Functional Observers for Dynamical Systems

225 Pages·2012·3.225 MB·English

by  Hieu Trinh, Tyrone Fernando (auth.)

Lecture Notes
in Control and Information Sciences 420
Editors:M.Thoma,F.Allgöwer,M.Morari
Hieu Trinh and Tyrone Fernando
Functional Observers for
Dynamical Systems
ABC
SeriesAdvisoryBoard
P.Fleming,P.Kokotovic,
A.B.Kurzhanski,H.Kwakernaak,
A.Rantzer,J.N.Tsitsiklis
Authors
Dr.HieuTrinh Dr.TyroneFernando
DeakinUniversity UniversityofWesternAustralia
SchoolofEngineering SchoolofElectrical,Electronicand
GeelongVIC3217 ComputerEngineering
Australia 35StirlingHighway
E-mail:[email protected] Crawley,WA6009
Australia
E-mail:[email protected]
ISBN978-3-642-24063-8 e-ISBN978-3-642-24064-5
DOI10.1007/978-3-642-24064-5
LectureNotesinControlandInformationSciences ISSN0170-8643
LibraryofCongressControlNumber:2011938116
(cid:2)c 2012SpringerBerlinHeidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthemate-
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broadcasting, reproduction onmicrofilmor inanyother way, andstorage indatabanks. Dupli-
cationofthispublicationorpartsthereof ispermittedonlyunder theprovisions oftheGerman
CopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalways
beobtainedfromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes
notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Typeset&CoverDesign:ScientificPublishingServicesPvt.Ltd.,Chennai,India.
Printedonacid-freepaper
987654321
springer.com
HieuTrinhdedicates thebook tohiswife NgocAnh and daughters
BaoHan andBao Anfortheirloveand support.
TyroneFernando dedicates thebooktohiswifeYasminand
daughtersMelissaandRochellefor theirloveandsupport.
Preface
In many practical situations, the physical state of a system cannot be determined
by direct observation. The problem of constructing the state vector and functions
ofthestatevectorisoneoftheimportanttopicsinControlEngineering.Sincethe
re-emergenceofthestate-spacemethodstoformadirectmultivariableapproachto
linearcontrolsystemssynthesisanddesign,ahostofcontrollersnowexisttomeet
thequalitativeandquantitativecriteriaincludingsystemstabilityandoptimality,and
oftenthecontrolsignalsarelinearfunctionsofthesystemstates.LinearFunctional
Observersestimatelinearfunctionsofthestatevectorofasystemwithoutneeding
toestimatealltheindividualstatesandsogreatlyreducetheorderandcomplexity
ofthedesignedobservers.Therehasbeenaconsiderableamountofresearchcarried
outonthesubjectoffunctionalstateestimationforwelloverfourdecadeseversince
D.G. Luenbergerintroduced the conceptin 1966.Due to their potential to reduce
thecomplexityandcostforcontrolsystems,therehasbeenaconsiderableinterest
inthetopic.
SincetheworkofLuenberger,theworkoncharacterizingsolutionstotheprob-
lemofdesigningfunctionalobservershasbeensteadilyprogressing.Thisbookat-
temptsto presentthe fundamentalconceptsrelevantto linear functionalobservers
developed over the years and complement the theory of state observers. The the-
ory of testing the existence and designing of linear functional observers is chal-
lenging when the system is subject to time-varying delays, unknown inputs and
nonlinearities.Thisbookhasdedicatedchaptersdetailingthedesignoflinearfunc-
tional observers when the system is subject to unknown inputs, time delays and
nonlinearities.
The theory of linear functional observers, which is the subject of this book, is
increasinglybecomingapopularresearchedtopicbecauseofthemanyadvantages
itpresentsinstateobservationandcontrolsystemdesign.Thisbookpresentsrecent
informationonthecurrentstateoftheartresearchinthisfield.Thisbookwillserve
as a usefulreference to researchersin this area of research to understandthe fun-
damentalconceptsrelevanttothetheoryoffunctionalobserversandtogathermost
recentadvancementsinthefield.Thisbookisusefultoacademicsandpostgraduate
students researching into the theory of linear functionalobservers. This book can
VIII Preface
also be useful for specialized final year undergraduatecourses in control systems
engineeringandappliedmathematicswitharesearchfocus.
TheauthorswishtothankFrederickCheeforproofreadingthisbookandPhan
ThanhNamforusefuldiscussionsonLinearMatrixInequalitieswhichformedthe
basisforstabilityanalysisoffunctionalobserversfortime-delaysystems.Bothau-
thorsalso wish to thanktheir families forthe loveandencouragementthroughout
thepreparationofthisbook.
Australia HieuTrinh
July2011 TyroneFernando
Contents
1 Introduction................................................... 1
2 StateObserversforDifferentTypesofSystems .................... 5
2.1 Introduction ............................................... 5
2.2 Full-OrderStateObserversforLinearSystems .................. 8
2.2.1 DesignAlgorithm.................................... 12
2.2.2 ANumericalExample ................................ 12
2.3 Reduced-OrderStateObserversforLinearSystems .............. 13
2.3.1 DesignAlgorithm.................................... 16
2.3.2 ANumericalExample ................................ 17
2.4 StateObserversforLinearSystemswithUnknownInputs......... 18
2.4.1 DesignofFull-OrderUnknownInputObservers .......... 19
2.4.2 DesignAlgorithm.................................... 21
2.4.3 ANumericalExample ................................ 22
2.4.4 DesignofReduced-OrderUnknownInputObservers ...... 22
2.4.5 DesignAlgorithm.................................... 27
2.4.6 ANumericalExample ................................ 27
2.5 Reduced-OrderStateObserversforTime-DelaySystems ......... 29
2.5.1 DesignAlgorithm.................................... 32
2.5.2 ANumericalExample ................................ 32
2.5.3 Reduced-OrderStateObserverswithoutInternalDelay .... 33
2.5.4 DesignAlgorithm.................................... 34
2.5.5 ANumericalExample ................................ 35
2.6 StateObserversforLipschitzNonlinearSystems ................ 36
2.6.1 DesignAlgorithm.................................... 40
2.6.2 ANumericalExample ................................ 40
2.7 Reduced-OrderStateObserversforDescriptorSystems........... 40
2.7.1 DesignAlgorithm.................................... 44
2.7.2 ANumericalExample ................................ 44
2.8 UnknownInputObservers ................................... 45
2.9 Conclusion................................................ 47
X Contents
3 LinearFunctionalObservers .................................... 49
3.1 Introduction ............................................... 49
3.2 FunctionalStateReconstructionProblem....................... 50
3.3 ExistenceConditions ....................................... 51
3.4 ScalarFunctionalObserversofOrder(v−1) ................... 55
3.4.1 DesignAlgorithm1:ScalarFunctionwith
Orderq=(v−1) .................................... 56
3.4.2 ANumericalExample ................................ 57
3.5 Multi-functionalObservers .................................. 58
3.5.1 DesignAlgorithm2:MultipleFunctionswith
Orderq=r ......................................... 62
3.5.2 ANumericalExample ................................ 62
3.6 AnUpperBoundfortheOrderofMulti-functionalObservers ..... 65
3.6.1 DesignAlgorithm3:MultipleFunctionswith
Orderq=q ........................................ 68
u
3.6.2 ANumericalExample ................................ 69
3.7 FurtherRemarks ........................................... 70
3.8 Conclusion................................................ 72
4 AParametricApproachtotheDesignofReduced-OrderLinear
FunctionalObservers........................................... 73
4.1 Introduction ............................................... 73
4.2 Preliminaries .............................................. 74
4.2.1 GeneralizedSylvesterMatrixEquation .................. 74
4.2.2 ObservableCanonicalForm ........................... 76
4.3 DesignofReduced-OrderScalarFunctionalObservers ........... 79
4.3.1 DesignAlgorithm1:ScalarFunctionalObserversofOrder
q<(v−1).......................................... 81
4.3.2 NumericalExamples ................................. 83
4.4 ExtensiontoLinearSystemswithUnknownInputs .............. 88
4.4.1 First-OrderFunctionalObservers....................... 90
4.4.2 DesignAlgorithm2:First-OrderObservers .............. 92
4.4.3 ANumericalExample ................................ 93
4.4.4 SecondandHigherOrderFunctionalObservers........... 94
4.4.5 ANumericalExample ................................ 98
4.5 Conclusion................................................ 99
5 Functional Unknown Input Observers: Further Results and
Applications...................................................101
5.1 Introduction ...............................................101
5.2 Delay-FreeFunctionalObserversforTime-DelaySystems ........102
5.3 Observer-Based Control of Multi-Agent Systems under
DecentralizedInformationStructure...........................104
5.4 MultipleFunctionalUnknownInputObservers..................107
5.4.1 ANumericalExample ................................115
Contents XI
5.5 ACommonDisturbanceDecoupledObserverforLinearSystems
withUnknownInputs .......................................116
5.5.1 ADecisionLogicSwitch..............................118
5.5.2 ConstructionofΩ ...................................119
1
5.5.3 ConstructionofΩ ...................................120
u
5.5.4 ExistenceConditionsfora
CommonFunctionalObserver .........................121
5.5.5 ANumericalExample ................................122
5.5.6 SimulationResults ...................................123
5.6 Conclusion................................................126
6 FunctionalStateandUnknownInput ObserversforNonlinear
Systems .......................................................127
6.1 Introduction ...............................................127
6.2 PartialStateEstimationforNonlinearSystems ..................128
6.2.1 DecompositionoftheNonlinearFunction................128
6.2.2 ConvergenceofObserver .............................130
6.2.3 SolvabilityoftheDesignMatrices......................131
6.2.4 ANumericalExample ................................133
6.3 PartialStateEstimationforNonlinearSystemswith
UnknownInputs ...........................................134
6.3.1 ExistenceConditionsandObserverDesignProcedure .....135
6.3.2 ANumericalExample ................................141
6.4 StateandInputSimultaneousEstimationforNonlinearSystems ...142
6.4.1 ObserverDesignProcedure............................145
6.4.2 DesignAlgorithm....................................149
6.4.3 ANumericalExample ................................149
6.5 Conclusion................................................152
7 SimultaneousStateandInputEstimation .........................153
7.1 Introduction ...............................................153
7.2 SimultaneousEstimationofStatesandUnknownInputs ..........153
7.2.1 ExistenceConditionsandanObserverDesignProcedure ...154
7.2.2 DesignAlgorithm:rOrderLinearFunctionalObserver ....158
7.2.3 SpecialCase1.......................................159
7.2.4 DesignAlgorithm:(n+l−p)OrderObserver ...........160
7.2.5 SpecialCase2.......................................161
7.2.6 ANumericalExample ................................163
7.3 SimultaneousStateandUnknownInputEstimationforUncertain
Time-DelaySystems........................................164
7.3.1 Reduced-OrderObserverDesign .......................165
7.3.2 DesignAlgorithm....................................171
7.3.3 ANumericalExample ................................172
XII Contents
7.4 SimultaneousFunctionalStateandUnknownInputEstimation ....173
7.4.1 ExistenceConditionsandaDesignProcedure ............173
7.4.2 StabilityConditionofN+N ..........................178
d
7.4.3 FunctionalObserverswithoutInternalDelays ............180
7.4.4 Delay-FreeFunctionalObservers.......................182
7.5 Conclusion................................................184
8 FurtherNotesonFunctionalObservers...........................185
8.1 Introduction ...............................................185
8.2 ARevisittoLinearFunctionalObserversfor
Time-DelaySystems........................................185
8.2.1 DesignAlgorithm....................................192
8.2.2 ANumericalExample ................................192
8.3 DelayedOutputInformation .................................193
8.3.1 DesignAlgorithm....................................197
8.3.2 ANumericalExample ................................198
8.3.3 FurtherNotesandOpenProblems ......................198
8.4 InterconnectedSystems .....................................200
8.5 Conclusion................................................204
References.........................................................205
Index .............................................................213