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1、5SingularandCyclicNormalSystems5.1SingularDiscreteSystemsandCyclicPairsConsiderthefollowingdiscretesystemEABxxu=+,i≠=D{0,1,...},(5.1.1a)ii++1iyxu=+CD(5.1.1b)iiinmpwherexi≠,ui≠,yi≠arethestate,inputandoutputvectors,respectively,atnìnnìmpìnpìmthediscreteinstanti,andE,A≠,B≠,C≠,D≠.Thesystem
2、(5.1.1)iscalledsingularifdetE=0andstandardifdetEò0.WeassumethatdetE=0anddet[EAzz-ò]0,forsome≠-(thecomplexnumbersfield).(5.1.2)Asystemoftheform(5.1.1)satisfyingthecondition(5.1.2)iscalledaregularsystem.Thetransfermatrixofthesystem(5.1.1)isgivenby-1TC()zz=-+[]EABD.(5.1.3)Thismatrixcanbew
3、ritteninthestandardformP()zT()z=,(5.1.4)dz()pìmpìmwhereP(z)≠[z]([z]isthesetofpolynomialmatricesofdimensionspìm),d(z)istheminimalmoniccommondenominatorofalltheelementsofthematrixT(z).256PolynomialandRationalMatricesApplyingelementaryoperationsonrowsandcolumns,wecanreducethepìmmatrixP(z)
4、≠[z]toitsSmithcanonicalformpmìPSr()zi=≠diag[12(),zi(),...,(),0,...,0ziz]<[]z,wherei1(z),…,ir(z)arethemonicinvariantpolynomialssatisfyingthedivisibilityconditionik+1(z)
5、ik(z),k=1,…,r-1(thepolynomialik(z)divideswithoutremainderthepolynomialik+1(z),k=1,…,r-1),andr=rankP(z).Theinvariantpol
6、ynomialsaregivenbyDz()kizk()==()Dz0()1,k=1,...,r,(5.1.6)Dz()k-1whereDk(z)isagreatestcommondivisorofallthek-thorderminorsofthematrixP(s).Thecharacteristicpolynomialj(z)=det[Ez–A]ofthepair(E,A)andtheminimalpolynomialY(s)arerelatedinthefollowingwayj()zY=()z.(5.1.7)Dz()n-1Definition5.1.1.(
7、E,A)iscalledacyclicpairifandonlyifY(z)=j(z).Itfollowsfrom(5.1.6)that(E,A)isacyclicpairifandonlyifDz()1orequivalently=n-1(5.1.8)iziz()====()?iz()1,()iz=Y=()zdz().12rr-1nìnTheorem5.1.1.(E,A)isacyclicpairifthematricesE=[eij]≠andnìnA=[aij]≠satisfyoneofthefollowingconditions=>0forji+1ej=>0,
8、forianda®i,j=1,...,n(5.1.9a)ijij¯ò=0forji+1or=>0forij+1ei=>0,forjanda®i,j=1,...,n.(5.1.9b)ijij¯ò=0forij+1Proof.Ifthecondition(5.1.9a)issatisfied,thentheminorMn1(obtainedbydeletionofthen-throwandfirstcolumn)ofthematrix[Ez–A]equalsMn1=a12a23…an-1,nò0.ThusDn-1(z)=1andfrom(5.1.7),wehavej
