Table Of Content(cid:2)
ACTIVE DISTURBANCE
REJECTION CONTROL
FOR NONLINEAR
SYSTEMS
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ACTIVE DISTURBANCE
REJECTION CONTROL
FOR NONLINEAR
SYSTEMS
AN INTRODUCTION
Bao-Zhu Guo
AcademyofMathematicsandSystemsScience,AcademiaSinica,People’sRepublic
ofChinaandUniversityoftheWitwatersrand,SouthAfrica
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Zhi-Liang Zhao
ShaanxiNormalUniversity,Xi’an,People’sRepublicofChina
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LibraryofCongressCataloging-in-PublicationData
Names:Guo,Bao-Zhu,1962-author.|Zhao,Zhi-Liang,author.
Title:Activedisturbancerejectioncontrolfornonlinearsystems:an
introduction/Bao-ZhuGuoandZhi-LiangZhao.
Description:Singapore;Hoboken,NJ:JohnWiley&Sons,2016.|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2016018239(print)|LCCN2016025091(ebook)|ISBN
9781119239925(cloth)|ISBN9781119239956(pdf)|ISBN9781119239949
(epub)
Subjects:LCSH:Damping(Mechanics)|Automaticcontrol.|Nonlinearsystems.
Classification:LCCTA355.G792016(print)|LCCTA355(ebook)|DDC
620.3/7–dc23
LCrecordavailableathttps://lccn.loc.gov/2016018239
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Setin10/12pt,TimesLTStdbySPiGlobal,Chennai,India.
1 2016
(cid:2)
(cid:2)
Contents
Preface ix
1 Introduction 1
1.1 ProblemStatement 1
1.2 OverviewofEngineeringApplications 6
1.3 Preliminaries 9
1.3.1 CanonicalFormofADRC 9
1.3.2 StabilityforNonlinearSystems 18
1.3.3 StabilityofLinearSystems 24
1.3.4 Finite-TimeStabilityofContinuousSystem 27
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1.3.5 StabilityofDiscontinuousSystems 32
1.3.6 ProofofTheorem1.3.11 34
1.4 RemarksandBibliographicalNotes 50
2 TheTrackingDifferentiator(TD) 53
2.1 LinearTrackingDifferentiator 55
2.2 NonlinearTrackingDifferentiator 59
2.2.1 Second-OrderNonlinearTrackingDifferentiator 60
2.2.2 High-OrderNonlinearTrackingDifferentiator 64
2.3 Finite-TimeStableSystem-BasedTrackingDifferentiator 69
2.3.1 ConvergenceofFinite-TimeStableSystem-BasedTD 70
2.3.2 ASecond-OrderFinite-TimeStableTrackingDifferentiator 75
2.4 IllustrativeExamplesandApplications 77
2.4.1 ComparisonofThreeDifferentiators 77
2.4.2 ApplicationstoFrequencyOnlineEstimation 81
2.4.3 ApplicationtotheBoundaryStabilizationofWaveEquation 85
2.5 SummaryandOpenProblems 88
2.6 RemarksandBibliographicalNotes 89
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vi Contents
3 ExtendedStateObserver 93
3.1 LinearExtendedStateObserverforSISOSystems 94
3.2 NonlinearExtendedStateObserverforSISOSystems 100
3.2.1 NonlinearESOforSISOSystems 101
3.2.2 SomeSpecialESO 109
3.3 TheESOforSISOSystemswithTime-VaryingGain 119
3.4 TheESOforMIMOSystemswithUncertainty 133
3.4.1 ESOforSystemswithTotalDisturbance 133
3.4.2 ESOforSystemswithExternalDisturbanceOnly 141
3.4.3 ExamplesandNumericalSimulations 144
3.5 SummaryandOpenProblems 150
3.6 RemarksandBibliographicalNotes 153
4 TheActiveDisturbanceRejectionControl 155
4.1 LinearADRCforSISOSystems 157
4.1.1 GlobalConvergenceofLinearADRCforSISOSystems 157
4.1.2 GlobalConvergenceforSystemswithExternalDisturbanceOnly 167
4.1.3 Semi-GlobalConvergenceofLADRC 176
4.1.4 NumericalSimulations 183
4.2 NonlinearADRCforSISOSystems 187
4.2.1 GlobalADRCforSISOSystemswithTotalDisturbance 187
4.2.2 GlobalADRCforSISOSystemwithExternalDisturbanceOnly 195
4.2.3 Semi-globalADRCforSISOSystemwithVastUncertainty 203
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4.2.4 ExamplesandNumericalSimulations 204
4.3 ADRCwithTime-VaryingTuningParameter 206
4.4 NonlinearADRCforMIMOSystemswithVastUncertainty 228
4.4.1 Semi-GlobalADRCforMIMOSystemswithUncertainty 231
4.4.2 GlobalADRCforMIMOSystemswithUncertainty 238
4.4.3 GlobalADRCforMIMOSystemswithExternalDisturbanceOnly 245
4.4.4 NumericalSimulations 250
4.5 IMPVersusADRC 252
4.6 HGCandSMCVersusADRC 263
4.7 ApplicationstoPMSMs 267
4.8 ApplicationtoWaveEquationwithUncertainty 270
4.8.1 ControlDesign 271
4.8.2 ProofofTheorem4.8.1 279
4.9 SummaryandOpenProblems 287
4.10 RemarksandBibliographicalNotes 289
5 ADRCforLowerTriangularNonlinearSystems 291
5.1 ESOforLowerTriangularSystems 291
5.1.1 ConstantHigh-GainESO 292
5.1.2 Time-VaryingGainESO 301
5.1.3 NumericalSimulation 306
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Contents vii
5.2 StabilizationofLowerTriangularSystemsbyADRC 312
5.2.1 ADRCwithConstantGainESO 313
5.2.2 ADRCwithTime-VaryingGainESO 327
5.3 NumericalSimulations 331
5.4 SummaryandOpenProblems 336
5.5 RemarksandBibliographicalNotes 338
References 341
Index 349
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Preface
The modern control theory came to mathematics via N. Wiener’s book Cybernetics or the
Science of Control and Communication in the Animal and the Machine, published in 1948.
In1954,H.S.TsienpublishedthebookEngineeringCybernetics,whichbroughtthecontrol
theorytoengineering.
Roughlyspeaking,themoderncontroltheoryhasbeendevelopedthroughthreestages.The
firststageistheclassicalcontrolorautomaticprincipleofcompensationdevelopedfromthe
1940stothe1960s.Duringthisperiodthesingle-inputandsingle-outputlineartime-invariant
systemswerestudiedbythefrequencydomainapproach.Thesecondstagewasfromthe1960s
tothe1980s,duringwhichtimethemulti-inputandmulti-outputsystemswerestudiedbythe
statespaceortimedomainapproach.Thestate-spaceapproachreliesheavilyuponthemath-
(cid:2) ematical models of the systems. After the 1980s, many control theories were developed to (cid:2)
copewithuncertaintiesinsystems.Severalpowerfulmethodsweredeveloped,includingthe
internalmodelprincipleforoutputregulation(whichstartedinthe1970s),aswellasadaptive
control, roust control, high-gain feedback control, and sliding mode control (which started
even earlier). In particular, the robust control theory was well-established by both the fre-
quencydomainapproachandthetimedomainapproach.Acommonfeatureforthesemethods
wastheworstcasescenarioregardingthedisturbance.Adifferentwayofdealingwithuncer-
taintymaybefoundinadaptivecontrolwheretheunknown parametersareestimatedunder
the“excitingpersistentcondition”andinoutputregulationwhereaspecialclassofexternal
disturbance is estimated through the observer and internal model and is compensated for in
thefeedback-loop.
Duringthelate1980sand1990s,JingqingHanoftheChineseAcademyofSciencespro-
posedapowerfulunconventionalcontrolapproachtodealwithvastuncertaintyinnonlinear
systems. This new control technology was later called the active disturbance rejection con-
trol(ADRC).TheuncertaintiesdealtwithbytheADRCcanbeverycomplicated.Theycould
includethecouplingoftheexternaldisturbances,thesystemunmodeleddynamics,thezero
dynamicswithunknownmodel,andthesuperaddedunknownpartofcontrolinput.Thekey
ideaoftheADRCconsidersthe“totaldisturbance”asasignaloftime,whichcanbeestimated
bytheoutputofthesystem.Basically,theADRCconsistsofthreemainparts.Thefirstpartis
thetrackingdifferentiator(TD)thatisrelativelyindependentandisactuallythoroughlydis-
cussedinthecontroltheory.TheaimoftheTDistoextractthederivatives ofthereference
signal and is also considered as transient profile for output tracking. The second part of the
ADRCistheextendedstateobserver(ESO)whichisacrucialpartoftheADRC.InESO,both
thestateandthe“totaldisturbance”areestimatedbytheoutputofthesystem.Thisremarkable
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x Preface
feature makes the ADRC a very different way of dealing with uncertainty. The ESO is the
generalizationofthetraditionalstateobserverwhereonlythestateofthesystemisestimated.
ThefinalpartoftheADRCistheextendedstateobserver-basedfeedbackcontrol.Sincethe
uncertaintyisestimatedintheESOandiscompensatedforinthefeedbackloop,thebarriers
betweenthetimeinvariantandtimevarying,linearandnonlinearhavebeenbrokendownby
consideringthetime-varyingpartandthenonlinearpartasuncertainty.Atthesametime,the
controlenergyissignificantlyreduced.Moreimportantly,inthisway,theclosed-loopsystems
looklikelineartime-invariantsystems,forwhichareliableresultcanbeapplied.
In the past two decades, the ADRC has been successfully applied to many engineering
controlproblemssuchashysteresiscompensation,highpointingaccuracyandrotationspeed,
noncircularmachining,faultdiagnosis,high-performancemotioncontrol,chemicalprocesses,
vibrationalMEMSgyroscopes,tensionandvelocityregulationsinWebprocessinglinesand
DC–DC power converters by many researchers in different contexts. In all applications of
processcontrolandmotioncontrol,comparedwiththehugeamountofliteratureoncontrol
theory in dealing with the uncertainty such as system unmodeled dynamics, external distur-
bancerejection,andunknownparameters,theADRChasshownitsremarkablePIDnatureof
analmostindependentmathematicalmodel,nomatterthehighaccuracycontrolofmicrometre
gradeortheintegratedcontrolonaverylargescale.
Ontheotherhand,althoughmanysuccessfulengineeringapplicationshavebeendeveloped,
thetheoreticalresearchontheseapplicationslagsbehind.Thisbookservesasanintroduction
totheADRCfromatheoreticalperspectiveinaself-containedway.InChapter1,somebasic
backgroundisintroducedonnonlinearuncertainsystemsthatcanbedealtwithbytheADRC.
Chapter2presentsconvergenceofthedifferenttypesoftrackingdifferentiatorsproposedby
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Haninhisoriginalpapers.Chapter3isdevotedtoconvergenceoftheextendedstateobserver
forvariousnonlinearsystems.Chapters2and3canbeconsideredasindependentsectionsof
thebook.Chapter4looksatconvergenceoftheclosed-loopbasedontheTDandESO.This
can be considered as a separation principle of the ADRC for uncertain nonlinear systems.
Thenumericalsimulationsarepresentedfromheretowheretoillustratetheapplicabilityof
theADRC.Finally,inChapter5,theESOandstabilizationforlowertriangularsystemsare
discussed.
Most of the material in this book is from the authors’ published papers on this topic.
However,theideaforthebookcomesfromideaspublishedinHan’smanyoriginalnumerical
experiments, engineering applications that appeared publicly in Chinese, and the numerous
worksdonebytheADRCgroup,inparticularthegroupledbyDr.ZhiqiangGaoatCleveland
StateUniversityintheUnitedStates.
Theauthorsaredeeplyindebtedtothosewhohelpedwiththeworkspresentedinthisbook.
TheseincludeZhiqiangGao,YiHuang,andWenchaoXue.ThisbookisdedicatedtoBao-Zhu
Guo’smemoryofProfessorJingqingHanwhopassedawayin2008.
This book is partially supported by the National Natural Science Foundation of China
(No.61403242)andtheNatureScienceResearchProgramofShaanxiProvince-KeyProgram
(No.2016JZ023).
Bao-ZhuGuoandZhi-LiangZhao
October,2015
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1
Introduction
Inthischapter,weintroducesomenecessarybackgroundabouttheactivedisturbancerejection
control(ADRC).Somenotationandpreliminaryresultsarealsopresented.
1.1 ProblemStatement
Inmostcontrolindustries,itishardtoestablishaccuratemathematicalmodelstodescribethe
systems precisely. In addition, there are some terms that are not explicitly known in mathe-
maticalequations and,ontheotherhand,someunknown externaldisturbances existaround
thesystemenvironment.Theuncertainty,whichincludesinternaluncertaintyandexternaldis-
turbance,isubiquitousinpracticalcontrolsystems.Thisisperhapsthemainreasonwhythe
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proportional–integral–derivative (PID) control approach has dominated the control industry
foralmostacenturybecausePIDcontroldoesnotutilizeanymathematicalmodelforsystem
control. The birth and large-scale deployment of the PID control technology can be traced
back to the period of the 1920s–1940s in response to the demands of industrial automation
beforeWorldWarII.Itsdominanceisevidenteventodayacrossvarioussectorsoftheentire
industry.Ithasbeenreportedthat98%ofthecontrolloopsinthepulpandpaperindustriesare
controlled by single-input single-output PI controllers [18]. In process control applications,
morethan95%ofthecontrollersareofthePIDtype[9].
LetuslookatthestructureofPIDcontrolfirst.Foracontrolsystem,letthecontrolinput
be u(t) and let the output be y(t). The control objective is to make the output y(t) track a
reference signal v(t). Let e(t)=y(t)−v(t) be the tracking error. Then PID control law is
representedasfollows:
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t
u(t)=k e(t)+k e(τ)dτ +k e˙(t), (1.1.1)
0 1 2
0
where k ,k , and k are tuning parameters. The PID control is a typical error-based control
0 1 2
method,ratherthanamodel-basedmethod,whichisseenfromFigure1.1.1foritsadvantage
ActiveDisturbanceRejectionControlforNonlinearSystems:AnIntroduction,FirstEdition.
Bao-ZhuGuoandZhi-LiangZhao.
©2016JohnWiley&SonsSingaporePte.Ltd.Published2016byJohnWiley&Sons,Ltd.
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2 ActiveDisturbanceRejectionControlforNonlinearSystems:AnIntroduction
Figure1.1.1 PIDcontroltopology.
ofeasydesign.Thenatureofindependentmathematicalmodelandeasydesignperhapshave
explainedthepartialityofcontrolengineerstoPID.
However,itisundeniablethatPIDisincreasinglyoverwhelmedbythenewdemandsinthis
eraofmodernindustrieswhereanunendingefficiencyispursuedforsystemsworkinginmore
complicated environments. In these circumstances, a new control technology named active
disturbancerejectioncontrol(ADRC)wasproposedbyJingqingHaninthe1980sand1990s
todealwiththecontrolsystemswithvastuncertainty[58,59,60,62,63].AsindicatedinHan’s
seminalwork[58],theinitialmotivationfortheADRCistoimprovethecontrolcapabilityand
performance limited by PID control in two ways. One is by changing the linear PID (1.1.1)
tononlinearPIDandtheotheristomakeuseof“derivative”inPIDmoreefficientlybecause
itiscommonlyrecognizedthat,inPID,the“D”partcansignificantlyimprovethecapability
(cid:2) andtransientperformanceofthecontrolsystems.However,thederivativeoferrorisnoteasily (cid:2)
measured and the classical differentiation most often magnifies the noise, which makes the
PIDcontrolactuallyPIcontrolinapplications,thatis,in(1.1.1),k =0.
2
Inautomaticprincipleofcompensation,thedifferentialsignalforagivenreferencesignal
v(t)isapproximatedbyy(t)inthefollowingprocess:
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s 1 1
yˆ(s)= vˆ(s)= vˆ(s)− vˆ(s) , (1.1.2)
Ts+1 T Ts+1
whereLˆ(s)representstheLaplacetransformofL(t),T isaconstant,and 1 vˆ(s)represents
Ts+1
theinertialelementwithrespecttoT (seeFigure1.1.2).
Thetimedomainrealizationof(1.1.2)is
1
y(t)= (v(t)−v(t−T)). (1.1.3)
T
Figure1.1.2 Classicaldifferentiationtopology.
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