A Note on Minimality of Positive Realizations
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676IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.45,NO.6,JUNE1998
inasystemparameterortheswitchinginstantareverysmall,leading
tominimaldisturbancetothecurrentcontrolledoperationofthe
converter.Wedonotmakeanyclaimsofgeneralityoroptimality
ofourcontrolalgorithms.Ourmainfocushasbeentoextractthe
embeddedfixedpointsinPEswitchingsystemsandtostabilizethem
withminimumcomputationalburden.
REFERENCES
[1]J.H.B.DeaneandD.C.Hamill,“Instability,subharmonics,andchaosinpowerelectronicscircuits,”IEEETrans.PowerElectron.,vol.5,no.3,pp.260–268,1990.[2]K.Chakrabarty,G.Podder,andS.Banerjee,“Bifurcationbehaviorofbuckconverter,”IEEETrans.PowerElectron.,vol.11,no.3,pp.439–447,1995.[3]K.Pyragas,“Continuouscontrolofchaosbyself-controllingfeedback,”Phys.Lett.A,vol.170,pp.421–428,1992.[4]G.ChenandX.Dong,“Onfeedbackcontrolofchaoticnonlineardynamicsystems,”Int.J.BifurcationandChaos,vol.2,no.2,pp.407–411,1992.[5]K.PyragasandA.Tama˘sevi˘cius,“Experimentalcontrolofchaosbydelayedself-controllingfeedback,”Phys.Lett.A,vol.180,pp.99–102,1993.[6]E.Ott,C.Grebogi,andJ.A.Yorke,“Controllingchaos,”Phys.Rev.Lett.,vol.64,no.11,pp.1196–1199,1990.[7]E.R.Hunt,“Stabilizinghigh-periodorbitsinachaoticsystem:Thedioderesonator,”Phys.Rev.Lett.,vol.67,no.15,pp.1953–1955,1991.[8]G.ChenandX.Dong,“Controlofchaos—Asurvey,”inProc.32ndConf.DecisionandControl,SanAntonio,TX,1993,pp.469–474.[9]G.Podder,K.Chakrabarty,andS.Banerjee,“Controlofchaosintheboostconverter,”Electron.Lett.,vol.31,no.11,pp.25,1995.
ANoteonMinimalityofPositiveRealizations
LucaBenvenutiandLorenzoFarina
Abstract—Awell-knownresultfromlinearsystemtheorystatesthattheminimalinnersizeofafactorizationoftheHankelmatrixHofasystemgivestheminimalorderofarealization.Inthisbriefitisshownthatwhendealingwithpositivelinearsystems,theexistenceofafactorizationoftheHankelmatrixintotwononnegativematricesisonlyanecessaryconditionfortheexistenceofapositiverealizationoforderequaltotheinnersizeofthefactorization.NecessaryandsufficientconditionsfortheminimalityofapositiverealizationintermsofpositivefactorizationoftheHankelmatrixarethenderived.
IndexTerms—Factorization,positivesystems,realization.
I.INTRODUCTION
Thepositiverealizationproblem[4],[7]isthefollowing:Givena
rationaltransferfunction
G(z)=bn01zn01+111+b0zn+an01zn01+111+a0=
k1gkz0k
ManuscriptreceivedMay28,1996;revisedJune9,1996,January7,1997,andJuly22,1997.ThispaperwasrecommendedbyAssociateEditorO.Raimund.TheauthorsarewiththeDipartimentodiInformaticaeSistemistica,Universit`adiRoma“LaSapienza,”00184Roma,Italy(e-mail:luca@riscdis.ing.uniroma1.it).PublisherItemIdentifierS1057-7122(98)02803-7.findatripleA2IRN2N+,b,c2IRN+forwhich
G(z)=cT(zI0A)01b
whereIRN2N+andIRN+denoteN2NmatricesandN-vectorswith
allentriesnonnegative.Thesystemdescribedbythetriple(A;b;c)
isalsoknownasapositivesystem[6],[9].Recently,Andersonetal.
[1]gaveanessentiallycompletesolutionoftheexistenceofsucha
realization(acompletesolutionisgivenin[3]).Aspointedoutin
[1],animportantopenquestionisminimality,i.e.,findingapositive
realizationofminimalorderN.Itisnoteworthythatitcanbeeasily
shown[8]thatNdoesnotgenerallycoincidewiththeMcMillan
degreen.
Awell-knownresultfromsystemtheorystatesthattheminimal
innersizeofafactorizationoftheHankelmatrixHgivestheminimal
orderofarealization.Since,obviously,theimpulseresponseofa
positivesystemisnonnegative,i.e.,gk0,thenHhasnonnegative
entries.Moreover,givenaminimalpositiverealization(A;b;c)of
orderN,thefollowinghold:
H
=cTbcTAbcTA2b111
cTAbcTA2bcTA3b111
cTA2bcTA3b...........
.
=cT
cTA
cTA2..
.(bAbA2b111)
=RS(1)
whereR2IR12N+andS2IRN21+.Asaconsequenceof
thepreviousconsiderations,itisinterestingtostudywhetherthe
factorizationofHintotwononnegativematricesofminimalinner
sizeNimpliestheminimalorderofapositiverealizationtobeN.We
shownext,bymeansofanexample,thatthisisnottrue.Considerthe
systemwithnonnegativeimpulseresponseg1+4i=10,g2+4i=8,
g3+4i=6,g4+4i=8fori=0;1;111,describedby
x(k+1)=100
0001
010u(k)
y(k)=(111)x(k)
forwhich,fromtheFrobeniustheorem[2],athird-orderpositive
realizationdoesnotexist.Nevertheless,apositivefactorizationof
theHankelmatrixhavinginnersizeequaltothreedoesexist.Tosee
this,considertheHankelmatrixofthesystem
H
=IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.45,NO.6,JUNE1998677
MatrixHijcanbefactorizedas
Hij
=533
426
355
462
Ht..
.x2x3...xn3
