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WeiandYangAdvancesinDierenceEquations2013,2013:20http://www.
advancesindifferenceequations.
com/content/2013/1/20RESEARCHOpenAccessAnewapproachtoquantizedstabilizationofastochasticsystemwithmultiplicativenoiseLiWei*andYuanhuaYang*Correspondence:weili@mail.
sdu.
edu.
cnSchoolofControlScienceandEngineering,ShandongUniversity,Jinan,ChinaAbstractAnewquantization-dependentLyapunovfunctionisproposedtoanalyzethequantizedfeedbackstabilizationproblemofsystemswithmultiplicativenoise.
Forconvenienceoftheproof,onlyasingle-inputcaseisconsidered(whichcanbegeneralizedtoamulti-inputchannel).
Conditionsforthesystemstobequantizedmean-squarepoly-quadraticallystabilizedarederived,andtheanalysisofH∞performanceandcontrollerdesignisconductedforagivenlogarithmicquantizer.
Themostsignicantfeatureistheutilizationofaquantization-dependentLyapunovfunction,leadingtolessconservativeresults,whichisshownboththeoreticallyandthroughnumericalexamples.
Keywords:multiplicativenoise;discrete-timesystems;mean-squarestability;logarithmicquantizer;Lyapunovfunction1IntroductionRapidadvancementofdigitalnetworkshaswitnessedagrowinginterestininvestigat-ingeortsofsignalquantizationonfeedbackcontrolsystems.
Theemergingnetwork-basedcontrolsystemwhereinformationexchangebetweenthecontrollerandtheplantisthroughadigitalchannelwithlimitedcapacitieshasfurtherstrengthenedtheimportanceofthestudyonquantizedfeedbackcontrol.
Dierentfromtheclassicalcontroltheorywheredatatransmissionisassumedtohaveaninniteprecision,transmissionsubjecttoquantizationorlimiteddatacapacityindigitalnetworks,thetoolsinclassicalcontroltheorymaybeinvalid,sonewtoolsneedtobedevelopedfortheanalysisanddesignofquantizedfeedbacksystems.
Thestudyofquantizedfeedbackcontrolcanbetracedbackto[].
Mostoftheearlyre-searchfocusesontheunderstandingandmitigationofthequantizationeects,whilethequantizationerrorisconsideredtoimpairtheperformance[].
Inmoderncontrolthe-orywherethequantizerisalwaysconsideredasaninformationencoderanddecoder,onemainproblemishowmuchinformationhastobetransmittedinordertomakethesystemachieveacertainobjectivefortheclosed-loopsystem.
Foradiscrete-timesystemwithasingle-inputchannel,whenthestaticquantizerisconsidered,[]showstheminimumdatarateforthesystemtobestabilizedisprovedtobecharacterizedbytheunstablerootsofthesystemmatrix,andthecoarsestquantizerislogarithmic.
[]considersthecasewhentheinputchannelsubjecttoBernoullipacketsdropouts,theminimumdatarateisrelatednotonlytotheunstablerootsofthesystemmatrix,butalsowiththepacketsdropoutprobability.
Asforadiscrete-timesystemwithsingleinputsubjecttomultiplicativenoises2013WeiandYang;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense(http://creativecommons.
org/licenses/by/2.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
WeiandYangAdvancesinDierenceEquations2013,2013:20Page2of11http://www.
advancesindifferenceequations.
com/content/2013/1/20in[],thecoarseststaticquantizerforthesystemtobequadraticallymean-squarestabi-lizedisprovedtobelogarithmicwithinnitelevels,andthequantizationdensitycanbeapproximatedbysolvingaRiccatiequation;comprehensivestudyonfeedbackcontrolsys-temswithlogarithmicquantizersisnotgiven.
Asectorboundapproachisproposedin[]tocharacterizethequantizationerrorcausedbyalogarithmicquantizer,bywhichmanyquantizedproblemcanbesolvedbytherobusttools.
Theresultsarealsoextendedtoadaptivecontrolin[,]andtheLQR-typeproblemin[].
Basedonthecharacterizationofthequantizederror,[]giveslessconservativeconditionsofthequantizationdensitytoachievestabilitybystudyingthepropertiesofthelogarithmicquantizerfurther;[]useamethodbasedonTsypkin-typeLyapunovfunctionstostudytheabsolutestabilityanal-ysisofquantizedfeedbackcontrolofadiscrete-timelinearsystem,lessconservativecon-ditionsthanthoseinthequadraticframeworkarederived.
[]showedthatanite-levellogarithmicquantizersucestoapproachthewell-knownminimumaveragedatarateforstabilizinganunstablelineardiscrete-timesystemundertwobasicnetworkcongu-rations,andexplicitnite-levellogarithmicquantizersandthecorrespondingcontrollerstoapproachtheminimumaveragedataratearederived.
Fornetworkedsystems,[]givesthequantizedoutput-feedbackcontrollerforthecontrolwithdatapacketsdropout.
Inthispaper,anewapproachtotheanalysisandsynthesisofquantizedfeedbackcontrolforstochasticsystemswithmultiplicativenoiseisproposed.
Usinglogarithmicquantizedstate-feedbackcontrol,resultsformean-squarestabilizationandH∞performanceanal-ysisaswellasthecontrollersynthesisaregiven.
Lessconservativeresultsarederivedbytheutilizationofaquantization-dependentLyapunovfunction,whichisshownboththe-oreticallyandthroughanumericalexample.
Notations:P>(P≥)meansPisasymmetricpositive(semi-positive)matrix.
PTstandsforthetranspositionofmatrixP.
Thespaceofasquaresummableinnitese-quenceisdenotedbyl[,∞),andforw={w(t)}∈l[,∞),itsnormisgivenbyw=∞|w(t)|.
2Stabilityandstabilization2.
1ProblemformulationConsiderthefollowinglineardiscrete-timesystemswithmultiplicativenoise:x(t+)=A+Aξ(t)x(t)+B+Bξ(t)u(t),x()=x,()wherex(t)∈Rnisthesystemstatevectorwithknowninitialstatex;u(t)∈Rmisthecontrolinput;ξ(t)∈RistheprocessnoisewithEξ(t)=,Eξ(t)ξ(j)=σδtj,andisuncor-relatedwithinitialstatex.
Asprovedin[],thecoarseststaticquantizerforthesystem()tobequadraticallymean-squarestabilizedviaquantizedstate-feedbackisprovedtobelogarithmic.
Supposeuisascalarthathastobequantized,thelogarithmicquantizerisinthefollowingform:q(u)=uiif+δui,ifu=,–Q(–u)ifu,()whereρisthequantizeddensityofthelogarithmicquantizerq,whichcanbecomputedusingtheapproachgivenin[],withδ=–ρ+ρ.
()Forthemulti-inputcasewithdierentquantizers,thestate-feedbackcontrolwithoutquantizationisintheformofv(t)=Kx(t)Kx(t)···Kmx(t),()whichhastobetransmittedthroughadigitalnetworksubjecttologarithmicquantizersasgivenin(),anddenotethequantizedcontrolasu(t)=qv(t)=q(Kx(t))q(Kx(t))···qm(Kmx(t)),()whereqi,i=,.
.
.
,marequantizerswithdierentquantizationdensity.
Withoutlossofgenerality,inthispaperonlyasingle-inputcasewithm=isconsideredforsimplicity,whichcanbegeneralizedtoamulti-inputcase.
Foraquantizerasgivenintheformof(),asillustratedin[],usingthesectorboundapproach,thequantizationerrore(t)canbecharacterizedase(t)=qv(t)–v(t)=fKx(t)–Kx(t)=(t)Kx(t),()where(t)∈[–δ,δ]withδgivenby(),sotheclosed-loopsystemwithquantizedfeedbackisgivenbyx(t+)=A+Aξ(t)x(t)+B+Bξ(t)+(t)Kx(t).
()Wemainlyfocusonthederivationoflessconservativesucientconditionsforthesystemtoachievecertainperformance.
Tomakethepaperself-contained,thedenitionsforthesystem()tobemean-squarestableandmean-squarepoly-quadraticalstableareintro-duced.
DenitionTheclosedsystem()iscalledmean-squarestablewithquantizedfeedbackcontrolintheformof()ifthereexistsacontrolLyapunovfunctionVP(x)=xT(t)Px(t)satisfyingEVPx(t+)–EVPx(t),Q>,VandVsatisfying–Q[A+(–δ)BK]TViσ[A+(–δ)BK]TViQi–Vi–VTiQi–Vi–VTi,Q>,VandVsatisfying()and().
()and()():SupposethereexistmatricesQ>,Q>,VandVsatisfying()and().
First,asQi>,wehave(Vi–Qi)TQ–i(Vi–Qi)≥,whichimplies–VTiQ–iVi≤Qi–VTi–Vi.
()From()and()wehave–Q[A+(–δ)BK]TViσ[A+(–δ)BK]TVi–VTiQ–iVi–VTiQ–iVi,Q>,VandKsatisfying–Q[AV+(–δ)BK]Tσ[AV+(–δ)BK]TQi–V–VTQi–V–VTandQ>,VandKsatisfying()and().
Fromthe(,)block,weknowthatQi–V–VTQi>,soVisnonsingular.
Performingdiag{V–T,V–T,V–T}anddiag{V–,V–,V–}to()and(),respectively,yields–V–TQV–V–T[AV+(–δ)BK]TV–σV–T[AV+(–δ)BK]TV–V–TQiV––V–T–V–V–TQiV––V–T–V–,Q>,VandVsatisfying()and(),andusingthecontrollergaingivenin(),thesystem()canachievemean-squarepoly-quadraticallystability.
TheoremisbasedonTheorembysettingV=V=V,whichincreasestheconser-vativeness;thefollowingtheoremgivesalessconservativecondition.
TheoremConsiderthesystemin()andthestatefeedbackcontrollawin().
Givenalogarithmicquantizerasin(),theclosed-loopsystemin()ismean-squarepoly-quadraticallystableifthereexistmatricesQi>,Xi>,ViandKsatisfying–Q[A+(–δ)BK]T[A+(–δ)BK]T–VTi–ViVTi–XiVTi–ViVTiXi,Q>,VandV.
WhenTheo-remisusedtocomputethecoarsestquantizationdensityδmaxsuchthattheclosed-loopquantizedsystemismean-squarepoly-quadraticallystable,thatis,()and()arebilin-earmatrixinequalities.
Inthiscase,alinesearch(suchasthebisectionmethod)hastobeperformedtothevariablesδin()and(),andndδmaxiteratively,whichcanbereferredto[–].
2.
4IllustrativeexampleInthispart,anexampleisgiventoshowthatthenewproposedLyapunovfunctioncanleadtolessconservativeconditionsofthequantizationdensityforthesystemtoachievestability.
ExampleForthestochasticdiscrete-timesystem(),considerthescalarcaseofthefollowingform:A=A=.
–.
–.
.
.
–.
,B=B=T,()Eξ(t)=,Eξ(t)=σ=.
.
Itcanbeprovedthatthesystemwithoutcontrolpartisunstableinthemean-squaresense.
Supposethatthestate-feedbackin()isgivenbyK=[.
–.
],andthequantizerweuseislogarithmicintheformof().
Wewanttodeterminethemaximumsectorboundδmaxbelowwhichthestochasticsystemwithquantizedstatefeedbackismean-squareasymptoticallystable.
TablegivesthemaximumboundofδmaxusingtheLyapunovfunc-tionrelatedtothequantizationdensityproposedinthispaperandthegeneralcontrolLyapunovfunction.
WeiandYangAdvancesinDierenceEquations2013,2013:20Page9of11http://www.
advancesindifferenceequations.
com/content/2013/1/20Table1ComparisonofquantizationdensityMethodsδmaxρinfQuadraticapproach0.
44500.
3841Quantizationdependentapproach0.
49960.
33373ExtensiontoH∞performanceanalysisForthesystemx(t+)=A+Aξ(t)x(t)+B+Bξ(t)u(t)+Gw(t),()z(t)=Cx(t)+Du(t)+Fw(t),()wherethestatex(t),theinputu(t)andthesystemnoiseξ(t)aredenedasthoseofthesystem(),z(t)∈Rnisthecontroloutput.
A,A,B,B,C,D,G,Faresystemmatriceswithproperdimensions.
Supposethequantizerisgiventobelogarithmicintheformof()andthequantizationdensityisknown,sotheclosed-loopsystemwiththequantizedstatefeedbackcontrolisgivenasfollows:x(t+)=A+Aξ(t)x(t)+B+Bξ(t)+(t)Kx(t)+Gw(t),()z(t)=Cx(t)+D+(t)Kx(t)+Fw(t),()where(t)∈[–δ,δ].
DeningW={w(t)}∈l[,∞),theobjectiveofthispartistoderivetheconditionsforthesystem()and()tobemean-squareasymptoticallystablewithanH∞disturbanceattentionlevelγ,thatis,z(t),Q=QT>,VandVsatisfying–Q[A+(–δ)BK]TVi[C+(–δ)DK]Tσ[A+(–δ)BK]TVi–γIGTViFTQi–Vi–VTi–IQi–Vi–VTi<,()–Q[A+(+δ)BK]TVi[C+(+δ)DK]Tσ[A+(+δ)BK]TVi–γIGTViFTQi–Vi–VTi–IQi–Vi–VTi<,i∈{,}.
()ProofThetheoremisprovenbasedontheLyapunovfunctiondenedin().
First,()and()imply()and(),whichguaranteestheclosed-loopsystemin()and()toWeiandYangAdvancesinDierenceEquations2013,2013:20Page10of11http://www.
advancesindifferenceequations.
com/content/2013/1/20bemean-squarestablebyTheorem.
ToprovetheH∞performance,assumezeroinitialconditionsandconsiderthefollowingindex:=∞EzT(t)z(t)–γEwT(t)w(t)≤∞EzT(t)z(t)–γEwT(t)w(t)+EVx(t),()whereEVx(t)=ExT(t+)Q(t+)x(t+)–xT(t)Q(t)x(t).
()Then,alongthesolutionsof()and(),wehave=∞ηT(t)η(t),()withη(t)=x(t)w(t),=,where=A++(t)BKTQ(t+)A++(t)BK–Q(t)+σA++(t)BKTQ(t+)A++(t)BK+C++(t)DKTC++(t)DK,()=A++(t)BKTQ(t+)G+C++(t)DKTF,()=GTQ(t+)G+FTF–γI.
()Ontheotherhand,bysimilarreasoningasintheproofofTheorem,wecanconcludefrom()and()that<.
Thenfrom()weknowthatTheproofiscompleted.
4ConclusionTheproblemofquantizedstate-feedbackcontrolforastochasticsystemwithmulti-plicativenoiseshasbeeninvestigatedthroughaquantization-dependentapproach.
Con-ditionsformean-squarepoly-quadraticalstabilityareobtainedbyintroducinganewquantization-dependentLyapunovfunctionapproachforlinearstatefeedbackwithalog-arithmicquantizer,whichareshowntobelessconservativethanthosederivedbyacom-monLyapunovfunction.
Moreover,H∞performanceanalysishasalsobeenproposedinthequantization-dependentframework.
However,itisworthpointingoutthatthoughlessconservativeconditionsareobtained,dierentfromthederivationofthecoarsestquan-tizer,theexplicitrelationofthesystemmatricesandquantizationdensityisnotgiven.
Theanalysisofrelationbetweenthequantizationdensityandthesystemmatricesandthestatisticalpropertiesofnoisesintheproposedquantization-dependentframeworkisasubjectworthfurtherresearching.
WeiandYangAdvancesinDierenceEquations2013,2013:20Page11of11http://www.
advancesindifferenceequations.
com/content/2013/1/20CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsLWcarriedouttheproofofthemainpartofthisarticle,YYcorrectedthemanuscriptandparticipatedinitsdesignandcoordination.
Allauthorshavereadandapprovedthenalmanuscript.
AcknowledgementsWewouldliketothanktheeditor-in-chief,theassociateeditorandthereviewersfortheirvaluablecommentsonthepaperwhichhaveledtosignicantimprovementonthepresentationandqualityofthepaper.
ThisworkissupportedbytheTaishanScholarConstructionEngineeringbyShandongGovernment,theNationalNaturalScienceFoundation(No.
61174141),andtheMajorStateBasicResearchDevelopmentProgramofChina(973Program)(No.
2009cb320600),YangtseRiveScholarBonusSchemes(No.
31400080963017),NationalNaturalScienceFoundation(No.
61034007).
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AdvancesinDierenceEquations20132013:20.