Linear and Nonlinear Multivariable Feedback Control: A Classical Approach

348 Pages·2008·6.83 MB·English

by  Oleg Gasparyan

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Linear and Nonlinear
Multivariable Feedback Control
LinearandNonlinearMultivariableFeedbackControl:AClassicalApproach OlegN.Gasparyan
(cid:2)C 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-06104-6
i
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Linear and Nonlinear
Multivariable Feedback Control:
A Classical Approach
Oleg N. Gasparyan
StateEngineeringUniversityofArmenia
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LibraryofCongressCataloging-in-PublicationData
Gasparyan,Oleg.
Linearandnonlinearmultivariablefeedbackcontrol:aclassicalapproach/OlegGasparyan.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-06104-6(cloth)
1.Controltheory. 2.Feedbackcontrolsystems. 3.Functionsofcomplexvariables. I.Title.
QA402.3.G372008
629.8(cid:3)36—dc22
2007044550
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AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN978-0-470-06104-6
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Tomybelovedfamily:Lilit,Yulia,andNikolay
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Contents
Preface xi
PartI LinearMultivariableControlSystems
1 CanonicalrepresentationsandstabilityanalysisoflinearMIMOsystems 3
1.1 Introduction 3
1.2 GenerallinearsquareMIMOsystems 3
1.2.1 TransfermatricesofgeneralMIMOsystems 3
1.2.2 MIMOsystemzerosandpoles 5
1.2.3 Spectralrepresentationoftransfermatrices:characteristic
transferfunctionsandcanonicalbasis 10
1.2.4 StabilityanalysisofgeneralMIMOsystems 19
1.2.5 Singularvaluedecompositionoftransfermatrices 31
1.3 UniformMIMOsystems 40
1.3.1 Characteristictransferfunctionsandcanonicalrepresentations
ofuniformMIMOsystems 41
1.3.2 StabilityanalysisofuniformMIMOsystems 43
1.4 NormalMIMOsystems 51
1.4.1 CanonicalrepresentationsofnormalMIMOsystems 51
1.4.2 CirculantMIMOsystems 53
1.4.3 AnticirculantMIMOsystems 62
1.4.4 Characteristictransferfunctionsofcomplexcirculantand
anticirculantsystems 70
1.5 Multivariablerootloci 74
1.5.1 RootlociofgeneralMIMOsystems 76
1.5.2 Rootlociofuniformsystems 89
1.5.3 Rootlociofcirculantandanticirculantsystems 93
2 PerformanceanddesignoflinearMIMOsystems 100
2.1 Introduction 100
2.2 Generalizedfrequencyresponsecharacteristicsandaccuracyoflinear
MIMOsystemsundersinusoidalinputs 101
2.2.1 FrequencycharacteristicsofgeneralMIMOsystems 101
2.2.2 FrequencycharacteristicsandoscillationindexofnormalMIMOsystems 117
2.2.3 FrequencycharacteristicsandoscillationindexofuniformMIMOsystems 121
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viii Contents
2.3 DynamicalaccuracyofMIMOsystemsunderslowlychanging
deterministicsignals 124
2.3.1 MatricesoferrorcoefficientsofgeneralMIMOsystems 124
2.3.2 Dynamicalaccuracyofcirculant,anticirculantanduniform
MIMOsystems 129
2.3.3 AccuracyofMIMOsystemswithrigidcross-connections 132
2.4 StatisticalaccuracyoflinearMIMOsystems 135
2.4.1 AccuracyofgeneralMIMOsystemsunderstationary
stochasticsignals 135
2.4.2 StatisticalaccuracyofnormalMIMOsystems 139
2.4.3 StatisticalaccuracyofuniformMIMOsystems 141
2.4.4 Formulaeformeansquareoutputsofcharacteristicsystems 145
2.5 DesignoflinearMIMOsystems 151
PartII NonlinearMultivariableControlSystems 171
3 Studyofone-frequencyself-oscillationinnonlinearharmonically
linearizedMIMOsystems 173
3.1 Introduction 173
3.2 Mathematicalfoundationsoftheharmoniclinearizationmethodfor
one-frequencyperiodicalprocessesinnonlinearMIMOsystems 181
3.3 One-frequencylimitcyclesingeneralMIMOsystems 184
3.3.1 Necessaryconditionsfortheexistenceandinvestigationofthe
limitcycleinharmonicallylinearizedMIMOsystems 184
3.3.2 StabilityofthelimitcycleinMIMOsystems 194
3.4 LimitcyclesinuniformMIMOsystems 199
3.4.1 Necessaryconditionsfortheexistenceandinvestigationoflimit
cyclesinuniformMIMOsystems 199
3.4.2 Analysisofthestabilityoflimitcyclesinuniformsystems 205
3.5 LimitcyclesincirculantandanticirculantMIMOsystems 214
3.5.1 Necessaryconditionsfortheexistenceandinvestigationoflimit
cyclesincirculantandanticirculantsystems 214
3.5.2 Limitcyclesinuniformcirculantandanticirculantsystems 229
4 Forcedoscillationandgeneralizedfrequencyresponsecharacteristics
ofnonlinearMIMOsystems 236
4.1 Introduction 236
4.2 NonlineargeneralMIMOsystems 244
4.2.1 One-frequencyforcedoscillationandcapturingingeneralMIMOsystems 244
4.2.2 Generalizedfrequencyresponsecharacteristicsandoscillation
indexofstablenonlinearMIMOsystems 251
4.2.3 Generalizedfrequencyresponsecharacteristicsoflimitcycling
MIMOsystems 260
4.3 NonlinearuniformMIMOsystems 265
4.3.1 One-frequencyforcedoscillationandcapturinginuniformsystems 265
4.3.2 Generalizedfrequencyresponsecharacteristicsofstable
nonlinearuniformsystems 268
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Contents ix
4.3.3 Generalizedfrequencyresponsecharacteristicsoflimitcycling
uniformsystems 271
4.4 Forcedoscillationsandfrequencyresponsecharacteristicsalongthe
canonicalbasisaxesofnonlinearcirculantandanticirculantsystems 274
4.5 DesignofnonlinearMIMOsystems 278
5 AbsolutestabilityofnonlinearMIMOsystems 284
5.1 Introduction 284
5.2 AbsolutestabilityofgeneralanduniformMIMOsystems 287
5.2.1 MultidimensionalPopov’scriterion 287
5.2.2 ApplicationoftheBodediagramsandNicholsplots 293
5.2.3 DegreeofstabilityofnonlinearMIMOsystems 296
5.3 AbsolutestabilityofnormalMIMOsystems 299
5.3.1 GeneralizedAizerman’shypothesis 301
5.4 Off-axiscircleandparaboliccriteriaoftheabsolutestabilityofMIMOsystems 304
5.4.1 Off-axiscirclecriterion 305
5.4.2 Logarithmicformoftheoff-axiscriterionofabsolutestability 309
5.4.3 Paraboliccriterionofabsolutestability 313
5.5 Multidimensionalcirclecriteriaofabsolutestability 314
5.5.1 GeneralandnormalMIMOsystems 316
5.5.2 Inverseformofthecirclecriterionforuniformsystems 319
5.6 Multidimensionalcirclecriteriaoftheabsolutestabilityofforcedmotions 321
Bibliography 327
Index 335
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Preface
Thistextbookprovidesaunifiedcontroltheoryoflinearandnonlinearmultivariablefeedback
systems,alsocalledmulti-inputmulti-output(MIMO)systems,asastraightforwardextension
of the classical control theory. The central idea of the book is to show how the classical
(frequency- and root-domain) engineering methods look in the multidimensional case, and
howapractisingengineerorresearchercanapplythemtotheanalysisanddesignoflinearand
nonlinearMIMOsystems.
Atpresent,thereisagreatnumberoffundamentaltextbooksonclassicalfeedbackcontrol
asappliedtosingle-inputsingle-output(SISO)systems,suchasthebooksbyDorfandBishop
(1992), K. Ogata (1970), Franklin, Powell and Emami-Naeini (1991), Atherton (1975) and
E.Popov(1973),thelasttwobeingdevotedtononlinearSISOsystems,andmanyothers.A
generalqualityofallthesebooksisaunitedconceptualapproachtointroducingtheclassical
control theory, as well as clearly indicated branches of that theory; in fact, a lecturer can
successfully use any of these books in teaching his course on related subjects. On the other
hand,therearemanyremarkabletextbooksandmonographsonmultivariablefeedbackcontrol,
butthesituationhereisnotsoplain.Historically,attheoutset,thedevelopmentofmultivariable
controltheorywasconductedindifferentwaysandmanners,varyingfrommassiveeffortsto
extend directly the basic classical methods and techniques, to no less massive attempts to
reformulateradicallyandeven‘abolish’theclassicalheritageofcontroltheory.Besides,the
initial stages of formation of multivariable control essentially coincided with the advent of
state-space methods and approaches, and with rapid development of optimal control theory,
equallydealingwithSISOandMIMOsystems.Atlast,ataroundthattime,theretherobust
control theory also applicable to both SISO and MIMO systems emerged. As a result, the
notion of ‘modern’ multivariable control is so manifold and embraces so many directions
andaspectsoffeedbackcontrolthatitisdifficulttolistthemallwithoutrunningtheriskof
missing something significant. Nevertheless, it is obvious that optimal, adaptive and robust
methods (and their variations) are predominant in the scientific and technical literature, and
advancesinthesemethodsconsiderablyexceedtheachievementsofthe‘classical’branchin
multivariablecontrol.Atthesametime,itshouldbeacknowledgedthatmodernMIMOcontrol
theoryjust‘jumpedover’manyimportantproblemsoftheclassicaltheoryandnowthereisan
evidentgapbetweenthetopicspresentedinmosttextbooksonSISOcontrolandthoseinmany
booksonmultivariablecontrol(SkogestadandPostlethwaite2005;Safonov1980;Maciejowski
1989,etc.).
Thegoalofthisbookistobridgethatgapandtoprovideaholisticmultivariablecontrol
theory as a direct and natural extension of the classical control theory, for both linear and
xi