the problem of realization

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4TheProblemofRealization4.1BasicNotionsandProblemFormulationConsideracontinuoussystemgivenbytheequationsx"=+ABxu,(4.1.1a)y=+CDxu,(4.1.1b)nmpwherex≠,u≠,y≠arethestate,theinputandtheoutputvectors,nìnnìmpìnpìmrespectively,andA≠,B≠,C≠andD≠.Thetransfermatrixofthesystem(4.1.1)isgivenby-1TC()ss=-+[]IABD.(4.1.2)ForthegivenmatricesA,B,CandDthereexistsonlyonetransfermatrix(4.1.2).Ontheotherhand,foragivenpropertransfermatrixT(s)therearemanymatricesA,B,CandDsatisfying(4.1.2).nìnnìmpìnDefinition4.1.1.Thequadrupletofthematrices:A≠,B≠,C≠andpìmD≠satisfying(4.1.2),iscalledarealisationofthegiventransfermatrixpìmT(s)≠(s).ItwillbedenotedRn,m,p(T)orbrieflyRn,m,p.Definition4.1.2.ArealisationRn,m,piscalledminimalifthematrixAhastheminimal(least)dimensionamongallrealisationsofT(s).AminimalrealisationwillbedenotedbyRn,m,p.Definition4.1.3.AminimalrealisationRn,m,piscalledcyclic(orsimple)ifthematrixAiscyclic.AcyclicrealisationwillbedenotedbyRˆn,m,p. 220PolynomialandRationalMatricesThematrixDforagivenpropertransfermatrixT(s)canbecomputedusingtheformulaDT=lim()s,(4.1.3)sçÑwhichresultsfrom(4.1.2),since-1lim[]IAs-=0.sçÑFrom(4.1.2)and(4.1.3),wehave-1TTDsp()ss=-()=-C[]IsAB.(4.1.4)HavingthepropermatrixT(s)andusing(4.1.4)wecancomputethestrictlypropermatrixTsp(s).Therealisationproblemcanbeformulatedinthefollowingway.pìmWithaproperrationalmatrixT(s)≠(s)given,computetherealisationRn,m,pofthismatrix.Theminimalrealisationproblemcanbeformulatedinthefollowingway.pìmWithaproperrationalmatrixT(s)≠(s)given,computeaminimalrealisationRn,m,pofthismatrix.Theproblemofcyclicrealisationisformulatedasfollows.WithaproperpìmrationalmatrixT(s)≠(s)given,computeacyclicrealisationRˆn,m,pofthismatrix.pìmInthecaseofastrictlypropertransfermatrixTsp(s)≠(s),therealisationproblemreducestothecomputationofonlythreematricesA,B,Csatisfying(4.1.4).4.2ExistenceofMinimalandCyclicRealisations4.2.1ExistenceofMinimalRealisationsThetheoremstatedbelowprovidesuswithnecessaryandsufficientconditionsfortheexistenceofaminimalrealisationRn,m,pforagivenrationalpropertransferpìmmatrixT(s)≠(s).Theorem4.2.1.Arealisation(A,B,C,D)ofamatrixT(s)isminimalifandonlyif(A,B)isacontrollablepairand(A,C)isanobservablepair.Proof.Wewillshowbycontradictionthatif(A,B)isacontrollablepairand(A,C)isanobservablepair,thentherealisationisminimal. TheProblemofRealization221nìnnnìLet(A,B,C),A≠and(A,B,C),A≠ofthematrixT(s).From(4.1.4),wehave-1-1CIABCIAB[]ss-=-»º¬¼(4.2.1)andiiCAB==CAB,i0,1,....(4.2.2)Fromtheassumptionthat(A,B)and(A,B)arecontrollablepairsandthat(A,C)and(A,C)areobservablepairs,itfollowsthatrankrankSH==n,(4.2.3a)rankrankSH==n,(4.2.3b)where»ºC…»CASBA==[]B>ABHn-1,…»,(4.2.3c)…»@…»n-1¬¼CA»ºC…»CAS=BAB…AB»ºn-1,H=…».(4.2.3d)¬¼…»@…»n-1…»¬¼CAFrom(4.2.2)wehaven-1»ºC»ºCBCAB>CAB…»…»2nCACABCAB>CAB…»n-1…»HS==»ºBAB…AB…»@¬¼…»@@B@…»…»CAn-1CABCABnn-1CA21()n-B¬¼…»¬¼>n-1»ºCBCAB>CAB»ºC…»2n…»CABCAB>CABCA…»…»n-1==»ºBAB>ABHS=…»@@B@…»@¬¼…»…»nn-121()n-n-1…»¬¼CABCAB>CAB…»¬¼CAand 222PolynomialandRationalMatricesrankHS=rankHS.(4.2.4)TherelationshipsrankHS=n,rankHS=nand(4.2.4)leadtoacontradictionsincebyassumptionnn>.Nowwewillshowthatif(A,B)isnotacontrollablepairor/and(A,C)isnotanobservablepair,then(A,B,C)isnotaminimalrealisation.If(A,B)isnotacontrollablepair,thenthereexistsanonsingularmatrixPsuchthat--11»ºAA12»ºB1A==PAP…»,B=PB===…»,CCP[]C12C,¬¼0A3¬¼0(4.2.5)ABA≠≠≠<<0…»001>0…»A=…»@@@B@.(4.3.1)F…»…»000>1…»---aaa>-a¬¼012n-1GivenT(s)andusing(4.1.3)wecancomputethematrixD,andinturnthestrictlyproperrationalmatrix-1L(s)TTDsp()ss=-()=-C[]IsAFB=.(4.3.2)ms()Thustheproblemisreducedtocomputingaminimalrealization(AF,B,C)≠Rˆn,m,ppìmofthestrictlypropermatrixTsp(s)≠(s). TheProblemofRealization227Thecharacteristicpolynomialm(s)ofthematrix(4.3.1),whichisequaltotheminimalpolynomialY(s),hastheformnn-1ms()=Y()s=det[IAs-]=sas+n-11+>+asa+0.(4.3.3)OnecaneasilyshowthatAdj[Is-AF]ofthematrix(4.3.1)hastheform»ºs-10>00…»0100s->…»Adj[]IAs-=FAdj…»@@@B@@…»…»000>s-1(4.3.4)…»aaa>asa+¬¼012nn--21»ºws()1nnì=≠…»<[]s,¬¼M()sks()wherews()(=»¬¼msmnn--12)(s)>ms1()º,T21n-ks()=»º¬¼ss>s,(4.3.5)nn--12mssasnn--11()=++++>a2sa1nn--23mssasnn--21()=++++>a3sa2mssa11()=+n-(n-1)ì(n-1)andM(s)≠[s]isapolynomialmatrixdependingonthecoefficientsa0,a1,…,an-1.-1Inordertoperformthestructuraldecompositionoftheinverse[Is-AF],wereducethematrix(4.3.4)totheform(3.4.14).Tothisend,wepre-multiplythematrix(4.3.4)by»º1011,n-U()s=…»(4.3.6a)¬¼-ks()In-1andpost-multiplyitbytheunimodularmatrix»º0n,--11In1V()s=…».(4.3.6b)¬¼1-ws()Nowweobtain 228PolynomialandRationalMatrices»º1011,n-UI()ssAdj[]-=AFV()s…»,(4.3.7)¬¼0n,-11M()s-ksws()()-1where[Is-AF]isanormalmatrix.Everynonzerosecond-orderminorisdivisiblewithoutremainderbym(s).ThuseveryentryofM(s)=M(s)–k(s)w(s)isdivisiblewithoutremainderbym(s).Therefore,wehaveMM()s=≠ms()()ˆˆs,M()s<()nn-ì-11()[s].(4.3.8)Takingintoaccountthat--11»º1011,n-»ws()1ºUV()ss==…»,()…»(4.3.9)¬¼ks()In-1¬Inn--110,1¼aswellas(4.3.8)and(4.3.7),weobtain--11»º1011,n-Adj[]IAUs-=F()ss…»V()=…»¬¼0n,-11ms()()Mˆs(4.3.10)=+PQFF()ss()m()ssGF(),where»º1»º1-1PUF()ss==()…»…»,¬¼¬¼0n,-11ks()-1QVF,()ss==»º¬¼1011n-()»º¬¼w()s1,(4.3.11)--11»º0011,n--»º011,n0GUF()ss==()…»V()s…».…»¬¼00n,--11MMˆˆ()ss…»¬¼()n,11From(4.3.2)and(4.3.10),wehaveLCIA(ss)=-=Adj[FF]BCPQBC(s)F(s)+m(s)GBF(s)=(4.3.12)=+PQ()()ssm()()ssG,where TheProblemofRealization229»º1PC()ss==PCF()…»,¬¼ks()QQB()ssw==F()»º¬¼()s1,B(4.3.13)GC()ss=GBF().LetCibethei-thcolumnofthematrixC,andBithei-throwofthematrixB,i=1,2,…,n.Takingintoaccount(4.3.13)and(4.3.5)weobtain»º1…»sPC()s=[]12CC>n…»…»@…»n-1¬¼snn-12=++CCss>>C=++++PPPssPs,12nn123»ºB1…»(4.3.14)BQ()s=»m()sm()s>ms()1º…»2¬¼nn--121…»@…»B¬¼n=++BB11msmsnn--()22()>++BBnn-1ms1()nn--12n-3=+++++BB11sa()nn--1BB2saa(21n-1B2B3)s21n->>+++()aa11BB22+=BQn1++++QQ23sss>QnwherePC==,forin1,2,>,,(4.3.15a)iiQBQ==,,aaBBQ+=B+aBB+,>,nn111--n1222n--n11n-23(4.3.15b)QBB=+++aa>aBB+.11122nn--11nWithQn,Qn-1,...,Q1knownwecanrecursivelycomputefrom(4.3.15b)therowsBi,i=1,2,…,nofthematrixBBQBQ==,,-aaBBQ=-B-aB,>,12nn--1n113n--2n21n-12(4.3.17)BQBB=----aa>aB.nn11122--11nFromtheaboveconsiderationswecanderivethefollowingprocedureforcomputingthedesiredcyclicrealisation(AF,B,C,D)ofagiventransfermatrixpìnT(s)≠(s). 230PolynomialandRationalMatricesProcedure4.3.1.pìmStep1:Using(4.1.3),computethematrixD≠andthestrictlypropermatrix(4.3.2).Step2:Withthecoefficientsa0,a1,…,an-1ofthepolynomialm(s)known,computethematrixAFgivenby(4.3.1).Step3:PerformingthedecompositionofthepolynomialmatrixL(s),computethematricesP(s)andQ(s).Step4:Using(4.3.15a)and(4.3.17),computethematricesCandB.Example4.3.1.UsingProcedure4.3.1,computethecyclicrealisationoftherationalmatrix3321»º--+ss12sss+++2T()s=…».(4.3.18)323232sss+++21¬¼ssssss++22252+++Itiseasytocheckthatthematrix(4.3.18)isnormal.Thusitscyclicrealizationexists.UsingProcedure4.3.1,wecomputeStep1:Using(4.1.3)and(4.3.2),weobtain»º-11DT==lim()s…»(4.3.19)sçѬ¼12and21»ºss++21TTDsp()ss=-()=32…».(4.3.20)sss+++21¬¼-1sStep2:Inthiscase,a0=1,a1=2,a2=1and»º010…»A=001.(4.3.21)F…»…»¬¼---121Step3:Inordertoperformthestructuraldecompositionofthematrix2»ºss++21L()s=…»¬¼-1sitsufficestointerchangeitscolumns,i.e.,topost-multiplyitby TheProblemofRealization231»º01V()s=…»¬¼10andcomputeP(s)andQ(s)22»ºss++21»º01»º1ss++2LV()()ss==…»…»…»¬¼--11ss¬¼10¬¼»º102»0º=+…»»º¬¼12ss++…32»,¬¼ss¬02----ss1¼thatis»º1022»1ºPQ()ss==…»,()»º¬¼1s+s+=2…»»º¬¼s+s+21.¬¼s¬10¼Step4:Takingintoaccountthat»º»º10PP()s=+=+12Pss…»…»¬¼¬¼01and22QQ(s)=++=123QQss[2110]+[]s+[10]s,from(4.3.15a)and(4.3.17),weobtain»º100»º»ºCP11==…»,CP22==…»,CP33==…»,BQ13==[]10,¬¼010¬¼¬¼BQB222=-=a1[][][]1011000,-=BQBB311=--=aa122[][][]2121001.-=HencethedesiredmatricesBandCare»º»ºB110…»…»»º100BB==…»…»2100,CCCC=[]23=…».(4.3.22)¬¼010…»…»B01¬¼¬¼3 232PolynomialandRationalMatricesItiseasytocheckthat(AF,B)(determinedby(4.3.21)and(4.3.22))isacontrollablepairand(AF,C)isanobservablepair.Thustheobtainedrealisationiscyclic.4.3.2ComputationofaCyclicRealisationwithMatrixAintheJordanCanonicalFormTheproblemofcomputingthecyclicrealisation(AJ,B,C,D)≠Rˆn,m,pofagiventransfermatrixT(s)withthematrixAJintheJordancanonicalformcanbeformulatedasfollows.pìmGivenanormalrationalmatrixT(s)≠(s),computetheminimalrealisation(AJ,B,C,D)≠Rn,m,pwiththematrixAJintheJordancanonicalform»ºJ100>…»00J>AJ==…»2diag»ºJ>J,(4.3.23a)J…»@@B@¬¼12p…»…»¬¼00>Jpwith»ºsi10>00…»010s>0…»iJÅ=≠…»@@@B@@s1…»i…»000>0s¬¼i(4.3.23b)»ºsi00>00…»100s>0…»iJÅ=≠…»01s>001s¬¼iwherei=1,2,…,p,ands1,s2,…,sparedifferentpoleswithmultiplicitiesm1.m2,……,mp,respectively,pƒmni=i=1ofthematrixT(s).WiththematrixT(s)given,andusing(4.1.3)wecomputethematrixD,andthenthestrictlyproperrationalmatrix(4.1.4). TheProblemofRealization233Theproblemhasbeenreducedtothecomputationoftheminimalrealizationpìm(AJ,B,C)≠Rn,m,pofthestrictlypropermatrixTsp(s)≠(s).Firstlyconsiderthecaseofpolesofmultiplicity1(m1=m2=…=mp=1)ofthematrixL()sT()s=,spms()wheremsssss()=-(12)(-)>>(ssss-nij),ò,forijijò,,=1,,,n(4.3.24)ands1,s2,…,snarerealnumbers.Inthiscase,Tsw(s)canbeexpressedinthefollowingformnTiTsp()s=ƒ,(4.3.25)i=1s-siwhereL(si)TTii=-lim()(ssssp)=n,in=1,>,.(4.3.26)ssçi¥()ssij-j=1jiòFrom(4.3.26)and(3.4.11)itfollowsthatrankT==1,in1,>,.(4.3.27)iWedecomposethematrixTiintotheproductofthetwomatricesBiandCiofrankequalto1TCB==,rankCrankB=1,in=1,>,.(4.3.28)iiiiiWewillshowthatthematrices»ºB1…»BAB==diag[]ss>>s,…»1,C=[]CCC(4.3.29)J12nn11…»@…»B¬¼nareaminimalrealisationofthematrixTsw(s). 234PolynomialandRationalMatricesTothisend,wecompute-1CIA[s-=J]B»ºB1…»=»º111…»B1[]CC11>>Cndiag…»ssss--ss-…»@¬¼12n…»B¬¼nnnCBTiii===ƒƒTsp()s.ii==11ss--iissThusthematrices(4.3.29)arearealisationofthematrixTsp(s).Itiseasytocheckthat»ºss-00>B11…»00ss->Brank[]IABs-=rank…»22=nJ…»@@B@@…»¬¼00>ssB-nnforalls≠,sincerankBi=1fori=1,…,n.Analogouslytotheabove»ºs-s100>…»00s-s>»ºIAs-…»2Jrank…»==rank…»@@B@n¬¼C…»00>s-s…»n…»CC>C¬¼12nforalls≠,sincerankCi=1fori=1,…,n.Thus(AJ,B)isacontrollablepairand(AJ,C)isanobservablepair.Hencetherealisation(4.3.29)isminimal.Thedesiredcyclicrealisation(4.3.29)canbecomputedusingthefollowingprocedure.Procedure4.3.2.Step1:Using(4.3.26)computethematricesTifori=1,…,n.Step2:DecomposethematricesTiintotheproduct(4.3.28)ofthematricesBiandCi,i=1,…,n.Step3:Computethedesiredcyclicrealisation(4.3.29).Example4.3.2.Giventhenormalstrictlypropermatrix TheProblemofRealization235»º11…»ss++111»ss+22+ºTsw()s==…»1112…12»,(4.3.30)…»()ss++()¬s+¼…»¬¼()sss+++121()computeitscyclicrealisation(AJ,B,C).Inthiscase,m(s)=(s+1)(s+2)andthematrix(4.3.30)hastherealpoless1=-1ands2=-2.UsingProcedure4.3.2weobtainthefollowing.Step1:Using(4.3.26),weobtain»º11»º11TT11=-lim()(sssps)=…»1=…»,ssç1…»1¬¼11¬¼s+2s=-1»ºss++22(4.3.31)…»ss++11»00ºTT22=-lim()(sssps)=…»=…».ssç2…»121s+-¬0¼…»¬¼ss++11s=-2Step2:Wedecomposethematrices(4.3.31)intotheproducts(4.3.28)»º11»1º»º00TC11==…»B1,CBTC1=…»,12=[]11,==…»2B2,¬¼11¬1¼¬¼-10»º0CB22==…»,[]10.¬¼-1Step3:Thusthedesiredcyclicrealisationofthematrix(4.3.30)is»ºs0»º-10»Bº»11º11AB==…»…»,,=…»=…»J¬¼0s¬¼02-¬B¼¬10¼22(4.3.32)»º10CCC==[]12…».¬¼11-IfthematrixTsp(s)hascomplexconjugatedpoles,thenusingProcedure4.3.2,weobtainthecyclicrealisation(4.3.29)withcomplexentries.Inordertoobtainarealisationwithrealentries,weadditionallytransformthecomplexrealisation(4.3.29)bythesimilaritytransformation.Lettheequationm(s)=0haverdistinctrealrootss1,s2,…,srandqdistinctpairsofcomplexconjugatedrootsa1+jb1,a1–jb1,…,aq+jbq,aq-jbq,r+q=n.Letthecomplexrealisation(4.3.29)havetheform 236PolynomialandRationalMatricesA=+diag[ss……sajbaj-baj+baj-b],J12r1111qqqq»ºB1…»@…»…»Br…»…»cj+d(4.3.33)11B=,…»cj-d…»11…»@…»cj+d…»qq…»cj-d¬¼qqCCC=+»º……Cgjhgj-hgj+hgj-h.¬¼12rq1111qqqInthiscase,thesimilaritytransformationmatrixPhastheform1»º1jnnìPD=≠diag1[]……111DC,D1=…».(4.3.34)2¬¼1-jUsing(4.3.33)and(4.3.34),weobtain-1AJJ==PAPdiag[ss11……rAAq],»ºB1…»@…»…»Br…»2c-1…»1BPB==,(4.3.35)…»2d1…»…»@…»2c…»q…»¬¼2dqCCPC==»º¬¼11……Crqg-hgh1-q,since»ºaj+-b0»abº-1kkkkAD==…»D…»,k11¬¼0aj-b¬ba¼kkkk(4.3.36)-1»ºcjkk+d»2ckºDD1…»=…»,.[]gjkkkk+-=-hgjh1[ghkk]¬¼cjkk-d¬2dk¼Thustherealisation(4.3.35)hasonlyrealentries. TheProblemofRealization237Example4.3.3.Giventhenormalmatrix1»º13s+Tsp()s=32…»2,(4.3.37)sss+++342¬¼-+ss42computeitsrealcyclicrealisation(AJ,B,C).Thematrix(4.3.37)hasonerealroots1=-1andthepairofthecomplexconjugatedrootss2=-1+j,s3=-1-jsince32(sssss-123)(-)(-=++-++=+++sssj)(11)()(sjsss1)342.ApplyingProcedure4.3.2,weobtainthefollowing.Step1:Using(4.3.26)weobtain1»º131s+»2ºTT11=-lim()(sssps)=2…»2=…»,ssç1ss++22¬¼-+ss42¬-12-¼s=-1TT=-lim()(sss)=22spssç2»º111»º13s+--1-j==…»…»22,(4.3.38)+++-+2…»()ssj11()¬¼ss42sj=-+1--jj12¬¼TT33=-lim()(sssps)=ssç3»º111»º13sj+--1+==…»…»22.2()ssj++11(-)¬¼-+ss42…»sj=--1¬¼jj12+Step2:Decomposingthematrices(4.3.38)intotheproducts(4.3.28),weobtain»º12»1ºTC11====…»B1,,C1…»B1[]12,¬¼--12¬-1¼»º11--1-j»º1»º11TC22=…»22===B2,,C2…»B2…»--1-j,…»¬¼22j¬¼2¬¼--jj12»º11--1+j»º1»º11TC==…»22B,,1C=…»B=…»--+j.33333…»¬¼-22j¬¼2¬¼jj12+Step3:Thedesiredcyclicrealisation(4.3.29)withcomplexentriesis 238PolynomialandRationalMatrices»ºs00»-100º1…»…»A==000sj-10+,J…»2…»…»00sj…00--1»¬¼3¬¼»º(4.3.39)…»12»ºB1…»…»11»º111BB==…»2…»--1,-j.CCCC=[]123=…»…»22¬¼--12j2j…»¬¼B…»311…»--1+j¬¼22Inordertocomputearealrealization,weperformthesimilaritytransformation(4.3.34)ontherealisation(4.3.39)»º100…»…»…»11PD==diag1[]10j.…»22…»11…»0-j…»¬¼22Using(4.3.35),weobtain-1»º»º100100…»…»…»»º-100…»-1…»11…»…»11APAJJ==P…»0022jj…»-1+0…»022j…»¬¼…»001--j…»1111…»00--j…»j…»¬¼22…»¬¼22»º-100…»011=--,…»…»¬¼011-»º»º10012…»…»…»…»»º12-1…»111…»1…»BPB==01jj---=--12,…»222…»2…»…»…»…»¬¼01-1111…»01---jj…»+…»¬¼222¬…»2¼ TheProblemofRealization239»º100…»…»»º111…»11»110ºCC==P…»0.j=…»¬¼--12jj2…»22¬--102¼…»11…»0-j…»¬¼22Letinageneralcasepmmmpmsss()(=-)(12ss-)…()ss-,mn=,12piƒi=1wheres1,s2,…,sparerealorcomplexconjugatedpoles.Inthiscase,thematrixTsw(s)canbeexpressedaspmiTijTsp()s=ƒƒmj-+1,(4.3.40)ss-iij==11()iwherej-1=-1d»ºmiTT()(sss).(4.3.41)ij-j-1¬¼isp|ss=()j1!dsiLetonlyoneJordanblockJioftheform(4.3.23b)correspondtothei-thpolesiwithmultiplicitymi,andthematricesBandChavetheform»ºB1…»B==…»B2,CCC»º…C(4.3.42a)…»¬¼12p@…»…»¬¼Bpwhere»ºBi1…»BB==…»i2,,CCC»º……Cip=1,2,,.(4.3.42b)i…»@iii¬¼12imi…»B…»¬¼imiTakingintoaccountthat 240PolynomialandRationalMatrices»º111……»ss-ss--2ssmi…»i()ii()…»11-1…»0…m-1[]IJs-=…»ss-()ss-i,1,2,ip=…,,(4.3.43)iii…»@@B@…»…»1…»00…¬¼ss-iwecanwrite-1CIJBii[]s-=i11mmii-11(4.3.44)=+ƒƒCBikik2CBikik+11+…+miCBiimi.ss-ikk==11()ss--ii()ssAcomparisonof(4.3.40)to(4.3.44)yieldsjTCij==ƒikBi,mjki-+,forip1,……,,jm=1,,i.(4.3.45)k=1From(4.3.45)forj=1,weobtainTCBii11=imi.(4.3.46)WiththematrixTi1given,wedecomposeitintothecolumnmatrixCi1andtherowmatrixB.Nowfor(4.3.45),withj=2,weobtainimiTCi2112=+iBi,mii-CBii,m.(4.3.47)WithTi2andCi1,Bi,mknown,wetakeasthevectorCi2thiscolumnofthematrixiTi2thatcorrespondstothefirstnonzeroentryofthematrixBi,mandwemultiplyitibythereciprocalofthisentry.Thenwecompute()1TTCiii222=-BCi,miii=11Bi,m-(4.3.48)andBi,mi-1fortheknownvectorCi1.From(4.3.45),forj=3,wehaveTCi312213=++iBi,miii--CBCBii,mii,m.(4.3.49) TheProblemofRealization241WithTi3andCi2,Bi,m-1known,wecancomputeiTTCB=-=+CBCB(4.3.50)i3321123iii,miii--ii,mii,mandthen,inthesamewayasCi2,wecanchooseCi3andcomputeBi,m-2.Pursuingitheprocedurefurther,wecancomputeCi1Ci2,…,Ci,mandBi1Bi2,…,Bi,m.iiIfthestructuraldecompositionofthematrixL(s)ofthefollowingformisgivenLPQ(s)(=+ssm)()(ss)G(),(4.3.51)thenmmL(s)()(s-=ssiTP)=+()()(sQGss-=ss)i()ip>(4.3.52)iswii,1,,,msi()wherems()Q(s)msii()==m,Q()s.(4.3.53)()ss-ims()iiTakingintoaccount(4.3.53),wecanwrite(4.3.41)inthefollowingformj-11dTPij==j-1»º¬¼()()sQisiss=ifor1,>>,,pj=1,,mi,(4.3.54)()j!-1dssincej-1d»º()(s-=ssmiG)0forjm=1,>>,,ip=1,,.dsj-1¬¼iiss=iFrom(4.3.54)itfollowsthatthematricesTijdependonlyonthematricesP(s)andQ(s)anddonotdependonthematrixG(s).KnowingP(s)andQ(s)andusing(4.3.54),wecancomputethematricesTijfori=1,…,pandj=1,…,mi.Itiseasytocheckthatforthematrices(AJ,B,C)determinedby(4.3.23)and(4.3.42),(AJ,B)isacontrollablepairand(AJ,C)isanobservablepair.Thusthesematricesconstituteacyclicrealisation.Ifthepoless1,s2,…,sparecomplexconjugated,then,accordingto(4.3.34),inordertoobtainarealcyclicrealisationonehastotransformthembythesimilaritytransformation.Fromtheaboveconsiderations,onecanderivethefollowingimportantprocedureforcomputingthecyclicrealisation(AJ,B,C)foragivennormal,strictlypropermatrixTsw(s)withmultiplepoles. 242PolynomialandRationalMatricesProcedure4.3.3.Step1:Computethepoless1,s2,…,spofthematrixTsp(s)andtheirmultiplicitiesm1,m2,…,mp.Step2:Using(4.3.41)or(4.3.54)computethematricesTijfori=1,…,pandj=1,…,mi.Step3:Usingtheprocedureestablishedabove,computethecolumnsCi1Ci2,…,Ci,mofthematrixCiandtherowsBi1Bi2,…,Bi,mofthematrixBiiifori=1,…,p.Step4:Using(4.3.23)and(4.3.42)computethedesiredrealisation(AJ,B,C).Example4.3.3.Giventhenormalmatrix221»º()ss+-11()+Tsp()s=22…»,(4.3.55)()ss++12()…»¬¼()ss++12()()s+2computeitscyclicrealisation(AJ,B,C).ApplyingProcedure4.3.3,weobtainthefollowing.Step1:Thematrix(4.3.55)hasthetwodoublerealpoles:s1=1,m1=2,s2=-2,m2=2.Step2:Using(4.3.41),weobtain2TT11=+()(ss1sp)ss=1221»º()ss+-11()+»º00==…»…»,2()s+2…»¬¼()ss++12()()s+2¬¼01s=-1d2T12=+»º¬¼()(sT1sps)ss=1ds½»º+-22+»ºd°°1()ss11()00==®¾…»…»,2ds°°¯¿()s+2…»¬¼()ss++12()()s+2¬¼11-s=-12TT21=+()(ss2sp)ss=2221»()ss+-11()+º»º11-=…»=…»,2()s+1¬…»()ss++12()()s+2¼s=-2¬¼00d2T22=+»º¬¼()(sT2sps)ss=2ds½»º+-22+»ºd°°1()ss11()00==®¾…»…».2ds°°¯¿()s+1…»¬¼()ss++12()()s+2¬¼-11s=-2 TheProblemofRealization243Step3:Using(4.3.46)and(4.3.47),weobtain»º00»0ºTC11==…»11B12,,C11==…»B12[]01,¬¼01¬1¼»º00TC==+…»BCB.1211111212¬¼11-Wechoose»º00»0º»º0»º00CC12==…»thus11B11T12-C12B12=…»…»-[]01=…»,¬¼--11¬1¼¬¼-1¬¼10B11=[]10,»º11-»1ºTC==…»B,,C=…»B=-[]11,2121222122¬¼00¬0¼»º00TC==+…»BCB.2221212222¬¼-11Wechoose»º00»0º»º0»º00CC22==…»thus21B21T22-C22B22=…»…»-[]1-1=…»,¬¼--11¬1¼¬¼-1¬¼00B21=[]00.Step4:Using(4.3.23)and(4.3.42),weobtainthedesiredrealisation»º»-1100»ºB10º11…»……»»0100-B01AB==…»…,,…»12=»J…»…0021-…»B00»21…»……»»¬¼¬0001--¬¼B11¼22»º0010CCCCC==[]11122122…».¬¼1101--Aquestionarises:Isitpossible,usingthesimilaritytransformation,toobtainacyclicrealisationfromanoncyclicrealisationandviceversa?Thefollowingtheoremprovidesuswiththeanswer.-1-1Theorem4.3.1.Arealisation(PAP,PB,CP,D)≠Rn,m,pforanarbitrarynonsingularmatrixPisacyclicrealisationifandonlyif(A,B,C,D)≠Rn,m,pisacyclicrealisation.-1-1Proof.AccordingtoTheorem4.2.2(PAP,PB,CP,D)isaminimalrealisationifandonlyif(A,B,C)isaminimalrealisation.WewillshowthatthesimilaritytransformationdoesnotchangetheinvariantpolynomialsofA.LetUandVbethe 244PolynomialandRationalMatricesunimodularmatricesofelementaryoperationsontherowsandcolumnsof[Is–A]transformingthismatrixtoitsSmithcanonicalform,i.e.,[IAUIAs-=]()ss[-]V(s).(4.3.56)S-1LetU(s)=U(s)PandV(s)=PV(s).U(s)andV(s)arealsounimodular-1matricesforanynonsingularmatrixP,sincedetU(s)=detU(s)detPand-1detV(s)=detPdetV(s),withdetPanddetPindependentofthevariables.We-1willshowthatthematricesU(s)andV(s)reducethematrix[Is–PAP]toitsSmithcanonicalform[Is–A]S.UsingthedefinitionofU(s)andV(s),and(4.3.56),weobtain--11-1U(s)[]Iss-=-=PAPV()U(s)PPI[ssAPPV]()=-=UIA()ss[]VIA()ss[]-.S-1Thusthematrices[Is–PAP],[Is–A]havethesameinvariantpolynomials.-1-1Hence(PAP,PB,CP,D)isacyclicrealisationifandonlyif(A,B,C,D)isacyclicrealisation.¢4.4StructuralStabilityandComputationoftheNormalTransferMatrix4.4.1StructuralControllabilityofCyclicMatricesnìnAmatrixA≠iscalledacyclicmatrixifitsminimalpolynomialY(s)coincideswithitscharacteristicpolynomial,Y(s)=det[Is–A].nìnDefinition4.4.1.A≠iscalledastructurallystablematrixifandonlyiftherenìnexistsuchapositivenumbere0thatforanymatrixB≠andanyesatisfyingthecondition|e|0.nìnnìn2.IfA≠hasrankA=r,thenrank[A+B]írforthematrixB≠satisfyingthecondition(4.4.1).NoncyclicmatricesarenotstructurallystablebutforanoncyclicmatrixnìnnìnA≠onecanalwayschooseamatrixB≠andasufficientlysmallnumbere(|e|>0)sothatthesumA+Beisacyclicmatrix.OnlyforaparticularchoiceofthematrixBandeisthesumA+Beanoncyclicmatrix.Asitisknown,amatrixintheFrobeniuscanonicalform»º010>0…»001>0…»A=…»@@@B@(4.4.2)…»…»000>1…»---aaa>-a¬¼012n-1isacyclicmatrixregardlessofthevaluesofthecoefficientsa0,a1,a2,..,an-1.Forexample,thematrix»º110…»A=010(4.4.3)…»…»¬¼00aisacyclicmatrixforallthevaluesofthecoefficientaò1,anditisanoncyclicmatrixonlyfora=1.nìnLetDA≠beregardedasadisturbance(uncertainty)tothenominalmatrixnìnA≠,andtakeeB=DA.Then,accordingtoTheorem4.4.1,sinceAiscyclic,thematrixA+DAisalsocyclic.4.4.2StructuralStabilityofCyclicRealisationAminimalrealisation(A,B,C,D)≠Rˆn,m,pwiththecyclicmatrixAiscalledacyclicrealisation.Theorem4.4.2.Let(A1,B1,C1,D1)≠Rn,m,pbeacyclicrealisationand(A2,B2,C2,D2)≠Rn,m,panotherrealisationofthesamedimensions.Thenthereexistsuchanumbere0>0thatalltherealisations()AA12++++eeee,,,BB12CC12DD12≠Rn,m,pfore