JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= 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 JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= Linear and Nonlinear Multivariable Feedback Control: A Classical Approach Oleg N. Gasparyan StateEngineeringUniversityofArmenia iii JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= Copyright(cid:2)C 2008 JohnWiley&SonsLtd,BaffinsLane,Chichester WestSussex,PO191UD,England National 01243779777 International (+44)1243779777 Email(forordersandcustomerserviceenquiries):[email protected] VisitourHomePageonwww.wileyeurope.comorwww.wiley.com AllRightsReserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinany formorbyanymeans,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptunderthe termsoftheCopyright,DesignsandPatentsAct1988orunderthetermsofalicenceissuedbytheCopyright LicensingAgencyLtd,90TottenhamCourtRoad,LondonW1T4LP,UK,withoutthepermissioninwritingofthe Publisher.RequeststothePublishershouldbeaddressedtothePermissionsDepartment,JohnWiley&SonsLtd, TheAtrium,SouthernGate,Chichester,WestSussexPO198SQ,England,[email protected],or faxedto(+44)1243770571. 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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 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN978-0-470-06104-6 Typesetin10/12ptTimesbyAptaraInc.,NewDelhi,India PrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,Wiltshire Thisbookisprintedonacid-freepaperresponsiblymanufacturedfromsustainableforestry inwhichatleasttwotreesareplantedforeachoneusedforpaperproduction. iv JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= Tomybelovedfamily:Lilit,Yulia,andNikolay v JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= 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 vii JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= 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 JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= 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 JWBK226-FM JWBK226-Gasparyan December18,2007 15:4 CharCount= 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