J.S.I.A.M.CONTROLSet.A,Vol.2,No.PrintedinU.S.A.,1963CONTROLLABILITYAND - PDF document

J.S.I.A.M.CONTROLSet.A,Vol.2,No.PrintedinU.S.A.,1963CONTROLLABILITYANDOBSERVABILITYINMULTIVARIABLECONTROLSYSTEMS*ELMERG.GILBERTt1.Introduction.Theimportanceoflinearmultivariablecontrolsystemsisevidencedbythelargenumberofpapers[1-12]publishedinrecentyears.Despitetheextensiveliteraturecertainfundamentalmattersarenotwellunderstood.Thisisconfirmedbynumerousinaccuratestabilityanalyses,erroneousstatementsabouttheexistenceofstablecontrol,andoverlysevereconstraintsoncompensatorcharacteristics.Thebasicdifficultyhasbeenafailuretoaccountproperlyforalldynamicmodesofsystemresponse.Thisfailureisattributabletoalimitationofthetransfer-functionmatrix--itfullydescribesalinearsystemifandonlyifthesystemiscontrollableandobservable.TheconceptsofcontrollabilityandobservabilitywereintroducedbyKalman[1.3]andhavebeenemployedprimarilyinthestudyofoptimalcontrol.Inthispaper,theprimaryobjectiveistodeterminethecontrol-labilityandobservabilityofcompositesystemswhichareformedbytheinterconnectionofseveralmultivariablesubsystems.Toavoidthelimita-tionsofthetransfer-functionmatrix,thebeginningsectionsdealwithmultivariablesystemsasdescribedbyasetofnfirstorder,constant-coefficientdifferentialequations.Later,theextensiontosystemsdescribedbytransfer-functionmatricesismade.Throughout,emphasisisonthefundamentalaspectsofdescribingmultivariablecontrolsystems.Detaildesignproceduresarenottreated.2.Definitionsandnotation.LetamultivariablesystemSberepresentedby2Ax+Bu()v=Cx/Duwhere:uu(t),p-dimensionalinputvector.vv(t),q-dimensionaloutputvector.xx(t),n-dimensionalstatevector,nistheorderofS.2(t),timederivativeofstateA,constantnthorderdifferentialtransitionmatrix.*ReceivedbytheeditorsJuly5,1962andinrevisedformNovember1,1962.PresentedattheSymposiumonMultivariableSystemTheory,SIAM,November1,1962atCambridge,Massachusetts.InstrumentationEngineering,UniversityofMichigan,AnnArbor,Michigan.Reference[14]givesahistoricalaccountofcontrollabilityandlistsotherrefer-ences.128 CONTROLLABILITYANDOBSEI:tVABILITY1.'29B,constant,nrow,pcolumn,inputmatrix.C,constant,qrow,ncolumn,outputmatrix.D,constant,qrow,pcolumn,transInissionmatrix.Ifn0thesystemissaidtobestatic.ThecharacteristicrootsXi,i1,n,ofAareassumedtobedistinct.Thisgreatlysimplifiestheproofoftheoremsandpreventsthemaincourseofthepaperfrombecomingobscured.Besides,therearefewpracticalsystemswhichcannotbesatisfactorilyapproximatedwithanAwhichhasdistinctroots.Letpbeann-thordernonsingularmatrixwhichdiagonalizesA:a(2)p-lAp-ADefinenormalcoordinatesasthecomponentsofthen-dimensionalstatevectory,(3)xpy.ThenthenormalformrepresentationofSisgivenby(4)Ay-t-fluv/yq-Du,where(6)p-lB,thenormalforminputmatrix,Co,thenormalformoutputmatrix.Thenormalcoordinatesarenotunique.Ifdesired,theymaybemadesobyarrangingtheXiinorderofincreasingmagnitude(rootswithidenticalmag-nitudesmaybetakeninorderofincreasingangle)andchoosingthecolumnvectorsofp,o,i1,n,tohaveunitEuclideanlength.Forthepurposeconsideredhere,thesystemSisstableifReXi0foralli.TherankoftheinputruisdefinedastherankofthematrixB(orequiva-lently,therankof).Itisthe"effective"numberofinputswhichcanSeeBellman[15,p.198.]Familiarresultsofmatrixtheorywillbeusedwithoutcomment.TheseresultscanbefoundinBellman[15]orotherstandardtexts. 130ELMERG.GILBERTinfluencethestatevector.Theinteger(pru)�_-0isthereforethenumberofineffectualinputs.Itispossiblewithnolossofgeneralitytoreducethenumberofcomponentsofuby(pr).TherankoftheoutputrisdefinedastherankofthematrixC(or7).Itistheeffectivenumberofoutputsavailableforobservingthestateofthesystem.Theinteger(qr)�-0givesthenumberofoutputs(componentsofv)whicharelinearlydependentifD0.ItispossiblewithoutlossofgeneralitytoreducethenumberofcolumnsofCby(qr).3.Observabilityandcontrollability.AsystemSiscontrollableif/hasnorowswhicharezero.Coordinatesyicorrespondingtonon-zerorowsof/arecalledcontrollable;coordinatescorrespondingtozerorowsof$arecalleduncontrollable.Uncontrollablecoordinatescaninnowaybeinfluencedbytheinputu.Thusasystemwhichisnotcontrollablehasdynamicmodesofbehaviorwhichdependsolelyoninitialconditionsordisturbanceinputs.Disturbanceinputsarenotindicatedin(1)andwillnotbetreatedinthisstudy.Sometimes,theymaybesatisfactorilyhandledbymeansofap-propriatelyintroducedinitialconditions.AsystemSisobservableif7hasnocolumnswhicharezero.Coordinatesycorrespondingtonon-zerocolumnsof7arecalledobservable;coordinatesycorrespondingtozerocolumnsof7arecalledunobservable.Unobservablecoordinatesarenotdetectibleintheoutputv.Thusasystemwhichisnotobservablehasdynamicmodesofbehaviorwhichcannotbeascertainedfrommeasurementoftheavailableoutputs.Afewgeneralremarksareinorder.First,thedefinitionofcontrollabilityisdifferentfromKalman's[14]:"Asystemiscontrollableifanyinitialstatecanbetransferredtoanydesiredstateinafinitelengthoftimebysomecontrolaction."However,undertherestrictionsoftheprevioussectionthetwodefinitionsareequivalent.Morerecently,Kalman[16]hastakenthesamepointofviewgiveninthispaper.ForsomeadditionalremarksseethenotebyHo[17].Second,thereisastrikingsimilarityinthedefinitionsofcontrollabilityandobservability,therowsofplayingthesameroleasthecolumnsof7.ThisisalsotrueofKalman'sdefinitions,andmeansthatremarkssimilartothoseofthepreviousparagraphcanbemadeaboutobservability.Moreimportantly,foreveryconclusionconcerningcontrollability,thereisacor-respondingoneconcerningobservability.Thiswillbeevidentinthestate-mentandproofoftheoremswhichfollow.Finally,thedefinitionsbecomemoreinvolvedwhenhecharacteristicUsuallyitisdesirabletoeliminateineffectualinputsandsuperfluouscolumnsofC.Exceptionsoccurwhenmplitudeconstraintsareimposedontheu(suchasuk,1,p)ornoiseispresentinthemeasurementofthev. CONTROLLABILITYANDOBSERVABILITY131ucUV+Fla.1.SystemSandispartitionedrepresentationrootsarenotdistinct.ThediagonalmatrixisreplacedbyaJordannormalformandtheconditionsonand,arenotsosimplystated.Inordertodealmoreconciselywiththeaboveconceptsconsider"THEOREM1.AsystemSmayalwaysbepartitionedintofourpossiblesub-systems(showninFigure1)"1)asystemS*whichiscontrollableandobservableandhasatransmissionmatrixD,2)asystemSeachofwhosenormalcoordinatesareobservableandunco.r-trollable,3)asystemSeachoj'whosenormalcoordinatesarecontrollableandun-observable,4)asystemSfeachofwhosenormalcoordinatesareuncontrollableandunobservable.AllsubsystemshavezerotransmissionmatricesexceptS*.Also,uuu,vv*+v,andnn*nt-n+ncn.TheproofofTheorem1followsdirectlyfromequations(4)bypartition-ingyaccordingtotherestrictions1through4).Asomewhatmoreinvolvedpartitioningmayresultwhenthecharacteristicrootsarenotdistinct.ThustheonlysubsystemwhichhastodowiththerelationshipofvtouisS*.TheobservablesystemSonlyaddsadisturbancevtothecontrolledpartoftheoutputv*.AlthoughS,SandSsappeartohavelittleim-portanceinsystemanalysisthisisnotnecessarilyso.Ifstatevariablesap-propriatetothedescriptionofS,Sc,andSsgetlarge,neglectednonlinearcouplingsmaybecomeimportantorphysicaldamageofthesystemmayresult.ThiscertainlywillbethecaseifS,SorSareunstable,i.e.therearehiddeninstabilities.FromTheorem1itisclearthatanecessaryandsufficientconditionfortheabsenceofS,S,andSisthatSbecontrollableandobservable.ItispossibletodetermineifSiscontrollableandobservablewithoutrecoursetothenormalformrepresentationbymeansofthefollowingtheorem.THEOREM2.Letbii1,p,bethecolumnsofBandcir,i1, 132ELMERG.GILBERTq,betherowsofC.AsystemSiscontrollable(observable)ifandonlyifthevectorsekiAkbi,i1,...,p,kO,...,nl(ek(Ar)(ci),i1,q,kO,n1)spanthen-dimensionalcoordinatespace.Thecontrollabilitypartofthistheoremhasbeenprovedusingthepre-viouslymentionedalternativedefinitionofcontrollability[14].Byduality[13,16]theobservabilitypartmaybeobtainedforanalternativedefinitionofobservability[13].Thefactthatthesameresultsareobtainedforthedifferentdefinitionsprovestheirequivalence.Proo.-f.Firstconsiderthecontrollabilitypartofthetheorem.ToprovenecessityassumeSiscontrollableandwrite(7)Sincefl[illfly]hasnozerorowitispossibletoformavectorfl+]cfl++k,flvnoneofwhosecomponentsiszero.Clearly,thevec-torse+patti+,k0,n1formasubspaceofthespacedefinedbythee.But(8)det[e0+en-+]ooo2+00(detp)(detV)({i32+...n-t-)0becausetheVandermondedeterminantVisnonzerofordistincth,pisnon-singular,andthe+areallnonzero.Thusthesubspaceisn-dimensional.Thereforetheemustspanthen-dimensionalspace.Toprovesufficiencyassumetheespanthen-dimensionalspace.ThenforanyrO,saytheinnerproduct(r,e)cannotbezeroforallkandi.Butk(9)(r,ea)(r,pAfli)(AkoTr,i)klliAssumeall1i0andacontradictionisobtained.Thusnotall0.ByThesuperscriptTindicatesthetransposeofamatrixorvector. CONTROLLABILITYANDOBSERVABILITY]33changingrtheargumentalsoworksonallotherrowsof.HenceSisco-trollable.ToprovetheobservabilitypartoftheTheoremnote(,risi-throwof7)ek(Ar)kc(pAp--1)T}I(/iTp--])T(10)(p-l)TilcpT(p--1Ti(p-l)rhk,.Since(10)issimilarto(7)theremain.igstepsarethesameasthoseinthecontrollabilitypart.4.Observabilityandcontrollabilityofcompositesystems.Inthissectionthecontrollabilityandobservabilityofcompositesystemsarerelatedtothecontrollabilityandobservabilityoftheirsubsystems.Theo-rems3and4treatrespectivelytheparallelandcascadeconnectionoftwosubsystems.Successiveapplicationofthesetheoremsextendstheresulttocompositesystemswhichconsistofmanysubsystemsconnectedinparallelandcascade.Theorem5isthecentraltheoremofthepaper.Itstatescon-ditionsforthecontrollabilityandobservabilityofageneralfeedbacksys-tem.THEOREM3.LettheparallelconnectionofsystemsSaandSb.formacom-positesystemS(seeFigure2).Then:i)nna-n,;ii)knlanaklbnbbiii)anecessaryandsucientconditionthatSbecontrollable(observable)isthatbothSaandSbecontrollable(observable).ToproveTheorem3letSandSbberepresentedinnormalform.ThenFIG.2.ParallelconnectionofSandS 134ELMERG.GILBERTfromthenotationinFigure2thenormalformofScanbechosensothat(11)3L3A"['/'a"Yb],DD+D.Simpleinspectionof(11)yieldsallpartsofthetheorem.THEOREM4..LetthecascadeconnectionofsystemSafollowedbySbformacompositesystemS(seeFigure3).Then"i)n--nanb;ii)),.,MXla,Maa,Xlb,Xnbbiii)anecessary(butinsufficient)conditionforthecontrollability(observ-ability)ofSisthatbothSaandSbbecontrollable(observable);iv)ifSaandSbarebothcontrollable(observable)anyuncontrollable(un-observable)coordinatesofSmustoriginate,whendesignatedaccordingtocharacteristicroot,inSSUsingthenormalformrepresentationsofSaandSyields--IAa01IaIIYal(12)3b'YaAbX2VbDaU,wherex--Yv[D%%Ixq-DDu.asthesetofequationsrepresentingS.Toputtheseequationsinnormalformdefine(13)x=Iy'y=_x,where--(/)Aa-Abb--3b'Ya,i.e.,(14)[o]kia-kjbdenotestheijelementof3bTa.TheassumptionofdistinctrootsU=UcIVb=V\V'/FIG.3.CascadeconnectionofSafollowedby& CONTROLLABILITYANDOBSERVABILITY135requiresh.bib0alliandj.Itiseasilyshownthat(15)Ay-t-fluv,yq-Du,where(16)Ab(--,+flbD,)/[(Db/.+5'b)'b],DD.D,.Resultsi)andii)followimmediatelyfrominspectionof(15).Considerthecontrollabilitypartsofresultsiii)andiv).From(14)and(16)itisobviousthatanullrowof,orfibwillresultinanullrowof.Thusthenecessityofiii)follows.Itisalsoclearthat-a+flbDamayhaveanullrowevenifaand/bdonot.Thusiv)andtheremainderofiii)hold.Cor-respondingreasoningappliedtothecolumnsof,yieldstheobservabilityresults.Formulas(16)canbeusedtodetermineifSiscontrollableorobservable.Unfortunately,afairamountofworkisinvolvedandthereappearstobenowayofgettingsimplersufScientconditionsforthecontrollabilityorob-servabilityofS.ItishelpfultoconsiderafewsimpleexampleswhereSisuncontrollableorunobservableeventhoughS,andSarecontrollableandobservable.LetSaandSbbegivenby:(17))laYlaYlYlbV2aYla.ThenifxlyandxydefinethestatevectorofS,o](18)0--2x+ulvl=[01]x.InthisexampleSisuncontrollableandunobservablebecausethematrices,De,Db,'Ya,b,which"couple"SaandSbaresuch(D,Db0,b'0)thattheinputulneverreachesthenormalcoordinateofSbandthenormalcoordinateofSaisnotpassedontotheoutputv.Thisparticularsituationcannothappeninsingle-input,single-outputsystems,sinceitwouldimplyeither0ortb0. ].36ELMERG.GILBERTForthesecondexampleletla--Yla"-Ul(.9)VlaYla-UlVlYlbUlb.TakingthestatevectorofSasxly,x2Ylgives(20)-2x+andforv=[1-1]x-u,11(21)P=1thenormalformrepresentationis(22)Y--0-2y+v-[0--1]y--ul.Equation(20)showsthatxylaandx2Ylbcanindividuallybecon-trolledandobserved.Yetfromequation(22),Sisclearlyuncontrollableandunobservable.Thisapparentparadoxisresolvedbyobservingthattheuncontrolled(andthereforeunalterable)coordiatey2xl-t-x:--Ya-Yib.Thereforey.taandyl_cannotindependentlybecontrolledorobserved.Athirdexamplearises,applicabletotheparallelconnectionofSaandSb,iftheassumptioninsection2ofdistinctcharacteristicrootsiswaived.Theniii)ofTheorem3becomesanalogoustoiii)ofTheorem4,inthatthestatedconditionisnecessarybutnotsufficient.LetSaandSbeidenticalfirstordersystems(23)ThenSisgivenby/--yo+UlcCa,b.1-10I[111(24)20-1x-i-u.Whileylxandyxarecontrollable,theyarenotindependentlycontrollable,sincetheirdifferenceisgivenbythesolutionof(25)(-)-(x-x).THEOREM5.SystemsSaandS.formrespectivelytheforwardandreturnpathsofafeedbacksystemS(seeFigure4).LetthecascadeconnectionofSaUl CONTROLLABILITYANDOBSERVABILITY137UVEI=VU/F.c,.4.FeedbacksystemSwithS,,intheforwardpathandSinthereturnpathfollowedbySbbeScandofSbfollowedbySabeSo.Assumethat(I-D,D)isnonsingular.Then:i)n=n+n,ii)anecessaryandsutcientconditionthatSbecontrollable(observable)isthatS(So)becontrollable(observable),iii)anecessarybutnotsucientconditionthatSbecontrollable(observ-able)isthatbothSaandSbecontrollable(observable),iv)ifSandSarebothcontrollable(observable)anyuncontrollableun-observable)coordinatesofSareuncontrollableunobservablecoordinatesofS(So)andoriginateinS.Beforegoingonwiththeproof,afewgeneralobservationsaremade.Thenonsingularityof(I-DD),whichisequivalenttothenonsingular-ityof(I-DbDa),isphysicallyreasonable,forifitisbrokenthestaticgainD(I-DaDb)-DD(I+DbD)-oftheclosed-loopsystemSisunde-fined.IntroductionofsystemsSandSoisanaturalconsequenceofprovingseparatelythecontrollabilityandobservabilitypartsofthetheorem.SincecontrollabilityinvolvesonlytheinfluenceoftheinputuonS,thesystemshowninFigure5asuifices.Similarly,determinationofobservabilityleadstothesystemofFigure5b.Statementsanalogoustoii)ofTheorems3and4arenotpossible,sincefeedbackalterscharacteristicroots.ByemployingaUYb(2)andtheequationsdescribingSandS,theequationsdescribingSareobtained.Inspectionoftheseequationsshowsi)istrue;however,theyaretoocomplextoyieldasimpleproofofii). 138ELMERG.GILBERT(a)(b)FIG.5.Systemsfordeterminingthecontrollability(a)andobservability(b)offeedbacksystem.Considerfirstthecontrollabilitypartofii).FromFigure53Yb)c(27)UcU-Yc.UsingtheseequationsandthenormalformequationsforStgives(xyc)(28)Ax+BuforS,where(29)Ac-Bc(30)Bflc(I+DbDa)-1.Itiseasilyshownfromthenonsingularityof(I+DbDa)-1thatarowofBwillbezeroifandonlyifthecorrespondingrowof/ciszero.ThusBhasnon-zerorowsifandonlyifSciscontrollable.Thesufficiencypartofii)isprovedbycontradiction.LetScbecontrol-lableandassumethatSisuncontrollable.ThenfromTheorem2thevectorseke,lc0,n1,i1,pcannotspanthen-dimensionalspace.Thatis,anon-zerovectorrexistssuchthat(r,e)0,(31)k-0,,n--1,i1,,p.Orequivalently,(32)r'AkBrr(Ac-B)BO,]0,...,n1.Evaluating(32)startingwithk,0givesrrB0(33)r(A-BS'c)BrAfi0rAcB(rB)cBrr(A.Bc)'-IB:-Brn,O.FromTheorem2itcanbeseenthatthecolumnsofthematricesAB, CONTROLLABILITYANDOBSERVABILITY139/c0,n1spanthen-dimensionalspaceifandonlyifBhasnozerorow.SincebythepreviousparagraphBhasnozerorow,thisand(33)implythatriszero.Thusthecontradictionisobtained.Thenecessitypartofii)isobviousfromthediscussionattheendofsection4.Theobservabilitypartofii)isprovedbystartingwithFigure5bandSo.ThenSisgivenby2=Ax(34)AAooC,C(I+DaDb)-l'yoYCxTheabovestepscanthenbeappliedtothecolumnsofCwiththedesiredresults.Theorem4appliedtothedeterminationofScandSogivesiii).Considerthecontrollabilitypartofiv).From(28),(29),and(30)itcanbeseenthatifthei-throwof/ciszero,2iXix.Thustheuncontrollablecoordinateyixiisunchangedbyfeedback.Moreover,byTheorem4thiscoordinatemustoriginateinSb.Similarargumentsgivetheobservabilitypartofiv).ThemostimportantresultofTheorem5isii).Itsaysthatclosed-loopcontrollabilityandobservabilitycanbeascertainedfromtheopen-loopsystemsScandSo.Thusoneisnotforcedtodealwithintricateclosed-loopequations.WhenSbisstaticanevensimplersituationexists.Theniv)impliesthatSiscontrollableandobservableifSaiscontrollableandobservable.FurtherinformationonuncontrollableandunobservablecoordinatescanbegleanedfromTheorems3,4,and5.LetSdenotethecombinationofsystemsS,S,andSz,thatis,thepartofsystemSwhichisnotcontrollableandobservable.FromTheorem1itisclearthatthecoordinatesof$2,S,areuncontrollableorunobservableinthecompositesystemS.ThusSu,S,arepartofSu.ToseewhathappenstotheremainingcoordinatesofSa,S,itissufficienttoexaminebymeansofTheorems3,4and5theinterconnectionofthecontrollableandobservablesystemSa*,S,*,AsanexampletakethefeedbacksystemofTheorem5"SconsistsofSaandSplusthecoordinatesofS*whichareuncontrollableinthesystemSa*followedbyS*andunobservableinthesystemSb*fol-lowedbyS*.5.Thetransfer-functionmatrix.Thetraditionalapproachtotheanalysisandsynthesisofmultivariablesystemsisbasedonthetransfer-functionmatrixratherthanthedifferentialequations(1).Toobtainatransfer-functionrepresentationofasystemS,itisassumedthattheoutputvectorv 140ELMERG.GILBERTisentirelyduetoinputforcingu,i.e.,initialconditionsarezero.LetLa-placetransformsbedenotedbyupper-caseletters.Thenw(s)wheresistheLaplace-transformvariableandH(s)[H.]istheqbyptransfer-functionmatrix.TheelementH(s)isthescalartransferfunctionwhichrelatesthei-thoutputandthej-thinput.Toobtainthetransfer-functionmatrixfromthedifferential-equationrepresentationconsider:THEOREM6.GivenasystemSdefinedbyequations1and(4),thetransjr-functionmatrixisH(s)C(IsA)-B+Dy(IsA)-I+D=1s-i*+DwherethematricesKhaveranlcone.ThefirsttwoexpressionsforHfollowdirectlyfromtheLaplacetransformof(1.)and(4)withx(0)y(0)0.Since(IsA)-1isdiagonal,thesecondexpressioncanbewrittenoutintermsofthecolumnsofy,y,andtherowsof5,5i':37II''-.D.Foranyicorrespondingoanuncontrollableorunobservablecoordinate,yor5'iszero.ThusthesumneedstobetakenonlyoverthecharacteristicrootsassociatedwithS*.K7,,,beingavectorouterproduct,isofrankone.Theimportant,andnotsurprising,conclusionofTheorem6isthatatransfer-functionmatrixrepresentsthecontrollableandobservablepartofS,S*.IthasbeennotedinTheorems4and5thatcontrollabilityandobservabilityofsubsystemsdoesnotassurethecontrollabilityandobservabil-ityofacompositesystem.Thustransfer-functionmatricesmaysatisfac-torilyrepresent&llthedynamicmodesofthesubsystemsbutfailtorepre-sentallthoseofthecompositesystem.Furthermore,thelossofhiddenresponsemodesisnoteasytodetectbecauseofthecomplexityofthetrans-fer-functionmatricesandmatrixMgebra.Sincedifferentialequationsofferasaferbasisfordescribinginultivariablesystemsitisvalidtoaskwhytransfer-functionmatricesshouldbeusedatall.Theansweristhatfre-(tuencydomaindesignproceduresandthesmallersizeofH(itisqXpratherthannXn)oftenmakecomputationsmoremanageable. CONTROLLABILITYANDOBSERVABILITY141Ifthetransfer-functionmatrixofaphysicalsystemisgivenitisgenerallyimpossibletoderivethecorrespondingdifferential-equationrepresentation.ThisisbecausethestatevariablechoiceisnotuniqueandallinformationconcerningsystemsSc,S,andSsismissing.Itispossible,however,tofindasetofdifferentialequations(1)or(4)whichyieldthesameH(s)asaprescribedH(s).Proceduresfordoingthisaredescribedbelow.Themainresultisstatedhereasatheoremandgivestherequiredorderofthedif-ferentialequations.THEOREM7.Givenarationaltransfer-functionmatrixH(s)whoseelementshavea.finitenumberoj'simplepolesats,il,minthefinites-plane.Letthepartial.fractionexpansionofHbe,(38)Hs_,K____-t-D,where(39)K,zlim(s-):)H(s),(40)DlimH(s).Lettheranofthei-thpole,ribedefinedastheran]cofKThenH(s)canberepresentedbydifferentialequations(l)or(4)whoseorderis(41)nEr,i.i-----1TheeigenvaluesofAandAaredistinctifandonlyifallr1.ItisimpossibletorepresentH(s)byadifferentialequationwhoseorderislessthann.FirstitwillbeshownhowH(s)canberepresentedbyasetofdifferentialequations.SincethematrixKisofrankrtherearer,linearlyindependentcolumnsinK.Leteel,j1,rbesuchasetofcolumns.TheneverycolumnofK,canbeexpressedsalinearcombinationofthee..Acompactnota-tionis(42)K,iE,iFwhereE,zisaqXrmatrixwhichhascolumnse..TodetermineFpre-multiply(42)byEr.Then(43)ErKErEF.ButthedeterminantofEfEistheGramdeterminant[19]ofthee.,andisnonzerobecausethec.,;arelinearlyindependent(thisisagoodtestfor 142ELMERG.GILBERTpickingalinearlyindependentseteii).ThusTE--1T(44)F(Ei)EK.OnceFisknownKcanbeexpressedaswhereffsisthej-throwofF.Thus(6)H()f='=8-XThisformulaissimilarto(37)exceptthattherearervectorouterproductsforeachX.ThusH(s)canberepresentedby(4)whereX1I1000(7)L0I-..El1ndLisidentitytrixoforderr.ThusthrootXisofmultiplioitynd:r,.Toshothatre,libationoflororderisopossible,theploetransformof(4)istken,definingtransfer-functionmtrix.However,tocover11possibilitiesitisessentialthatAtkitsmosttnrlform,Jordnormalform.Foro5motristicrootofultiplioityf,thisnstktthnumberofJordanblocksiththisohmotristiorootisnotxd,onlytStlltheblookstkentogetherformmtrixoforderf.Hoeeer,shosthatllJordanblocksmustbeoforderoneifistohvesimplpoles(unlesssomemodesreunoontrollbleorunobservble,hiohonlyincreasestheorderof(4)).Furthermore,therankoftheresidueofXisnotreaterthnthultiplioityofX.ThusifistohvthformofHin(3S)thdierntilqutios()usthvmiimuorderri.i=lIftheequations()hveordergrtrthnn,thrlitioniseitherunoontroilble,unobserwhle,orboth.Theor7providessolutionofthsynthesisproblem,sinoeonce CONTROLLABILITYANDOBSERVABILITY]_43differentialequations(1)or(4)areknown,theycanberealizedasaphysi-calsystem(example,anelectronicdifferentialanalyzer).Furthermore,thesynthesizedsystemusesaminimumnumberofdynamicelements.Theas-sumptionofsimplepolescanberelaxed,butattheexpenseofconsiderableadditionalcomplexity.Kalman[16]givesanalternativeprocedurefordeterminingn.Theorem7andasimpleexampleillustratehowtheorderofasystemrepresentedbyatransfer-functionmatrixmaybeunderestimated.Lets+ls-t-1(4s)H()=|---11s+l+(s-1)(s-2)s-t-2s+2Afirstglanceitmightbeguessedthatthesystemhasordertwo,butrl-t-r22-t-13,sotheminimumorderisthree.OnerealizationofanequivalentthirdordersystemisshowninFigure6.Itispossiblethattheactualorderofthesystemmaybegreaterthanthree.Forexample,inFigure7theorderisfive.Underestimationofsystemorderisthereasonwhymosterroneousstabilityanalyseshavegoneunnoticed.Inastabilityanalysisthenumberofcharacteristicrootsconsideredshouldatleastbeequaltothesumoftheminimumordersofallthesubsystems.ThisiseasilycheckedbymeansofTheorem7--anderrorsinmanyreferencesgavebeennoted.Ifatransferfunctionmatrixhasanypolesofrankgreaterthanone,theassumptionofdistinctcharacteristicroots,whichwasmadeinallpriordevelopments,isviolated.Ifsuchtransferfunctionsareencountered,anapproximatingsystemmaybesetup(useapproximationtoequations(47))whichhaspolesofrankone.Thenallthepreviousresultscanbeused.FromtheabovediscussionitisclearthateachelementofH(s)isanintegralpartofthewholedescription.Thusitisgenerallynotpermissibletopartitionatransfer-functionmatrixintoseveraltransferfunctionmatricesandtreattheresultingmatricesasthoughtheydescribedistinctsystems.Yet,thishasbeendoneconsistentlyintherepresentationofplantswhichhavemoreinputsthanoutputs[9,12].Asaconsequenceer-roneousstatementshavebeenmadeconcerningtheexistenceofstablefeedbacksystems.McMillan[18]definesthedegreeofasquarerationalmatrix,whichisequivalentton,butthedevelopmentismorecomplicatedbeingbasedontheSmithnormalformofapolynomialmatrix.Healsoshowsthatifthematrixisanimpedancematrix,itmaybesynthesizedbyapassivenetworkwithn,andnofewer,reactiveelements.Seeforexample[10,11].Thishasbeennotedbytheauthorinadiscussion[12]. 144-ELMERG.GILBERTU2(s)-t-v(s)FI(.6.ThirdorderrepresentationofH(s).U2(s)Fl.(.7.Fiftltorderrepresenta.tionofH(s)6.Transfer-functionrepresentationofmultivariablefeedbacksystems.Oncethelimitationsoftransfer-functionmatricesarerecognized,itispossibletoapplythemsuccessfullytotheanalysisandsynthesisoffeedbacksystems.Inwhatfollowsitwillbeassumedthatlltransfer-functionmatriceshavesimplepolesofrankone.Thiswillkeepthetransfer-functionrepresentationsconsistentwiththedifferential-equationrepresentationsspecifiedearlier.LetHaandHbetransfer-functionmatricesrepresentingSaandSinFigure4.Thenthedevelopments,(49)(50)and(1)(52)Ua--V-Vb--U-HVU-HbHaU,,(I+H,Ha)-'U,VHaUaHa(I--HbHa)-'IU,VH(U-Vb)HaU-HaHbV(1n-HHb)-HaU, CONTROLLABILITYANDOBSERVABILITY145givealternativeexpressionsforthetransfer-functionofthefeedbacksys-temS,(53)tiII(I+HbHa)-1(I-t-HaIIb)-lHa.HrepresentsthecontrollableandobservablepartofS,S*.TheremainingpartofS,S,wasconsideredattheendofsectiont.SystemsSandSnaturallyaremissingfromtherepresentationHbe-causetheyarenotrepresentedinHaandH.ThecoordinatesofS*whicharenotcontrollableand/orobservableinScorrespondtothepolesofHwhichdonotappearinHHand/orHaHb.InthederivationforHitiseasytoseewherethepolesofHarelost:(49)givesthoseofHbHaand(51)givesthoseofHH.Itisnotsoeasytoseethatnoadditionalpolesarelost,adifficultywhichhastodowithcomplexitiesinevaluatingtheinverseofamatrixofrationalfunctions.Thisisoneofthereasonsthatledtothemorecarefultreatmentofsection4.SupposethatallthesubsystemswhichmakeupHmdIIarecon-trollableandobservable.Thisisareasonableassumptioniftransfer-functionmatricesaretobeused.Thenfromtheprecedingitisplainthatthecharacteristicrootsofthefeedbacksystemaregivenby:1)thepolesofH(theserootscorrespondtothedynamicmodesinSwhicharecon-trollableandobservable),2)thepolesofHwhichdonotappearaspolesofHaHn,and/orHHa,3)thepolesofthetrm'sferfunctionsrepresentingthesubsystemsofSaandSwhichdonotappearrespectivelyinHaandHb.Inthecourseofsystemsynthesisandstabilityanalysisallcharacteristicrootsofthefeedbacksystemmustbeconsidered.Proceduresforhandlingthecharacteristicrootsincategory1)havebeendevelopedreasonablywellintheliterature.Therefore,additionaleffortherewillbedirectedat2)and3).Inparticular,theproblemofpolecancellationinmultiplyingtwotransfer-functionmatriceswillbeexplored.Thisproblemappliesdirectlyto2),andoftento3),sincethesystemsSaandSareusuallyacascadeconnectionofsubsystems.IfSorSarethemselvesfeedbacksystemstheymustfirstbeanalyzedasfeedbacksystemsbeforeprogresscanbemadeontheanalysisoftheoverallfeedbacksystem.7.Polecancellation.ConsiderthecascadeconnectionofthecontrollableandobservablesystemsSandS(nottheSaandSoftheprevioussee-tion,seeFigure3).Thetransfer-functionrepresentationgives(54)HIfHhasfewerpolesthanthesumofpolesinHaandH,polecancellationThespecialcasewhereSisstaticandSisthecascadeconnecionoftwosub-systemshasbeenconsideredin[9,12].Theresultsobtainedarenotasgeneralasthoseofthenextsection. 1.46ELMNRG.GILBERThasoccurredandthesystemSisuncontrollableorunobservable.Togofurther,amoredetailednotationisrequired.LetHabewrittenas()HohawherehaisthecharacteristicpolynomialforSa,(56)haa(tXla)(t'Xnaa)]aO.SinceithasbeenassumedthatHahassimplepolesofrankone,hahasnorepeatedlinearfactors.1-Theelementsofthematrix5Caarepolynomialsins.Suchamatrixissaidtohaveafactor,ifeveryelementofthematrixhasthesamefactor.SinceSaiscontrollableandobservable3Cahasnofactorscommonwithha.SimilarremarksapplytoHb.Usingthenotation(57)and(58)hhbha,systemSiscontrollableandobservableifhand3edonothavecommonfactors.AnylinearfactorofhcancelledinHbyalikefactorof5ecorre-spondstoanuncontrollableorunobservablemodeinS.Unlesstheele-mentsof5Caand3ebareinsomewayrelated,thepossibilitythathand5Cwillhavecommonfactorsisremote.Themostcommonsituationwhichcausesaand5CtoberelatedisthatofcompensationwhereeitherHaorHisfixed,andtheother(thecom-pensator)ischosentomakeHequaladesiredtransfer-functionmatrixH.Clearly,ifhedoesnotequalhhathecompensatedsystemSwillbeuncontrollableorunobservable.ThuscertainconstraintsmustbeimposedonHeifSistobecontrollableandobservable.OftenitissuttieienttorequirethatonlytheunstablemodesofSaandSbecontrollableandobservableinS.Thisreducesthenumberofconstraints.Thefollowingtreatmentofconstraintsassumespre-eompensation(Hfixed)andHbsquare(Pbqb).TheassumptionthatSaiscontrollableandobservableisreasonablebecauseaminimumorderrealizationofHamustbecontrollableandobservable.Also,itispointlesstoconsideranI-Iwhichcorrespondstoanuncontrollableorunobservablesystem.Formally,compensationrequires(59)HaH-I[Ie.10Herethetermfactormeansanon-consta,ntfactor. CONTROLLAB]LITYANDOBSERVABILITY147ThereforeItblmustnotbeidenticallyzero..ThisisassuredifBbandChaverankp.Expansionof(59)yields(60)It.[adjH]Hh[adjb]Letthegreatestcommondivisorofthenumeratoranddenominatorbeg.ThenandbecauseandhcannothaveacommonfactorifSistobecontrollubleandobservable.From(57),(58),(61),and(62)gild(64)hhh3CgSincehg-istheonlyfctorcommontobothhnd(andhdonothvecommonfetors)itslinearfctorsgivellthemodeswhichreuncontrollablendunobservbleinS.Supposeallunstablemodesofretobecontrollablendobservable.Thenlllinearfactorsofhwhichgotozerointheright-hlfs-planemustbeincludeding,orequiva-lentlyscommonfactorsofh3ndh[dj]3.Thishppensonlyif1)hincludestheright-hlf-plnefctorsofh,nd2)[dj]includestheright-hlf-plnefctorsof.Veryoftentheconstraintscannotbeimposedasindicated.ConsidertheexampleBothpoleshverankone.Constraint1)requireshtohvethefctor(s1).Suppose2)istobestisfiedwith3Cdiagonal.Then(66)[adjb]Cd122d--,11d822dJ"Since1(s"1),eachelementof(66)mustincludethefactor(s1),whichinthiscasemea,nsbothand(havethefactor(s1).But(s1)cannotbeacommonfactorof5Candh.Thesameproblemalsooccursif.f,0.Aswillbeseenshortly,thedicultycanbe 148ELMERG.GILBERTresolvedonlybylettingHehaveapoleofranktwoats1.Thisisthesameconstraintwhichwouldresultfromproceduresdescribedintheliterature[9,12].IthasnotbeennotedpreviouslythatitmayberelaxedifHeisnotdiagonal,afactwhichisofinterest,sincepresentdesignpro-ceduresarebasedondiagonalizationoftheopen-looptransfer-functionmatrix.TheaboveanalysecamotbeextendedreadilywhenH,H,orHehavepolesofrankgreaterthanone,becausethencommonfactorsinthenumeratorandden.onfinatorofHa,H,and.Hedonotnecessarilyimplythatthesystemsareuncontrollableorunobservable.Theorem7offersasatisfactoryalternativeapproach.SystemSiscontrollableandobservableiftheorderofSasdetermiedfromHHeisequaltotheorderofSplustheorderofS.Tosimplifytheapplicationofthisstatementthefollowingassumptionsaremade:i)HaHb,andHeallhavesimplepoles,ii)Hhaspolesofrankone,iii)S,.mdSarecontrollableandobservable,iv)He,isdiagonal.ThenSiscontrollablendobservableifndonlyif(7)Definernk[lira(s)tt0,]-t--rank[lim(sX)H,Irank[lim(sX)H,I,(68)G.H-[g,g,]andlet(0ifh.isanalyticats),,lifh.,hasasimplepoleats.Using(59)andii),(67)canbewrittenasrank[lim{(sk)hneg.'"(sh)hg,}]p,ilhk,,'"(69),i=,.pi=1Oncetheg,arecomputed,constraintsontheh.,:,zsuchthat(69)issatisfiedareeasilyfound.Forexample,withH,asdefinedby(65),(7o)g-'=-1g=sConsider1.Clearly,(69)holdsonlyif.f.f1.Thusbothhxandh,ehavesimplepolesats1.Thesameresultistrueats--1.OthervaluesofsXwhichmustbeconsideredarethosewhereH(andalsoH)haspoles.Sincefors1,gandgarelinearlyindependent,(69)will CONTROLLABILITYANDOBSERYABILITY1.49besatisfiedautomatically.Ifgl.org2hadpolestheywouldhavetobeincludedaszerosofhlldorh22d.ASbefore,itisoftensuttieienttoimposetheconstraintsonlyatXvalueswhichhavepositiverealparts,lettingsomestablemodesbeuneontrollab!eorunobservab!e.Inthisexample,theonlyactiveconstraintwouldthenbethathll,andhehavesimplepolesats1.Usually,thegareanalyticat,sX1,Xnb.Infactwithii),anecessaryandsufficientconditionforanalytieityatsXisthatK.andpEKa'bhavecolumnswhichspanthep-dimensionalcoordinatespace.Further-more,ifG(s)isanalyticatsXthenitcanbeshownthatG(X)isofrankp1.ThusforXXequation(69)issatisfiedifh,i1,phavesimplepolesatsX.Manytimes,aconsiderablylessseverecon-straintissufficient.Forexample,ifgl(M)0(g,g,arelinearlyindependent)onlyh,requiresapoleatsXb.Orsupposeg,(X)lclg.t(X,,)q-]c,g(Xb)where]cand]carearbitraryconstants;then(69)istrueif.g=1,i=1,2,3and.0,i=4,..-,p.Fin,lly,consideranexamplewhereG(s)isnotanalyticatsX.(71)Hb(72)G2s-(s-:t)(s-1)(sq-1)2(sq-1.)-(s-.)-2(s-.)(s-t-1)(sq-1)(s+3)(+3)|.I--2(s--1.)--4s!TakeX1.If.,.1(theusualconstraint[9,12]),(69)isnotsatis-fied(Sisnotcontrollableandobservableevenifamultip!epoletreatmentisconsidered);butitwillbesatisfiedif,1andh2ahasazeroats1.Ifstablemodesaretobecontrollableandobservable1421a46)k1.andhldandhmusthavezerosats-3.Thoughtheaboveisalimitedtreatment,itdoesallowsolutionofmanycompensationproblemsandindicatesthecomplexityofthesituation.Withobviousmodificationstheeaseofpos--eompensation(Hfixed)canbehandled. ]50ELMERG.GILBERT8.Conclusion.Fromtheforegoingitshouldbeconcludedthattoogreatanemphasisonoperationalmethods(thetransfer-functionmatrix)isun-wise.Differentialequations(1)arisenaturallyinrelatingthephysicalpropertiesofasystemtoitsresponsecharacteristics,andanymathematicalprocedurewhichneglectsinformationcontainedintheseequationsshouldbeviewedskeptically,itItissurprisingthatphysicalconsiderationshaveotraisedmoredoubtsaboutthetransfer-functionrepresentationearlier.Certainly,theerrorsofunderestimatedorderwouldnothaveoccurredifanyefforthadbeemadetorelatethemathematicalrepresentationtothephysicalworld--forexample,bymeansofsystemsimulation.Finally,itshouldbenotedthatthesynthesisofamultivariablefeedbacksystemistrulyaformidabletask.Unwieldycalculations,complexcompen-sationconstraints,anddifficultiesinevaluatingtheeffectofdisturbanceinputsandparametervariationsallcomplicatethesearchforsatisfactorydesignprocedures.Theresultsdevelopedaboveshouldatleastprovideasoundbasisforthissearch.Acknowledgment.TheauthorwishestothankEdwardO.GilbertandBernardS.Morganforhelpfulsuggestionsduringthewritingofthepaper.TheworkwaspartiallysupportedbytheNationalAeronauticsandSpaceAdministrationundercontractnumberNAS-8-1569.REFERENCES[1]A.S.BOKSENBOMANDI.IIoo,),GeneralAlgebraicMethodAppliedtoControlAnalysisofComplexEngineTypes,NationalAdvisoryCommitteeforAeronautics,TechnicalReport980,Washington,D.C.,1949.[2]M.(]OIOMBANDE.l.[SI)IN,Atheoryofmultidimensionalservosystems,J.Frank-linInst.,253(1952),pp.29-57.[3]It.S.TSIEN,EngineeringCybernetics,McGraw-HillBookCompany,Inc.,NewYork,1954,Ch.5,pp.53-69.[4]D.J.POVEJSIIANDA.M.Fcctts,Amcthod.forthepreliminarysynthesisqfacomplexmultiloopcontrolsystem,Trans.Amer.Inst.Elec.Engrs.,74,PartII(1955),pp.129-34.[5]R.J.KAVaNA(}-,Theapplicationofmatrixmethodstomulti2ariablecontrolsys-tems,J.FranklinInst.,262(1956),pp.349-67.[6]R.J.KAVANAG,Noninteractioninlinearmultivariablesystems,Trans.Alner.Inst.Elec.Engrs.,76,PartII(1957),pp.95-100.[7]R.J.KAVANAGH,Multivariablecontrolsystemsynthesis,Trans.Amer.Inst.Elec.Engrs.,77,PartII(1958),pp.425-429.[8]H.FREEMAN,Asynthesismethodformultipolecontrolsystems,Trans.Amer.Inst.Elec.Engrs.,76,PartII(1957),pp.28-31.[9]H.FREEMAN,Stabililyandphysicalrealizabilityconsiderationsinthesynthesis11Similarremarksapplytosampl.ed-.datasystems,exceptthevectordifferentialequationsarereplacedbyvectordifferenceequationsandtheLaplacetransformisreplacedbythez-transform. CONTROLLABILITYANDOBSERVABILITY51.ofmultipoleconlrolsystems,Trtns.Amer.Inst.Elec.Engrs.,77,PurrII(1958),pp.1-5.[10]I.M.HOROWITZ,Synthesisoflinear,multivariablefeedbackcontrolsystems,Trans.I.R.E.,Prof.GrouponAutomaticControl,AC-5,no.2(1960),pp.94-105.[11]E.V.BOHN,Stabilizationoflinearmultivariablefeedbackcontrolsystems,Trnns.I.R.E.,Prof.GrouponAutomaticControl,AC-5,no.4(1960),pp.321-327.[12]K.CN,R.A.MATHIAS,AND]).M.SAUTER,1)esignofnon-inleraclingcontrolsystensusingbodediagrams,Trans.Amer.Inst.Elec.Engrs.,80,Pa.rt,II(1(.)61),pp.336-346.1.13]R.E.KALMAN,Onthegeneraltheoryofcontrolsystems,Proc.Fi,'stInternationalCongressofAutomaticControl,Moscow,USSR,1960.[141R.E.KAMAN,Y.C.Ho,ANDK.S.NAtZN)RA,Controllabilityoflineardy.nami-cal,ystems,Contrii)utionstoI)ifferentialEquations,1,1.961.Vol.1,No.(1961),JohnWiley,NewYo'k.[151I[CtAI)BEILMAN,IntroductiontoMatrixAnalysis,McGrw-HillBookCom-pany,Inc.,NewYork,1960.1.1611R.E.KALM*N,Mathematicaldescriptionoflineardynamicalsystems,J.Soc.Indust.A1)pl.Mth.Set.A:OnControl,Vol.1,No.2(1963),pp.152-192.[171.C.Ho,Whatconstitutesacontrollablesystem?,Trtns.I.R.E.,Prof.GrouponAutomttticContr()l,AC-7,n().3(1902),p.76.[18]BROCKWAYM(M'AN,Introductiontoformalrealizabilitytheory,BellSystemTech.J.,31(1(.)52),Pl?.217--279andpp.541.-600.[]9]R.COURANTAND|).HILBERT,Methods(t]'Mathematical]'hy.ics,1,Inter-SciencePut)lishers,Inc.,NewYork,1953.