeedbackduringthereviewprocessof thisbook,andspecialthankstoWiley-IEEEPressfortheirexceptional professionalism,dedication,industryknowledgeandseamlesscoordination thatexceededmyexpectations.Lastbutnotleast,Iamalsogratefultomyfamily fortheircollaborationandsupport,allowingmetodedicatemostofmyholidays, weekendsandeveningstocompletingthisbook.
ListofAcronyms
ASSCAdaptiveswitchingsupervisorycontrol
BIBOBounded-inputbounded-output
CFDLCompact-formdynamiclinearisation
CSPConcentratedsolarpower
DALDehghani–Anderson–Lanzon
DDKPCData-drivenKoopmanpredictivecontrol
DeePC Data-enabled Predictive Control
DFTDiscreteFouriertransform
DMDDynamicmodedecomposition
EDMDExtendeddynamicmodedecomposition
ETFEEmpiricaltransferfunctionestimate
FFDLFull-formdynamiclinearisation
GLAGeneralisedLaplaceanalysis
GPCGeneralisedpredictivecontrol
ICLAIncreasingcostlevelalgorithm
IFACInternationalFederationofAutomaticControl
LFTLinearfractionaltransformation
LICLALinearlyincreasingcostlevelalgorithm
LLCLinearisationlengthconstant
LMILinearmatrixinequality
LQGLinearquadraticGaussian
LQRLinearquadraticregulator
LSTMLongshort-termmemory
LTILineartime-invariant
MFACModel-freeadaptivecontrol
MMUASCMulti-modelunfalsifiedadaptiveswitchingcontrol
MMUASC-RMMUASCwithresettime
MMUCGPCMulti-modelunfalsifiedconstrainedGPC
MPCModelpredictivecontrol
MPUMMostpowerfulunfalsifiedmodel
PEPersistenceofexcitation,persistentlyexciting
PFDLPartial-formdynamiclinearisation
PGPseudo-gradient
PIDProportionalintegralderivative
PJMPseudo-Jacobianmatrix
PPDPseudo-partialderivative
PPJMPseudo-partitioned-Jacobianmatrix
PSOParticleswarmoptimisation
QFTQuantitativefeedbacktheory
SAStochasticapproximation
SCLIStablycausallyleftinvertible
SICESocietyofInstrumentandControlEngineering
SIHSAScale-independenthysteresisalgorithm
SISOSingle-input-single-output
SNRSignal-to-noiseratio
SPSASimultaneousperturbationstochasticapproximation
SVDSingularvaluedecomposition
THSAThresholdhysteresisalgorithm
UASCUnfalsifiedadaptiveswitchingcontrol
UASC-RUASCwithreset-time
UASSCUnfalsifiedadaptiveswitchingsupervisorycontrol
VRFTVirtualreferencefeedbacktuning
1.1.1TheEarlyDevelopments
Theadventofmodelsincontrolsystemstheoryanddesignisrootedintheseminal paperofMaxwell OnGoverners [1].NorbertWiener,inintroducingtheword cyberneticsinRef.[2]describestheMaxwellpaperas‘… thefirstsignificantpaper onfeedbackmechanismsisanarticleongovernors,whichwaspublishedbyClerk Maxwellin1868’andinRef.[3],Maxwellisrecognisedasthe‘fatherofcontrol theory’.TheMaxwellmagicwastointroducedifferentialequationsinmodelling thebehaviouroftheflyballgovernorfeedbackcontrolsysteminventedbyJames Wattin1788.Thisground-breakingcontributionbyMaxwellintroducedthe conceptofmathematicalmodellinginthestabilityanalysisofaclosed-loopcontrolsystem,anideathatsoonfoundmanyapplicationsandadvocatesandsolved manyuntilthenunsolvedstabilityanalysisproblems.Thedifferentialequations encounteredintheflyballgovernormodelwerenonlinear.Bylinearisingthese nonlinearequations,Maxwellmanagedtointroducethenotionsofwhatistoday calledrealpoles,imaginarypolesandthesignificanceofpolepositionintheright halfplane.Thismodel-basedapproachtotheanalysisofacontrolsystemthrough thedifferentialequationsofmotionwasperformedforthefirsttimeinthehistory ofcontroltheory.Hence,itisplausibletointroduceMaxwellasthepioneerofthe model-basedcontroltheory.
Intheearlytwentiethcentury,controlsystemdesignmethodologiessuchas the classicalcontrol techniquesinitiatedbyBode,Nyquist,EvansandNichols wereallmodel-basedapproachestocontroldesignsincethetransferfunction knowledgeofthecontrolledsystemisrequired.Thetransferfunctioncanbe derivedfromasetofalgebraicanddifferentialequationsthatanalyticallyrelate inputsandoutputs,oritcouldbeobtainedfromsimpletestsperformedonthe plantwiththeassumptionsoflinearityandtimeinvariance.Laterinthe1960s,
AnIntroductiontoData-DrivenControlSystems,FirstEdition.AliKhaki-Sedigh. ©2024TheInstituteofElectricalandElectronicsEngineers,Inc.Published2024byJohnWiley&Sons,Inc.
Kalmanintroducedthemodel-basedstate-spaceapproachthatwasmoredetailed andmathematical.
Theonlynotabledata-driventechniqueofthefirsthalfofthelastcentury istheZiegler–Nicholsproportional-integral-derivative(PID)parametertuning proposedinRef.[4],whichbecameawidelyusedcontroltechniqueinthe industry[5].ItisstatedinRef.[4]that“Apurelymathematicalapproachtothe studyofautomaticcontroliscertainlythemostdesirablecoursefromastandpoint ofaccuracyandbrevity.Unfortunately,however,themathematicsofcontrolinvolves suchabewilderingassortmentofexponentialandtrigonometricfunctionsthatthe averageengineercannotaffordthetimenecessarytoplowthroughthemtoasolution ofhiscurrentproblem.”Thisstatementfromtheeminentcontrolengineersof thattimeshowsthelong-lastinginfluenceofmathematicalmodel-baseddesign techniquesonthecontrolsystemsdesigncommunity.Indescribingtheirwork, theyimmediatelystatethat‘thepurposeofthispaperistoexaminetheactionof thethreeprincipalcontroleffectsfoundinpresent-dayinstruments,assignpractical valuestoeacheffect,seewhatadjustmentofeachdoestothefinalcontrol,andgive amethodforarrivingquicklyattheoptimumsettingsofeachcontroleffect.The paperwillthusfirstendeavortoanswerthequestion:“Howcanthepropercontroller adjustmentsbequicklydeterminedonanycontrolapplication? ”’Thisstatement canenlightenaspectsofthephilosophyofthedata-drivencontrolsystemsthat evolvedinthelatetwentiethcenturyonwards.
1.1.2Model-basedControlSystemStatusQuo
Model-basedcontrolsystemdesignisthedominantparadigmincontrolsystem educationanddesign.Thisapproachisbasedonderivedanalyticalmodelsfrom basicphysicallawsandequationsormodelsfromanidentificationprocess.Models areonlyapproximationsofrealityandcannotcaptureallthefeaturesandcharacteristicsofaplantundercontrol.High-frequencyun-modelleddynamicsarean example,asinroboticandspacecraftapplicationswheretheresidualvibration modesarenotincludedinthemodel[6].Thestructureofamodel-basedcontrol systemisshowninFigure1.1.Inthecaseofadaptivecontrolstrategies,theapproximateplantmodelisupdatedusingtheinput–outputdata.
AsisshowninFigure1.1,theplantmodel,derivedfromfirstprinciplesoridentifiedfromplant-measureddata,isusedtodesignafixed-ordercontrollersatisfying thespecifiedclosed-looprequirements.However,thedesignedcontrollerdoesnot necessarilysatisfythepre-definedrequirementswhenconnectedtotherealplant, andtheclosed-loopperformanceislimitedbythe modellingerrors.Modelling errorscanhavemanyrootcauses,suchasun-modelleddynamics,unknownor varyingplantparametersresultingfromchangingoperatingpoints,equipment ageingorfaultsandinappropriatemodelstructures.
Figure1.1 Thestructureofamodel-basedcontrolsystemdesign.
Modellingerrorsduetoun-modelleddynamicsarejustifiedinthestandardpracticeofmodel-basedcontroldesignwhenthesystemiscomplexandisofa high order,andalow-ordermodelisemployedtofacilitatethecontroldesign.Onthe otherhand,therecanbeatendencytoincreasethemodelordertofindasuitable model.ItisshowninRef.[7]thatthisisnotgenerallytrueifthemodelhastobe usedforcontroldesign.Infact,the order ofarealsystemisabadlydefinedconcept, andeventhemostaccuratemodelsareonlyanapproximationoftherealplant. Intherealworld,a full-ordermodel doesnotexist,andanydescriptionis,bydefinition,anapproximation[7].Model-basedcontroldesigncanonlybeemployedwith confidenceinreal-worldapplicationsifthemodelstructureisperfectlyknown.
Theissueofmodel-basedcontrolsystemdesignandtheparadigmshiftstoand frommodel-basedapproachesisfurtherelaboratedinChapter2.
1.1.3ChallengesofModelsinControlSystemsDesign
Theintroductionofthestate-spaceconceptbyKalmanin1960,togetherwiththe newlyestablishednotionofoptimality,resultedinaremarkabledevelopmentof model-basedcontroldesignmethods.BeforeKalman’sstate-spacetheory,most ofthecontroldesignwasbasedontransferfunctionmodels,asisintheBode andNyquistplotsortheroot-locusmethodandtheNicholschartsforlead–lag compensatordesign.
Inthecaseswherereliablemodelswereunavailable,orinthecaseofvarying parametersandchangingoperatingconditions,theapplicationofthemodel-based controlwasseverelylimited.Inthemid-1960s,thesystemidentificationstrategy evolved.TheproposedMaximumLikelihoodframeworkfortheidentificationof
input–outputmodelsresultedinthepredictionerror-typeidentifiers.Theadvent ofidentificationtheorysolvedtheproblemofcontrollingcomplextime-varying plantsusingmodel-basedcontroldesignmethodologies.
Initially,controlscientistsworkingontheidentificationmethodsaimedat developingsophisticatedmodelsandmethodologieswiththeelusivegoalof convergingtothe truesystem,undertheassumptionthatthetruesystemwas inthedefinedmodelset.Later,theyrealisedthatthetheorycouldbestachieve anapproximationofthetruesystemandcharacterisethisapproximationin termsofbiasandvarianceerrorontheidentifiedmodels.Finally,system identificationwasguidedtowardsacontrol-orientedidentification.Inallthe modellingstrategies,modellingbyfirstprinciplesorbyidentificationfromdata, modellingerrors areunescapable,and explicitquantification ofmodellingerrorsis practicallyimpossible.Hence,themodellingstrategiescannotprovideadequate practicaluncertaintydescriptionsforcontroldesignpurposes.Therefore,thefirst modellingprinciplegiveninRef.[8],thatarbitrarilysmallmodellingerrorscan leadtoarbitrarilybadclosed-loopperformance,isseriouslyalarmingforcontrol systemsdesigners.
Applicationofthe certaintyequivalenceprinciple (seeChapter2)wasbasedon theearlyoptimisticassumptionthatitispossibletoalmostperfectlymodelthe actualplantandthemathematicalmodelobtainedfromthefirstprinciplesor byidentificationfrominput–outputdataisvalidenoughtorepresentthetrue system.However,applicationsinreal-worldproblemsdidnotmeettheexpectationsofthecontrolscientistsanddesigners.Therefore,anobviousneedprevailed toguaranteeclosed-loopstabilityandperformanceinthemodel-basedcontrol designapproaches.Thisledtothedevelopmentofthemodel-basedapproaches offixed-parameterrobustcontrolandadaptivecontrolsystemdesign[9].
Themathematicalmodelsderivedfromthephysicallawshavebeeneffectively usedinpracticalapplications,providedthatthefollowingassumptionshold:
● Accuratelymodeltheactualplant.
● Prioriboundsonthenoiseandmodellingerrorsareavailable.
Also,identificationmodelshavebeenemployedinmanypracticalapplications. Theidentifiedmodelcancapturethemainfeaturesoftheplant,providedthat
● Compatibilityoftheselectedmodelstructureandparameterisationwiththe actualplant’scharacteristicsisassumed.
● Theexperimentdesignisappropriate;thatis,forcontrolproblems,theselection oftheinputsignalisinaccordancewiththeactualplant’scharacteristicsorthe persistenceofexcitation(PE)condition.
Itisimportanttonotethateveninthecaseofanaccuratelymodelledplant, iftheassumptionsabouttheplantcharacteristicsarenotmet,themathematical
theoremsrigorouslyprovingclosed-looprobuststabilityandperformanceand parameterconvergencearenotofpracticalvalue. Hence,insummary,if
● Anaccuratemodelisunavailable,or
● Theassumptionsregardingtheplantdonothold, thedesignedmodel-basedcontroller,validatedbysimulations,canleadtoan unstableclosed-loopplantorpoorclosed-loopperformance.
1.1.4AdaptiveandRobustControlMethodologies
Adaptiveandrobustcontrolsystemshavesuccessfullycontrolledmanyreal-world andindustrialplants.However,bothstrategiesrequiremanypriorplantassumptionstobemandatedbythetheory.Thekeyquestionsaretheclosed-looprobust stabilityandrobustperformanceissuesinpracticalimplementations.Theassessmentofthesespecificationsisnotpossibleapriori,asunforeseeneventsmayoccur inpractice.Hence,thecontrolengineermustresortto adhoc methodsforasafe andreliableclosed-loopoperation.Thisisoftendonebyperformingmanytestsfor manydifferentvariationsofuncertaintiesandoperatingscenariosintheMonte Carlosimulations.However,withthegrowingplantcomplexityandthepossible testsituations,thecostoftheseheuristictestsincreases.Hence,thelimitations inherentintheadaptiveandrobustcontrollersareclearlyobserved.Parameter adjustmentsandrobustcontrolandtheirsynergisticdesignpackagesaretheultimatesolutionsofthemodel-basedcontrolscientistsfortheutmostguaranteeof safeandreliableclosed-loopcontrol.
Aclosed-loopsystem’sperformancedegradationandeveninstabilityarealmost inevitablewhentheplantuncertaintyistoolargeorwhentheparameterchanges orstructuralvariationsaretoolargeoroccurabruptly.Theadaptiveswitching controlwasintroducedasoneoftherobustadaptivecontroltechniquesto handlesuchsituationsandlessentherequiredpriorassumptions.Thisledto theswitchingsupervisorycontrolmethods,whereasupervisorcontrollerselects theappropriatecontrollerfromacontrollerbank,similartotheirancestor,the gain-schedulingmethodology,whichhasbeenandstilliswidelyusedinmany applications.Thismindsetandtherecentlydevelopedselectionprocessbasedon thefalsificationtheoryareregardedasthefirstattempttowardstrulydata-driven, almostplant-independentadaptivecontrolalgorithms[10].
1.2Data-drivenControlSystemDesignApproach
Tocircumventthenecessityofderivingofflineoronlineplantmodels,analternativeapproachtocontrolsystemdesignistousetheplantdatatodirectlydesign
thecontroller.Thisisthe data-driven approach,whichappearedattheendofthe 1990s.Manyplantsregularlygenerateandstorehugeamountsofoperatingdataat specificinstantsoftime.Suchdataencompassalltherelevantplantinformation requiredforcontrol,estimation,performanceassessment,decisionmakingsand faultdiagnosis.Thisfacilitatesthedesignofdata-drivencontrolsystems.Theterm data-drivenwasinitiallyusedincomputerscienceandhasenteredthecontrolsystemscienceliteratureinthepasttwodecades.Althoughdata-drivencontrolwas actuallyintroducedinthefirstdecadesofthetwentiethcentury(seeChapter2), theapproachwasnotcalleddata-drivenatthattime.Thedata-drivencontroland data-basedcontrolconceptsaredifferentiatedinRef.[11].Also,Ref.[12]haselaboratedonthedifferencebetweendata-basedanddata-drivencontrol.Itisstated inRef.[12]that‘data-drivencontrolonlyreferstoaclosedloopcontrolthatstartingpointanddestinationarebothdata.Data-basedcontrolisthenamoregeneral termthatcontrollersaredesignedwithoutdirectlymakinguseofparametricmodels, butbasedonknowledgeoftheplantinput-outputdata.Sortedaccordingtotherelationshipbetweenthecontrolstrategyandthemeasurements,databasedcontrolcan besummarizedasfourtypes:post-identificationcontrol,directdata-drivencontrol, learningcontrol,andobserverintegratedcontrol.’
Themainfeaturesofthedata-drivencontrolapproachescanbecategorisedas follows:
● Controlsystemdesignandanalysisemployonlythemeasuredplant input–outputdata.Suchdataarethecontrollerdesign’sstartingpoint andendcriteriaforcontrolsystemperformance.
● Noprioriinformationandassumptionsontheplant’sdynamicsorstructureare required.
● Thecontrollerstructurecanbepredetermined.
● Theclosed-loopstability,convergenceandsafeoperationissuesshouldbe addressedinadata-drivencontext.
● Adesigner-specifiedcostfunctionisminimisedusingthemeasureddatato derivethecontrollerparameters.
Thestructureofadata-drivencontrolsystemdesignisshowninFigure1.2. Severaldefinitionsfordata-drivencontrolareproposedintheliterature.The followingdefinitionfromRef.[11]ispresented.
Definition1.1
Data-drivencontrolincludesallcontroltheoriesandmethodsin whichthecontrollerisdesignedbydirectlyusingonlineorofflineinput–output dataofthecontrolledsystemorknowledgefromthedataprocessingbutnot anyexplicitinformationfromamathematicalmodelofthecontrolledprocess andwhosestability,convergenceandrobustnesscanbeguaranteedbyrigorous mathematicalanalysisundercertainreasonableassumptions.
Control design with NO prior assumption on
Figure1.2 Thestructureofadata-drivencontrolsystemdesign.
Thethreekeypointsofthisdefinitionarethedirectuseofthemeasured input–outputdata,datamodellingratherthanfirstprinciplesmodellingor identifiedmodelling,andtheguaranteeoftheresultsbytheoreticalanalysis.
1.2.1TheDesignerChoice:Model-basedorData-drivenControl?
Ingeneral,systemsencounteredincontrolsystemsdesigncanbecategorisedas simple,complicated,complexandcomplexadaptive(seeChapter2fordefinitions andmoredetails).Inreal-worldapplications,thecontrolledplantsandallthe conditionsthattheymayconfrontintermsofmodelsandassumptionscanbe categorisedintothefollowingclasses:
Class1:Inthisclass,itispossibletoderiveaccuratemathematicalmodelsfrom thefirstprinciplesortheidentification-basedschemes,anditcanbeanticipated thatthetheoreticallyindispensableplantassumptionshold.Thisclassincludes simpleplantsandcertainwell-modelledcomplicatedsystems.
Class2:Inthisclass,forsomeevensimpleplants,manycomplicatedsystems, andafewcomplexsystems,modelsderivedfromthefirstprinciplesorthe identification-basedschemesarecrudelyaccurate,butuncertaintiescanbe usedtocompensateforthemodellingerrorwithknownbounds,anditcanbe anticipatedthatthetheoreticallyindispensableplantassumptionshold.
Class3:Inthisclass,conditionsaresimilartothoseofclass2,butwiththe differencethatthetheoreticallyindispensableplantassumptionsmaynotbe guaranteedtohold.
Class4:Inthisclass,forsomecomplicatedsystemsandmostcomplexsystems, modelsderivedfromthefirstprinciplesortheidentification-basedschemes modelsarecrudelyaccurate,andtheuncertaintiesusedtodescribethemodellingerrorsaredifficulttoobtainaccurately,anditcanbeanticipatedthatthe theoreticallyindispensableplantassumptionsmaynothold.
Class5:Inthisclass,forafewcomplicated,somecomplex,andcomplex adaptivesystems,derivationofmodelsfromthefirstprinciplesorthe identification-basedschemes,andreliableuncertaintydescriptionsaredifficult orpracticallyunavailable,anditcanbeanticipatedthatthetheoretically indispensableplantassumptionsdonothold.
Theplantsfallingintotheclass1categoryhavebeensuccessfullycontrolledby thewell-establishedandwell-documentedmodel-basedcontrolstrategiesfrom theclassicalandstate-spaceschoolsofthought.Fortheplantsfallingintotheclass 2category,bothadaptiveandrobustcontrolschoolsarewelldevelopedandhave beensuccessfullyimplementedinpractice.Althoughtherearestillmanyopen problemsintheadaptiveandrobustcontrolapproachestoreliablycontrolallsuch plants,solutionsareconceivableinthefuturewiththepresenttheoreticaltoolsor someextensionsandmodifications.Adaptiveandrobustcontrolmethodologies mustbeselectedwithmuchhesitationforthecontrolofthereal-worldplants fallingintotheclass3category.Insuchcases,adata-drivenapproachwould betherecommendedchoice.Fortheplantsfallingintotheclass4category,the data-drivenapproachisthestronglyrecommendedchoice.Althoughsomeof thepresentadaptiveandrobustcontroltechniquesmaybeemployedinafew class4categoryplants,theirapplicationisdifficult,time-consumingandwith noguaranteedsafetyandreliability.Inthecaseofplantsfallingintotheclass 5category,data-drivencontrolisthesolechoice.Manyofthefuturereal-world plantsaregoinginthisdirection[13],andthecontrolscientist’scommunity mustbeequippedwithawell-establishedandstrongsufficienttheoretical backgroundofdata-drivencontroltheorytohandlethesecontrolproblems.The finalpointtonoteisthatpracticalcontrollersshouldnotbetoocomplex,difficult ornon-economicaltouse.
Tosummarise,themaincharacteristicsofdata-drivencontrolsystemsthatmake themappealingtoselectionbyadesigneraregivenasfollows:
● Inthedata-drivencontrolapproaches,thedesignmethodologiesdonotexplicitlyincludeanypartsorthewholeoftheplantmodelorarenotrestrictedby theassumptionsfollowingthetraditionalmodellingprocesses.Hence,theyare basically model-free designs.
● Thestabilityandconvergencederivationsofthedata-drivenapproachdonot dependonthemodelanduncertaintymodellingaccuracy.
● Inthedata-drivencontrolframework,theinherentlybornconceptsof un-modelleddynamicsandrobustnessinthemodel-basedcontrolmethodologiesarenon-applicable.
1.2.2TechnicalRemarksontheData-DrivenControlMethodologies
Thefollowingremarksareimportanttoclarifycertainambiguitiesandconcepts inthecurrentdata-drivencontrolliterature:
Remark1:Inthecontrolliterature,thecontroldesigntechniquesthat implicitly utilisetheplantmodel,suchasthedirectadaptivecontrolandthe subspace-identification-basedpredictivecontrolmethods,aresometimes categorisedasdata-drivencontrol.However,theircontrollerdesign,stability andconvergenceanalysisarefundamentallymodel-basedandalsorequire fulfillingstrongassumptionsondifferentmodelcharacteristicssuchasthe modelorder,relativedegree,timedelay,noise,uncertaintycharacteristicsand bounds.Hence,thisbookcategorisessuchtechniquesasmodel-basedrather thandata-driven.
Remark2:Indealingwithmathematicalmodels,issuessuchasnonlinearity, time-varyingparametersandtime-varyingmodelstructurescauseserious limitationsandrequirecomplextheoreticalhandling.However,suchissuesat theinput–outputdatalevelarenon-existent.Infact,atrulydata-drivencontrol approachshouldbeabletodealwiththeabovecontrolproblems.
Remark3:Theconceptsofrobustnessandpersistencyofexcitationthatappearin adaptiveandrobustcontrolmethodologiesaregeneralnotionsthatmustalsobe dealtwithinthedata-drivencontrolapproach.However,newdefinitionsand frameworksarenecessarytopursuetheseconceptsinthedata-drivencontrol context.
Remark4:Thetheory–practicegapinthemodel-basedapproachesisgreatly alleviatedinthedata-drivenapproachastheimplementationsaredirectly field-based.
Remark5:Averyrichliteratureonthemathematicalsystemtheoryand immenselyvaluableexperiencesintheimplementationofmodel-basedcontrol techniquesisavailable.Itwouldnotbedesirableorwisetoignoresuchvaluable information.Thefactisthatplantmodelscanplayavitalroleinthedesign ofcontrolsystems.Oneaspectistheapplicationofmodel-basedcontroller designtechniques,ifpossible.Theotheraspectwouldbethecooperationof data-drivencontrolwithothercontroltheoriesandmethods.Therelationship betweendata-drivenandmodel-basedcontrolshouldbecomplementary, anddata-drivenapproachescanlearnandbenefitfromtheestablished model-basedconcepts.Theeffectiveemploymentofexistingaccurateinformationabouttheplantbythedata-drivenapproachisanopenproblemforfurther research.
Remark6:Data-drivencontrolispredictedtobethedominantparadigmofcontroldesignscience,complementingandsubstitutingthepresentmodel-based paradigm.
1.3Data-DrivenControlSchemes
Inthissection,sixdifferentdata-drivencontrolschemesarebrieflyintroduced.A classificationandabriefsurveyontheavailabledata-drivenapproachesaregiven inRef.[11].
1.3.1UnfalsifiedAdaptiveControl
UnfalsifiedcontrolwasproposedbySafonovin1995[14].TheunderlyingphilosophyofunfalsifiedcontrolisPopper’sfalsificationtheoryproposedforthedemarcationprobleminthephilosophyofscience(seeChapter2).
Unfalsifiedcontrolisadata-drivencontroltheorythatutilisesphysicaldatato learnorselecttheappropriatecontrollerviaafalsificationoreliminationprocess. Intheunfalsifiedfeedbackcontrolconfiguration,thegoalistodetermineacontrol law C forplant P suchthattheclosed-loopsystem T satisfiesthedesiredspecifications,wheretheplantiseitherunknownoronlypartiallyknown,andthe input–outputdataareutilisedinselectingthecontrollaw C.Intheunfalsified control,thecontrolsystem learns whennewinput–outputinformationenablesit toeliminatethecandidatecontrollersfromthecontrolbank.Thethreeelements thatformtheunfalsifiedcontrolproblemareasfollows:
● Plantinput–outputdata.
● Thebankofcandidatecontrollers.
● Desiredclosed-loopperformancespecificationdenotedby T spec consistingofthe 3-tuplesofthereferenceinput,outputandinputsignals(r , y, u).
Definition1.2[15] Acontroller C issaidtobe falsified bymeasurement informationifthisinformationissufficienttodeducethattheperformance specification(r , y, u) ∈ T spec ∀ r ∈ ℝ wouldbeviolatedifthatcontrollerwerein thefeedbackloop.Otherwise,thecontrollaw C issaidtobe unfalsified
Figure1.3showsthegeneralstructureoftheclosed-loopunfalsifiedcontrolsystem.Theinputstothefalsificationlogicandalgorithmaretheplantinput–output data,thesetofcandidatecontrollersinacontrolbankorset,andthedesired closed-loopperformance.Thecontrollersareverifiedusingafalsificationlogicand algorithmwithperformancegoalsandphysicaldataasitsinputs.Notethatno
