RIASI6426476fTECHNICALREPORTdP647ONANEWCHARACTERIZATIONOFLINEAR - PDF document

1 RIASI_64-26476///".*fTECHNICALREPORT__,_
RIASI_64-26476///".*fTECHNICALREPORT__,_--dP64-7ONANEWCHARACTERIZATIONOFLINEARPASSIVESYSTEMSAPRIL1964ByR.E.Kalman1964016562 ONANEWCHARACTER17_TIONOFLINEARPASSIVESYSTEMSbyR.E.EalmanApril196_CENTEBFORCON_OLTHEC_YResearchl,mtituteforAdvancedStudies(RIAS)ADivisionoftheMartinCo.7212Bellon_AvenueB_ltlmore,M_ryl_nd212121964016562-002 andSystemTheory7November,"_uj.R.E._ellmanCenterforControlTheoryResearchInstituteforAdvancedStudies(RIAS)Baltimore12,Md.i.DynamicalE_uationsofanN-Port.ConsiieranN×NmatrixfunctionZ(-)ofthecomplexvari-ables.Assumethatitistheimpedancematrixofafinite,time-invarlaut,passive_(resistor3inductor,capacitor_idealtransformer,gyrator)N-port.Then3asiswellknowntZ(-)hasthefollowingproperties(1)EveryelementofZ(.)isaratiooftworelativelyprimepolynomialswithrealcoefficients.(ii)ForeveryswithRes•01theheraetlanmatrix(1.1)Z(s)+Z'(_)('=transpose,=complexconJu@_te)isnonne_tivedefinite.Ifgyratorsarenotallowed,i.e.,iftheN-portisreciprocal,thenwemusthavealso(lii)Z(')issymmetrica

2 l,i.e.,zij(.)-zji(.).Weshallalsoassume:(
l,i.e.,zij(.)-zji(.).Weshallalsoassume:(iv)Z(m)=O,i.e.,thedegreeofeverynumeratorpol_alcmialinZ(')islessthanthedegreeofthecorrespondingdenonin_tor.Requirement(i_)isnotreallyrestrictivebutitwillsimplifyconsiderablytheformulaswhicharetofollow.Thisresearchwass_pportedinpartbytheUSAirForceundercortractsAF33(697)-8599(AeronauticalS_mte_._Division)andAF49(638)-12(_(OfficeofScientificResearch))._[bxthe_ationalAeronauticalandSlm_ceAdminis-trationunderContractNASw-718.1964016562-004 -2-ThematrixfunctionZ(-)representstheinput-outputrelationsoftheN-port.1%isnaturaltoexaminether_lationbetweenthis"external"descriptionoftheN-portandthe"interr_l"descriptionintermsofdynamicalorstate_riables.Thatis,howdoesoneassociateastatewiththeN-portdescribedintermsofitsimpedancematrix?Thisquestionofrepresentationhasbeensettledrecentlybythewriter(seepa_tlcui_rly[i]).Itturnsoutthatevezy"transferfunction"matrixZ(')whichhasproperties(i)and(iv)3butnotnecessarily(ii)and(iii),onemayassociateasystemofvectordiffe

3 rentialequationsofitheform(1.2)/dt=Fx+(1
rentialequationsofitheform(1.2)/dt=Fx+(1.3)y(t)=Hx(t).iHerex,thesta_,isann-vector;u(.),the_(current)isanN-vector,andy(.),theoutp_(voltage),isalsoanN-vector.F,G_Hareconstantlineartransformations.Wecall(1.2-3)afinitedimensiona_constantlineard_namicalsystem[1].Ifequations(1.2-3)areknown,thematrixZ(s)canbewrit+endownbyinspectionbytakingtheformallaplacetransformof(1.2).Theresultisexpressedbytheformula(1.4)Z(s):H(sI-F)'IG(I:unitmatrix).GivenZ(-),thedeterminationofF,G,andHin(1.2-3)ismuchlesstrivial.Someset(F,G,H)satisfying(1.4)alwaysexists.Moreover,thereisasmallestintegernsuchthatrelations(1.2-4)holdsimultaneously.OGenerallyspeaking,thissmallestdimensionnoisacomplicatedfunctionofthematrixZ(.).Ifnin(1.2-3)islargerthanno,thenthedynamicalsystem(1.2-3)issaidtobered_ueible.Then_bernoisidenticalwiththeLJ1964016562-005 -3-so-calleddegreeofZ(.)asdefinedbyMcMillan[2-4].Anumericalmethodformachinecomputingnowasgivenin[I].Alternately,Mc_llan'sdefini-tionyieldnviatheso-calledSmithcanonicalform

4 ofpolynomialmatrices_"o[5].Fromthemathem
ofpolynomialmatrices_"o[5].Fromthemathematicalpointofview,equations(1.2-3)maybeviewedas_presentlnganabstractdynamicalsystemdefinedwithrespecttoanabstractvectorspaceX.Tog_vetheseequationsconcretemeaning,wemustchooseaspecificcoordinatesystemorbasisinXandexpressthe.abstractvectorxandtheabstractlineartransformationsF,G,andHinnumeri-calform.Oncethishasbeendone,xbecomesann-tupleofrealnumbersandF,G,Hbecomen×n,n×N,andN×narrays(matrices)ofrealnumbers.Anysystem(1.2-3)giveninnumericalform,asJustdescribed,iscalledarealizationofZ(.)(seeIll).M_thematically,theterm"realization"meansthatwepassfromtheabstracttotheconcrete(numerical)description.Physically,theterm"realization"ismotivatedbythefact[_]thatanynumericallygivensystemofequation_(1.2-3)maybeinterpretedastheprogramforananalogcomputerwhichsimulatesthegivenN-port.Eachrealizationcorrespondstoaspecificchoiceofacoordinatesystemforthestatevector.Ourultimateobjectiveistoobtainthatsub-classofrealizationswhichcanbeidentifiedwithapassivenetwork

5 ,notmerelywithananalogcomputerprogram.Th
,notmerelywithananalogcomputerprogram.ThenextproblemconcernsthestudyoftherelationshipsbetweenvariousrealizationsofZ(.).Thlsisindeedthemainideamotivatingtheresearchdiscussedhere.Theproblemisclearlyofagrouptheoreticalnature.Weask:Whatisthegroupoftransformationswhichleavethepropertiesofagivenrealizationinvarlant?Supposewepicktwobasesforrepresentingtheabstractvectorx.Inthefirst,thevectorisxdescribedbythentm_rlcaln-tuple=(_l'"'"_n)andintheseconditisrepresentedbythen-tupleA=dl,...,_n).ItIswellknown[7,P.82]that_l_arerelatedbyanonsingularlineartransformation,sothat1964016562-006 AAn(1.9)_=T_or_i=Zt_,i=i,...,ni=liJjwhereT=[tij]isaconstant3real_nonsingularmatrix.ThematricesT_formthegenerallineargroup.^iItisconvenienttoabusenotationandemploythesymbolsx,x;F_^_F_...alsoforthen-tuplesrepresentingthevectorxandmatricesrepre-isentingthelineartransformationFwithrespecttocertainspecifiedbases,iNowifthese_IF,G,H]specifiesthedyrm_icals_teminthefiz_tbasis,i!?thentheset[8,__]whichspecifiesthesame

6 s.vstemwithrespecttotheisecondbasisisrel
s.vstemwithrespecttotheisecondbasisisrelatedtothefirstsetbytherelationsiL_=_._-l:^(1.6)a=__=HT-1w_chareeasilyaeri_a_i_(z.2-3)and(1.9).(_e[I].).DifferentchoicesofbasescorrespondtodifferentrealizationsofthesameZ(.).Thereforeonewouldcel_uainlyexpectthatZ(.)isinvariarrtwithrespecttoachangeinbasis.Thisisverifiedwiththeaidof(1.6):Z(S)=H(sI-F)G=HT'IT(sI-F)-_'ITG_(sx_-i)-_,^^=-o=_(sl-F)'_G.Conversely,onemayask:_awhatwaydosak7tworealizationsofZ(.)differfromoneanother?Theansweris[1]thatiftheyareirreducible(n=no),thentheydifferonlybyachoiceofbasis.Thecriterionforirreducibilityisthatthetriple[F.G_H]becompletelycontrollableand1964016562-007 mcompletelyobservable[i].Thusiftwocompletelycontrollableandcom-pletelyobsex_bletriples[F3G,H)and[9,MGj_)yieldthesametrans-ferfunctionmatrixZ('),thentheyarenecessarilyconnectedbythere-lations(1.6).Notethatthisisanabstractresult;inpracticalcasesismaybequitedifficulttofindthetransformationT.Wenowrephrasethisimportantfactinsuchawayastoemphasizeitsgroupth

7 eoretic_lcharacter:THEGREM.An_twoirreduc
eoretic_lcharacter:THEGREM.An_twoirreduciblerealizations(1.2-3)ofatransferfunctionm_trixZ(.)havingproperties(i)and(iv)areequivalentunderthegenerallineargrou_.NowifZ(')istheimpedancematrixofapassiveN-port,itwillhavecertainotherproperties(namely(ii-ili)above)inadditiontothoseneededtoestablishthistheorem.Onewou/dthere.foreexpecttofindmorerestrictedtypesofrealizationswhichareinvariantwithrespecttocertainsubgroupsofthegenerallineargroup.Thedeterminationof_hese"networksubgroups"isidentical_ththeproblemofstudyingallpos&.blenetwork_JrealizationsofagivenZ("),whichisalsocalledtheproblemofnetworkequivalence.Animportantadvantageofthegrouptheoreticapproachwewishtoexplorehe1_isthatitprovidesaunifiedwayofstudyingtheproblemofsynthesisbydifferentclassesofelements.Forinstance,theRIL'2andR_synthesisproblemscanbestudiedsimultaneously.(See[5].)Itisimportanttobearinmindtheconceptualdistinctionbetweenthe"impedance"transferfunctionandthe"statevariable"po_n_sof_iewinnetworktheory.Transfezandimpedan

8 cefunctionsarecoordinate-freenotions.The
cefunctionsarecoordinate-freenotions.Theyaremostusefulinstudyingpropertiesofnetworksregardlessoftheirin-ternalst_xcture.Thisisthedeeperreasonwhyexistencecriteria(suchaspositiverealness)arestatedmoreconvenientlyintermsofZ(.)thenintermsofthetriple(F,G,H).Thisobservationisnotconfinedtonetworktheory[9-10].1964016562-008 -6-Ontheotherhand,dynamicalequations(1.2-3)alwaysInvclvecoor-dinates.Theseequationsaremostusefulinthedetaile,istudyoftheinternalstructureofanetwork.Suchconsiderations_vebeengenerallyabsentfromclassicalnetworktheory,whichmayex_,12inin_artthediffi-cultlesencounteredinresolvingn_tworkequivalenc_problems.Thegroup-theoz_ticalapproachsuggestedhereiscompletelyanalo-goustothefamousErlangerProgrammofFelixF1ein.TherehaveneverbeenanysystematicefforttoapplyKlein'sideasconet,_orktheory,asfarasthewriterisaware.Itshouldbe_intedout,however,thattheworkonnetworkequivalenceofCauer[93seepartlcularlypagesxviil,49,andChapter10]wascertainlyaconsciousstepint_samedirection.Muchmorecanbe

9 donealongtheselines.2.RestrictionsDuethe
donealongtheselines.2.RestrictionsDuethePassJ_ityandR_.ThefactthatZ(')representsa;assiveN-portimposescertainre-strictionsonthematricesF,G,andH.Theserestrictionsarethecounter-partofproperties(ii-ili)oft-heimpedancematrix.Firstofallitisnecessarytoidentifythecomponentsxiofthestaten-tuplexwithphysicalvariablesinthenetwork.Weadoptthefollow-ingconvention,whichisbothstandardandconvenient[12].LetasconsideranN-portwhichcontai-_nLinductorsandnC=n-nLcapacitors.Thenwedefinexi=currentthroughi-thinductor,wheni=l,...,nL_(2.1)xj=voltageacrossi-thcapacitor,whenJ=nL+l,...,u.Assumingforamomentthatnoneoftheindactorsar_ooupledwitheachother,andthesameforthecapacitors,itfollowsfromllr_aritythatthedynamicalequationsoftheretworkmaybewrittenintheform1964016562-009 -7-,_nNILidxl/dt=7alkxk+Zbi_Jl(t)i=l,...,nLk=lL=I(2.2)nN,Cjdx#dt4=Za_x,+ZbJ(t)J=_r4+l,...,nk=ldmml=lJlIDUl(t)=Zh_kXk(t),I=i,...,N.k=lTheJl(t)arethecurrentsenteringtheportsandtheUl(t)areth_volt-agesacrosstheports.Toderiv_thenumbersaikandbi!

10 wemayreplace,foraninstant3aninductorbyac
wemayreplace,foraninstant3aninductorbyacurrentsourceandacapacitorbyavoltagesource.Wethenobtainthefollowinginterpretation.Letil,i2beintegersbelongingto[i,nL],Jl'J2integer_be-longingto[nL,n],andlanintegerbelongingto[i,N].aili2=voltageacrossll-thindtmtor_enallcapacitorsareshortcircuited,all!_,tsandallinductorssavethei2-thareopencircuited,andthe12-Lhinductorisreplacedbyaunitcurrentsource.ailJ2=voltageacrosstheil-thinductorwhenallportsandallinductorsareopencircuited,allcapacitorssavetheJ2-thareshortcircuited,andtheJ2-thcapacitorisrepl_.-edbyaunitvoltagesource.bil!=voltageacrosstheil-thinductorwhenaA1capacitorsareshortcircuited,allinductorsareopencircuited,allportssavethel-thareopencircuited,_ndaunitcurrentsourceisconnectedacrosswithl-thport.Theotherquantitiesaredefinedanalogously.1964016562-010 ItIsclearthatthematricesAandBdependonlyonthatpartofthenetworkwhichcontainstheresistorsandidealtransformers.Partlonlngthesematricesaccordingtothenumberingschemeintro_:_edabove,wecaneasilys

11 eewhatrestrictionsareimposedbypassivitya
eewhatrestrictionsareimposedbypassivityand/orreciprocity.Tfwewrite-!-AI62i!(2.3)A=---I---!!C1,A2IBmthenA1hasthedimensionofresistance,A2hasthedimensionofcon-ductance,whileC1andC2aredimensionless.PassivityimpliesthatA1andA2arenonpositlve(butnotnecessarilys._mmetric)u_trlcesi.e.,theirquadraticformsarenonposltlve.M_reover;thequadraticfo._m0C1X!XCe0mustbeidentically0,i.e.,C2--C_.ReciprocityimpliesthatA1aml_2aresymmetric(butnotnecessarilynonpositive).AsfarasBandHareconcerned_If]B=,H=.B2If__0thenZ(_)_0andassumption(iv)oftheprevioussectionIsviolated.H1--0forthesamereason.Passivityrequiresthat_R2benonnegat_vedefinite.ReciprocityrequiresB2=_.1964016562-011 m_tfterremovingallInductorsandcap_citoz_theN-portbecomesan(N+n)-port.This(N+n)-portdoesnotnecessa1_lypossesseitheranimpedanceoradmittancematrixbeca_metheremaybeopencircuitsorshortcircuitsatcertainports.Insuchcases,oneadoptsasuitablemixedimpedance-admittancedescriptionofthelastnports.Wedonotwishtodwelluponthecomplications_alltrivial

12 --whichresultinsuchcases.Supposewewritet
--whichresultinsuchcases.SupposewewritetheenergystoredintheN-portas%n11212E=_x,_=_zLixi+_zcjxj.i--Ij--%+iTheneq_t_ons(9.2)takethesimpleform,!(2.2)Pdx/dt=Ax+BJ(t),U(t)=Hx(t).Theseequationsarevalidevenifthereiscouplingbetweentheind__.torsandcapacitors.ThusingeneralPwillP_v_theformIii:I(2.5)P=-'-.!C_I!Theoff-dlagonaltermsofPare0becauseinconventionalnetworksthereisnocouplingbetweenelectricandmagneticfi_ldsoSincethestoredenergymusth_apositivedefinitequadraticform,weassumethatLandCarepositivedefinitematrices.Reciprocityrequiresthattheybesymmetrlcalmatrices.IntheconventionalRLCTcasesLissymmetric,positivedefinite,whileCisamatrixwithpositiveentriesonthediagonalandzeroselsewhere.1964016562-012 -10-ThecasewhenLismerelynonnegativedefiniteindicatesthepre-senceofidealtransformers.Luthiscasethen-amherofstatev_riablesistoolarge.Weshallnotdiscusstheresultiugcomplications.Ifthedynamicalequations(2.2')oftheN-portarereducible,i.e.,maybereplacedbyasmallersetofequationshavingthesameimpedancematr

13 ixZ,thenthenetworkcontairmdynamicalmodes
ixZ,thenthenetworkcontairmdynamicalmodeswhicharenotspeci-fiedbytheimpedancematrixZbutarisesolelyasaresu/tofthesyn-thesisprocedure.Forinstance,theDarlingtonand_heBoht-Duffinproce-drupesintroducesuchextraneousmodes°Althoughthepresenceofsuchaddi-tionalmodesmaybenecessarytocarryoutcertaintypesofsynthesisprocedures,inthis_aperweshallbeconcernedonlywiththeirreduciblecase.Inotherwords,itwillalwaysbeassumedthattheN-portalwayscontainsaminimalnumbernofreactiveelements.o3.CharacterizationofPassivity.NowweshallstatearelationbetweentheimlY._danceZandthestatevariabledescriptionofanN-port.Thisrelationwasdiscoveredinthecourseofstudyingtheso-calledLur'eproblemofconstructionLy_punovf',_nctionsfordynamicalsystemswhicharelinearsaveforasinglenonlinearelement_].Theresultto"DestatedbelowismoregeneralthantheM_inlemmain[9],inthatweadmitNXNratherthan1X1impedancematricesandwedroptheassumptionthatalleigenvaluesofFhavenegativerealparts.Ontheotherhand,wewillassumehere,asamatterofconvenience,thatZ(®)=0.T

14 hisisanunessentialrestrictionwhichwasnot
hisisanunessentialrestrictionwhichwasnotneededintheM_inlemmaof[9].Forafullti_atmentofthegeneralproblem,incliningproofs,see[13,14].CHARACTERIZATIONC_PASSIVITYTHEOREM.letZ(-)bea_.___nNXNmatrixofratio_lfunctionsofthecomplexvariables,withZ(_)=0.let[F,G,HIbe_triplesuchthat(1.2-3)isanirreduciblerealizationofZ(.).Let_(y)heacontinuousp-vectorfullctionofthep-vectorys__hthat_(0)=0andy'_(y)-_0forally.1964016562-013 -II-Thenthefollowingstatementsareequivalent:(I)Z(')isnonneg_tivereal,i.e.,Res2_0impliesZ(s)+Z'(s)=nonnegativedefinitehermetianmatrix.(II)Thereexistsas_etric,positivedefinitematrixPandasymmetric,nonnegativedefinitematrixQsuchthat(3.1)FF+F'P=-2Q,(3.2)_=_.(AmatrixFsatisfying(3.1-2)cannothaveaneigenval_withpositiverealpart;initsJordanformallimaginaryeigenvai_sarecontainedin1x1blocks;th_nullspaceofthematrixQisnecessarilycontainedi_theeigen_ceofFsparmedbytheeigenvectorscorres.9ondingtoimaginaryeigenvalues.)(III)V=x'PxisaLyapunovfunctio_nfortheautonomousdyrmmicalsystemdx/dt=Fx-GY(

15 Hx)suchthatV(x)-_0.Letusgiveanindication
Hx)suchthatV(x)-_0.Letusgiveanindicationoftheproof.(II)implies(I)bydirectsubstitution.Given(1),thenonnegativerealcharacterofZ(_)allowsittobefactoredasZ(_)+Z'(-i_)=W'(_)W(_).Everysuchfactoriza-tiongivesrisetoaQ.Theirreducibilityoftherealizationisthenutilizedtoverifythat(5.1-2)hold.(II)implies(III)because,_(x)=m,F_,_-_,a'P_(m_)---2[_Qx+y"_(y)]_-0by(3.1-2).Finally,(llI)implies(I)bydirectcomputation.1964016562-014 -le-Remarks.(i)Ina1998paper,Desoer[15]pointedoutthatatthattimethestabilityofpassivenonreciprocalnetworkshasnotyetbeenprovedandgaveanargumentshowingtha_allsuchnetworksare,infact,(Lyapunov)stable.Thisresultisnowconfirmedinthestrongestpossible%By.SincethenonnegativerealnessofZ(')(asstatedabove,i.e.,notrequiringZ(')=Z'(.))isanecessary,conditionforrealizingapassiveRLCTFN-port,itisclearfromtheparentheticalremarkin(II)thatFcarmothaveeigenvalueswithpositiverealparts,norcanithavesolutionsofthetypetkcos(_t+(_),k�0.(2)Fornetworks3V=x'Pxcanalwaysbeidentifiedwiththestoreden

16 ergyintheinduct.orsandcapacitors.Hence(I
ergyintheinduct.orsandcapacitors.Hence(Ill)showsthatinunterminatedpassivenetworks(Y=0)theenergyisanonincreasingfunc-tionoftime.Forarbitrarypassiveresistiveterminations,whichmaybeeitherlinearornonlinear,thesameconclusionholds.Inthenonlinearcasethisresultrepresentsaconsiderableimprovementoverwhatwasknownbefore.(Tomyknowledge,thepreviousbestgeneralres1_lthereisthatofDuffinThecharacterizationtheoremexpressesfactswhichare%muallytakenforgrantedfrunanintuitivephysicalpointofview.Asaresult,theaverageengineermaybetemptedtoJumptotheconclusionthatnothingnewhasbeendone.Buttodothiswouldbeagrossmisunderstandingoftheprocessesofscientificresearch.Thetheoremi_amorepreciseandmoren_characterlzamlonofpassivitythanwasheret,_foreavailable_TheandtheKeneralityofthisresultinturnleadstodeepinsightintoproblemsofnetworksynthesis[8,14],whichcannow_st,tiedwithasimplicityandexplicitnessthatwasimpossiblepre_iously.Thistheoremisaveryconvlnc_additionalpieceofe_Idencethatpassivityandnetworksynthesisareintima

17 telyrelated.In[8]Ishallpre-sentsomepreli
telyrelated.In[8]Ishallpre-sentsomepreliminaryresultswhichshowthatthestructuralpropertiesoftheclassofallnetworkswhichrealizeagivenimpedancefunctionZ(.)areinl-1correspondencewithcertainalgebraicInvariantsofZ(.).1964016562-015 -13-4.Applications.Weshallil/u_tz&tetheusefulnessofthecharacterizationtheoremandthegrouptheoreticideasrelatedtoitbytwoclassicalexaapl_ss(i)LC1-portsynthesisaccordlngtoFoster,and(ii)synthesisoflossless2-ports.EXAMPLEi.WewishtoprovethatA_cessar_andsufficientconditionino_erthatthescalarImpe-dancefunctionz(o)berealizableast_].-portshowninFi_.1isthat(i)z(.)isnoqne_tlverealand(i±)anpolesorz(.)a__lr_ry.Necessity"Ifthestat_variablesforthenetworkareassignedasshowninFig.i,itfollowsbyinspectionthattheA,B,H,andPmatricesare"!I!"01,!I""t"""""|--"!IiI,0i{,|110,.'!(4.1)A:',"II!.tTherearer2X2I•!,blocksonthediagonal_.|_..I.IIInltiI_JoIiI,_0IIIr.a-01i0,111964016562-016 -14-(43)[i'''i]•H=_0l'•••'0|!lCiIj"'IIiii_o,I'I1I0CltI(4.4)p=_tII!I.'ItI•"_....I---fItIL0IIIrIIIIrJI,I0C

18 LIfIItisclearthatthequintupleF=p-1Ah,G=P
LIfIItisclearthatthequintupleF=p-1Ah,G=P-,H=H,P=P,Q=0satisfies(3.1-2).Hencetheimpedancefunctionofthe1-port,wkichisgivenby(4.9)z(s)=Co_+k=lZ--Ck(B2-+_k2)'=Ckisanonnegatlverealfunction.1964016562-017 Sufficiency:Supposethatz(-)isnonne_tlvereal.Thenre-lations(3.1-2)aresatisfied.If,inaddition,allpolesofz(')areima-glnary_thenQin(3.1)isnecessarilyzero.WeshallnowcarryoutaseriesofchangesofbaslswhichwilleventuallyexhibitthematricesA=FF,B=PG,H=H,andP=Pintheform(4.1-4),afterwhichtheexistenceofthenetworkshowninFig.1isobviousbyinspectSon.Ste_2_.WepickabasisthatP(1)=I.Thenby(3.1)F(I)=-Fil).ThisisalwayspossiblebecauseunderachangeofBasis(1.9)wehaveA(1.6')P=T'PTSincePispositivedefinite,itcanalwayshew_-IttenasP=T'IT,Tnonslngular,sothat_=P(1)=I.S_____.WepickasecondbasiswhichleavesPinvariant(Pt2_ti=I)buttransformsF(I)intothecanonicalform"_II-01II..L.._-;._IIII_0IIII"I"._....u.'L...._I0a_Ii!rIIi/I'I-(DrJ,0_I,I1964016562-018 -£6-Thisisalwayspossiblebecause[17]everyskewsymmetricmatrixcanbetransfo

19 rmedintothecanonicalform(4.6)bymeansofan
rmedintothecanonicalform(4.6)bymeansofanorthogonaltransformation.Notethattheelement0intheupperleft-handcornerofF(2)correspondstotheelgenv_l_0(ifF(I)_ppenstohavethateigenvalue),_ilethe2X2blocksalongthediagonalcorrespondtotheeige.nvalues+-_k'k=I,...,r.(Sincez(')hasonlyimagi_zrypoles,Fhasonlyimaginaryeigen_alues9bynonnegativerealnessalleigenvaluesmustbesimple.ThusF(2)isasitshouldbe.)Inthebasis(2)3thecolumnvectorsB,H'havetheform_omIm%°(2)=B(2)=_(2)=....•G_r_rSincethetriple[F,G3H'}mustbeirreducibleandthereforecompletelycontrollableandcompletelyobservableif(If)ofthecharacterizationtheoremistohold,itiseasilyverifiedthat_o�0and_+_�0,k=ij..._r.Step3.Anyproper(det=l)orthogonaltransformation(rotation)appliedtothe2X2matrix01964016562-019 -17-1eavesiti;_vari_nt.WeapplyrsuchrotationscorrespondingtotherblocksinF,ins_cha_ythatthevectorsinBandH'are.rotatedintothevectorsIjk=11"--3r.0LrkBythecommentmadeattheendoftheprecedingparagraphweknowthatTk--_+__o._'_"_F(3)--F(a)'P(3)-P(2)-

20 I_n_Wn_ot_Jm0Y1o(3):B(3)-0l"rSte_4.Now_o
I_n_Wn_ot_Jm0Y1o(3):B(3)-0l"rSte_4.Now_obtainchangetoabasiswhereA,BandHhaveonly0,i,-iaselements.Fortunatelythiscanbeaccomplishedwiththeaidofadiagonalmatrix.Ifx(3)=Ax(4),whereAisdiagonal,thenP(4)=AP(3)A--A2,1964016562-020 ISB(4)=P(4)°(4)--_(3)_-1G(3)--_(3)andA(4)=P($)F(4)=AP(3)AAIF(3)A=AA(3)A.Wechoosemi/_°',',!t!Y_/_I_0!I0i/ri""-IIIA=tl"It•IIiI_!YJ_)r0i'It_I0i/y!t,ItiSeasilycheckedthatA(4),B(4],H[4]assumetheform(4.1-4.])._JIfwelet=1/Bo,CoCk=1/_k,k=l,,.._rj{=yk/_k,k=l,...,r]thenthesamecanbesaidalsoaboutP(4)"Theproofiscompleted.1964016562-021 -_9-Itmayappeartothereaderwell-versedinclassicalsynthesistheorythattheusualFostersolutionjobt_i1_edbyapartialfractionexpansionof(4._)_isamuchsimplerroadtoobtainingthenetwork.Actuaily_thetotalnumberofart_entsisnotreallysmallerthaninthematrixcase.Oua"presentmethod_however,hastheveryimportantadditionalfeaturet.hatitgivsinsightintothegroupstructureofl_ssive1-ports.0nlyafewmoreargument_areneedeitocompletelydescrioeinthiswayallpassivel-por+swhi

21 chcontainam_uimalnumberofreactiveelement
chcontainam_uimalnumberofreactiveelements.ThepartialfractionmanipulationsusedintheFostertheorycanbeinterpretedascal_lationsbasedonthe__rpuprepresentationprovide_bythelaplacetransform.EXA/4PLE2.Letusconsidertheimpedancematrixms2+is2+iz(.)=k12sk22s!"-t22s+is+iWewishtorealizethismatrixwitha(lossless]reciprocal2-portcontainingaminimumnumberofreactiveelements.Reciprocityrequiresthatk21=k12.Passivityrequires_asmaybeeasilycheckedusingthecharacterizationtheorem)A=kllk22.k122a0,whichispopularlyknownasthe"residuecondition".Infact,thematrixofresiduesofZ(.)ateitherpoles=+iisgiven0yK=Ikllkl21kl2k221964016562-022 -20-sa_d&=detK.Twopossibilitiesmayarise:Either&�0or&=0.Themlnim.mnnumberofreactiveelementstobeusedisequaltotheminimumnumbernoofstatevariableswhicharenecessaryandsufficienttorealizeZ(')asadynamicalsystem(1.2u3).ThenumberncanbeOcomputedwiththealdofatheoremofE.G.Gilbert[Theoremll],whichstates(inthespecialcasewhenZ(-)hasdistinctpoles)n=ZranksofresiduematricesofZ(').oHencen=4_

22 henZ_�0andn=2when&=0.Itsho_uldben
henZ_�0andn=2when&=0.Itsho_uldbenotedoOthatnoisalsoequaltotheMcMillandegreeofZ(.)[4].Letusconsiderthecase_�O.In[1,Sect.8]twomethodsweregivenforconstructingrealizationsofagiventransferfunctionmatrix.UsingMethod(B),itisfoundthatthefollowingmatricesprovidearealiza-tion,_fZ(-):i0]r01F(1)='O(1)='_(i)--I0K].ziI0[-Ifwelet[°iP(_)='Q(1)=0_0Kthenthequlntu_le{F(1),G(1),H(1),P(1)'Q(1)]satisfies(3.1-2).How-ever,thesematricesdonotcorrespondtoanetworkz_alizationofZ(').Inordertoobtainsum.harealization,weintroduceanewbasisdefinedby_(1)=t_(_),whe_1964016562-023 -21-l°U=0K-IjThenr0[01IReferringtcthediscussionofSect.i,itiseasilyseenthatthesematricesrepresenttworeso_ntLCcircuitsinwhichthereiscot_ling(i_presentedbythem_triceEKandK-Irespectively)betweenbot__htheindictorsand_al_-citors.Sinceco%_pledcapacitorscanalwaysberealizedbyordir_rycapaci-torsandidealtransformers,wehaveproved:Whe____n_�0)Z(')canberealizedasanLCTnetworkwithaminimalnumberofreactiveelements.Byfurthereasymani

23 pulationsofthematricesinvolved,itcanbesh
pulationsofthematricesinvolved,itcanbeshown--asiswellknown[19,p.4A7-493]--thatasingleidealtransformerwillalwayssuffice.Thequestionthenariseswhetherornotthe2-portcanberealizedusingonlycoupledinductors(butnotcoupledcapacitors)andwithoutidealtransformers;further,whetherornotthe2-portcanberealizedwithoutanyinductiveozcapacitivecot_ling.Theanswertobothquestionsisin_neralno.Weshallnotgivetheproof(straightforward)butmerelystatetheresult:_.A�O,Z(')_nberealizedasanLCnetworkwithaminimalnumberofreactiveelements,wltho_tidealtransformers)andwithout_yco_llnKbetweenInductorsorcapacitorsifa_donl_ifeither(i,k:m=O,.or(__t)h2=_'n'or(tit)kZ2=k22._se1964016562-024 -22-conditionscannotbeweakenedifeithercoupledinductorsorcoupledca__citors(butnotboth)areadmitted.Inthefirstcasethe2-portc_nsistsoftwouncom_ected1-ports.Inthesecondandthirdcasethe2_por_canberealizedasanL-section.Idon'tknowhowwellknownthisoasyresultis.Itisquiteuse-fultheoreticallyber_auseitshowst_hattransformerlesssynthesisimplie

24 sverystrongconstraintsonZ(')inadditionto
sverystrongconstraintsonZ(')inadditiontononnegatlverealness.ThisfacthasadirectbearingonthetransformerlesssynthesisofHIE1-portsbecausesuchproblemscanalwaysbereducedtothesynthesisoflosslessN-ports.(Bee[8].)iThecaseA=0maybetreatedsimilarly.5.Conclusions.(i)MethodsbasedontheImpedanceconceptarecoordinate-free.Theydonotdisplaydirectlythestructuralpropertiesoftherealization.(2)State-variablemethodsont_otherhand,arecloselyrelatedtothestructuralpropertiesofnetworks.(I)Everytransferf_ictionmatrixadmitsrealizations.Someoftheserealizationsmaycorrespondtonetworks,whileothersmayrequireactiveelements(analogcomputers).(4)Nonnegativerealimpedancematricesal_ys_dmltpassiverealizations,i.e.realization_imposerestrictionsonF,G,H,P,Qinadditiontothoseimpliedbypassivity.Inallcasesthattheserestric-tionsareknowntheyareexpressibleintermsoftheimpedancematrix.Thisiswherethestorystandsatthemoment.Iwouldliketosug-gestthefollowingprogramforthefuture:Toclassifyalltypesofrealizationsofa_ivenabstractdynamical

25 systemandtoexpresstherealizabilit_condit
systemandtoexpresstherealizabilit_conditionsascoordinate-freeproper-ties(suchas_--------ssivit_,reclproclty,etc.)ofabstractd_amlcals_rstems.1964016562-025 -23-Thisproblemsuggestsapartnershipbetweenmathematicsandnet-worktheorywhichwillbeintellectuallyexcitingandpracticallyprofit-able.Dyr_unicalsystemsarethebuildingblocksofmoderntechnology.Theresolutionoftheproblemposedwilltelluswhattechnologyiscapableofdoingatpresent.Itwillalsosuggestdevelopingnewcomponentstorealizethatwhichis__nowntobepossibleformathematicalreasons.6.Refereuces.[i]R.E.Kalman,i'M_thematicaldescriptionoflineardyrm_Icaisystems"J.control,i(1965)152-192.[2]Brockw_yM_Mil/B_,"Introd_tlontoformalrealiz_bilitytheory",BellS_temT_cbnl_lJournal,_!(1952)217-e79,_i_30.[4]R.E.F_ima_"Irreduciblerealizationsandthedegreeofamatrixofrationalftmetlo_s",toappear._]F.R.Gantmakher,THEORY0FMATRICES(book,2volumes),Chelsea,1999.[6]R.E.K_!man_"Analysisanddesignprinciplesofsecondandhigherordersaturatingservomechanlsms",Trans.AIEE,_II(1

26 999)_93-310.[7]P.R._almos,FINITE-DIME_IO
999)_93-310.[7]P.R._almos,FINITE-DIME_IONALVECTCaSPACES(book),VanNostrand,1958.[8]R.E.Ea]man,"AnewlookattheRLCEFsynthesisproblem",Proc.FirstAllertonConf.onCircuitandSystemTheory,1965.[9]R.E.Kalman"Ly_punovfunctionsfortheproblemofLur'eInautomaticcontrol",Proc.Nat.Acad.Sci.(L_A),49(1965)201_q05.[10]R.E.Kalman,'_henisalinearcontrolsystemoptimal?",J.BasicEngr.1964(toappear).1964016562-026 -24-[11]WilhelmCauer,SYNTHESISQFLINEARCOMMUNICATIONNETWORKS(book),_Graw-Kill,1998.[12]R.E._Iman"Networkanal_sisbythestate-variablemethod"_toappear.[13]R.E.Kalm_n_"ResolutionoftheproblemofLur'e",toappear.[14]R.E.Ealman,"Synthesisof_N-ports",toappear.[15]C.A.Desoer,"Onthecharacteristicfrequenciesoflosslessnon-r_ciprocalnetworks",Trans.IREPGCT,_(1958)374-375.[16]R.J.Duffin,"NonlinearnetworksI",Bull.Am.M%th.Soc._2(19 )833 98.[17]SeeChapteri_VolumeIIofReference5.[18]R.E.K_Iman#':OnthestructureoflossiessN-ports",toappear.[19]LouisWeinberg,NETWCI_KANAL_ISANDSYNTHESIS(book),McGraw-Hill,1962.1964016562