EECE 574 - Adaptive Control - Other Approaches to Adaptive Control

64 Pages·2013·1.91 MB·English

by  Guy Dumont

EECE 574 - Adaptive Control
OtherApproachestoAdaptiveControl
GuyDumont
DepartmentofElectricalandComputerEngineering
UniversityofBritishColumbia
January2013
GuyDumont(UBCEECE) EECE574-OtherApproachestoAdaptiveControl January2013 1/64
Contents
1 Introduction
2 Youla-KuceraParametrization
Open-LoopInversionRevisited
AffineParametrization: TheStableCase
AllStabilizingControllers
3 TheWindsurfer’sApproach
4 MultiModelAdaptiveControl
5 L1 AdaptiveControl
6 ChallengesinAdaptiveControl
7 L1 AdaptiveControlTheory
8 PerformanceMonitoring
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Introduction
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Youla-KuceraParametrization
Outline
Novelwayofexpressingcontrollertransferfunction
Providesnewinsightintocontroldesign
Thekeyfeatureofthenewparameterizationisthatitrenderstheclosed
loopsensitivityfunctionslinearor(morecorrectly,affine)inadesign
variable
Wecallittheaffineparameterization
Theso-calledYoula-Kuceraparametrizationnowplaysacentralrolein
control,identificationandadaptivecontrol
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Youla-KuceraParametrization Open-LoopInversionRevisited
Open-Loop Inversion Revisited
Recallthatcontrolimplicitlyandexplicitlydependsonplantmodel
inversion. Thisisbestseeninthecaseofopenloopcontrol.
Inopenloopcontroltheinput,U(s),isgeneratedfromthereference
signalR(s),byatransferfunctionQ(s),i.e.U(s)=Q(s)R(s).
Thisleadstoaninput-outputtransferfunctionofthefollowingform:
T (s)=P(s)Q(s)
0
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Youla-KuceraParametrization Open-LoopInversionRevisited
Open-Loop Inversion Revisited
Thissimpleformulahighlightsthefundamentalimportanceofinversion,
asT (jω)willbe1onlyatthosefrequencieswhereQ(jω)invertsthe
0
model. Notethatthisisconsistentwiththeprototypesolutiontothe
controlproblemdescribedearlier.
AkeypointisthatT (s)=P(s)Q(s)isaffineinQ(s).
0
Ontheotherhand,withaconventionalfeedbackcontroller,C(s),the
closedlooptransferfunctionhastheform
P(s)C(s)
T (s)=
0
1+P(s)C(s)
TheaboveexpressionisnonlinearinC(s).
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Youla-KuceraParametrization Open-LoopInversionRevisited
Comparingthetwopreviousequations,weseethattheformeraffine
relationshipholdsifwesimplyparameterizeC(s)inthefollowing
fashion:
C(s)
Q(s)=
1+C(s)P(s)
WecanthendesignintermsofQ(s)andthenobtainC(s)fromQ(s)and
P(s)
Thisistheessenceoftheideapresentedhere.
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Youla-KuceraParametrization AllStabilizingControllers
Affine Parameterization. The Stable Case
WecaninverttherelationshipgivenonthepreviousslidetoexpressC(s)
intermsofQ(s)andP(s):
Q(s)
C(s)=
1−Q(s)P(s)
WewillthenworkwithQ(s)asthedesignvariableratherthanthe
originalC(s).
NotethattherelationshipbetweenC(s)andQ(s)isone-to-oneandthus
thereisnolossofgeneralityinworkingwithQ(s).
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Youla-KuceraParametrization AllStabilizingControllers
Stability
Actuallyaveryhardquestionisthefollowing:
GivenastabletransferfunctionP(s),describeallcontrollers,C(s)that
stabilizethisnominalplant.
However,itturnsoutthat,intheQ(s)form,thisquestionhasavery
simpleanswer,namelyallthatisrequiredisthatQ(s)bestable.
Thisresultisformalizedinthelemmastatedonthenextslide.
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Youla-KuceraParametrization AllStabilizingControllers
All Stabilizing Controllers
Lemma(Lemma15.1: Affineparameterizationforstablesystems)
ConsideraplanthavingastablenominalmodelG (s)controlledinaone
0
d.o.f.feedbackarchitecturewithapropercontroller.
Thenthenominalloopisinternallystableif,andonlyif,Q(s)isanystable
propertransferfunctionwhenthecontrollertransferfunctionC(s)is
parameterizedas
Q(s)
C(s)=
1−Q(s)P(s)
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