procedurelet美人2

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
52,NO.
9,SEPTEMBER20071731Finite-Time-ConvergentDifferentiatorBasedonSingularPerturbationTechniqueXinhuaWang,ZengqiangChen,andGengYangAbstract—Anite-time-convergentdifferentiatorispresentedthatisbasedonsingularperturbationtechnique.
Themeritsofthisdifferentiatorexistinthreeaspects:rapidlynite-timeconvergencecomparedwithothertypicaldifferentiators;nochatteringphenomenon;andbesidesthederiva-tivesofthederivablesignals,thegeneralizedderivativesofsomeclassesofsignalscanbeobtained—forexample,thegeneralizedderivativeofatriangularwaveissquarewave,etc.
Thetheoreticalresultsareconrmedbycomputersimulations.
IndexTerms—Chatteringphenomenon,differentiator,nite-time-con-vergent,generalizedderivative,singularperturbation.
I.
INTRODUCTIONDifferentiationofsignalsisanoldandwell-knownproblem[1]–[3]andhasattractedmoreattentioninrecentyears[4]–[6].
Rapidlyandaccuratelyobtainingthevelocitiesandaccelerationsoftrackedtargetsiscrucialforseveralkindsofsystemswithcorrectandtimelyperfor-mance,suchasthemissile-interceptionsystemindefencesystems[7]andunderwatervehiclesystems[8].
Therefore,theimprovementsoftheconvergenttimeandtheprecisionofthedifferentiatorareverysignicant.
Severalworkshavebeenintroduced.
Thesimplestoneislineardifferentiator[1]–[4].
Kahlil[4]designedalinearhigh-gainob-server,whichprovidestheestimatesofderivativesoftheoutputuptoordern01.
Becausetheconvergentvelocitiesofthestatevariablesareslowinthenonlinearregionofthesystemdynamics,thetime-laggingphenomenonisinevitable.
Levant[5],[6]proposedadifferentiatorviasecond-order(orhigh-order)slidingmodesalgorithm.
TheinformationoneneedstoknowonthesignalisanupperboundforLipschitz'scon-stantofthederivativeofthesignal.
Thisconstrainsthetypesofinputsignals.
Forthisdifferentiator,thechatteringphenomenonisinevitable.
Thedevelopmentofsingularperturbationtechniqueprovidesanef-fectivemethodforrapidlytrackingcontrol[9]–[14].
Inaddition,therehavemanyresultsonnite-timestability.
Aclassofboundedcontin-uoustime-invariantstabilizingfeedbacklawisgivenfortheintegrator[15].
Andthemainresultin[16]–[18]isgivenontheconstructionofcontinuoustime-invariantnite-timecontrollersforaclassofnonlinearsystems.
Thisclassofcontinuousnonsmoothcontrollercouldimprovethetransientbehaviorandrobustnesspropertyofconsideredsystems.
Inthispaper,wepresentanalgorithmofnite-time-convergentdif-ferentiatorbasedonsingularperturbationtechnique.
Inthistypeofdif-ferentiator,thenite-time-convergentfeatureisbasedonthetheoryoftimeoptimization[15]–[18]andthewholesystemisinfasttimescale;therefore,therapidconvergenceisguaranteed.
Bycomparingotherapproaches[4]–[6],thepresentednite-time-convergentdifferentiatorManuscriptreceivedJune10,2005;revisedMarch2,2006,July19,2006,andMay28,2007.
RecommendedbyAssociateEditorD.
Nesic.
ThisworkwassupportedbytheNationalNaturalScienceFoundationofChinaunderGrant60504005.
X.
WangwaswiththeDepartmentofAutomation,TsinghuaUniversity,Bei-jing100084,China.
HeisnowwiththeDepartmentofAutomaticControl,Bei-jingUniversityofAeronauticsandAstronautics,Beijing100083,China(e-mail:wangxinhua04@gmail.
com).
Z.
CheniswiththeDepartmentofAutomation,NankaiUniversity,Tianjin300071,China(e-mail:chenzq@nankai.
edu.
cn).
G.
YangiswiththeDepartmentofAutomation,TsinghuaUniversity,Beijing100084,China(e-mail:yanggeng@tsinghua.
edu.
cn).
DigitalObjectIdentier10.
1109/TAC.
2007.
904290cankeepmorerapid-convergentvelocityandcanbeadaptedtothesys-temsrequiringrapidconvergence.
Evenwithusingsaturatingvariousterms,nite-timestabilityisstillguaranteedandthechatteringphe-nomenoncanbeavoided.
Moreover,forthispresenteddifferentiator,thegeneralizedderivativesofsomeclassesofsignalscanbeobtained.
Thispaperisorganizedinthefollowingformat.
InSectionII,a-nite-time-convergentdifferentiatorisdesignedbasedonsingularper-turbationtechniqueandnite-timestability.
InSectionIII,thesimula-tionsaregiven,andourconclusionsaremadeinSectionIV.
II.
FINITE-TIME-CONVERGENTDIFFERENTIATORInordertodecreaseconvergenttimeandavoidthechatteringphe-nomenon,wedesignatypeofdifferentiatorbasedontheilluminationofsolvingforahigh-orderdifferentialequation.
Thebasicideais:con-structahigh-orderdifferentialequationandmakeaprocedureasshowninFig.
1.
Denition1[19]:"Generalizedderivative"denotesthattheleftandrightderivativesofapointinafunctiontrajectorybothexist,andtheymaybenotequaltoeachother.
Denition2[15]:Consideratime-invariantsystemintheformof_x=f(x)f(0)=0;x2Rn(1)wheref:D!
RniscontinuousonopenneighborhoodDRnoftheorigin.
Theoriginissaidtobeanite-time-stableequilibriumoftheabovesystemifthereexistsanopenneighborhoodNDoftheoriginandafunctionTf:Nnf0g!
(0;1),calledthesettling-timefunction,suchthatthefollowingstatementshold.
1)Finite-time-convergence:Foreveryx2Nnf0g,xistheowstartingfromxanddenedon[0;Tf(x)),x(t)2Nnf0gforallt2[0;Tf(x)),andlimt!
T(x)x(t)=0.
2)Lyapunovstability:ForeveryopenneighborhoodU"ofzero,thereexistsanopensubsetUofNcontainingzerosuchthat,foreveryx2Unf0g,x(t)2U"forallt2[0;Tf(x)).
Theoriginissaidtobeagloballynite-time-stableequilibriumifitisanite-time-stableequilibriumwithD=N=Rn.
Thenthesystemissaidtobenite-time-convergentwithrespecttotheorigin.
Assumption1:Supposetheoriginisanite-time-stableequilibrium[15,Th.
4.
3]of_z1=z2111_zn01=zn_zn=f(z1;z2;zn)(2)andthesettling-timefunctionTfiscontinuousatzero,wheref()iscontinuousandf(0)=0.
LetNbeasinDenition2andlet2(0;1).
ThenthereexistsacontinuousfunctionVsuchthat1)Vispositivedeniteand2)_VisrealvaluedandcontinuousonNandthereexistsc>0suchthat_V+cV0:(3)Assumption2:ThereexistsaLipschitzLyapunovfunctionVsatis-fying(3)withLipschitzconstantM.
Assumption3:For(2),thereexisti2(0;1],i=0;1;n01,andanonnegativeconstantasuchthatjf(z1;z2;zn)0f(z1;z2;zn)jani=1jzi0zij(4)wherezi;zi2R,i=1;n.
0018-9286/$25.
002007IEEE1732IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
52,NO.
9,SEPTEMBER2007Fig.
1.
Finite-time-convergentdifferentiator.
Remark1:Thereareanumberofnonlinearfunctionsactuallysatisfyingthisassumption.
Forexample,onesuchfunctionisxsincejx0xj210jx0xj,i2(0;1].
Moreover,therearesmoothfunctionsalsosatisfyingthisproperty.
Infact,itiseasytoverifythatjsinx0sinxj2jx0xjforanyi2(0;1].
Assumption4:v(t)isacontinuousandpiecewisen-orderderiv-ablesignalwiththefollowingproperties.
Thederivativesofv(t)uptoordern02existonthewholetimedomainandv(t)isnot(n01)-orderdifferentiableatsomeinstantstj,j=1;.
.
.
;k.
How-ever,the(n01)-orderleftderivativev(n01)0(tj)andrightderivativev(n01)+(tj)exist[19],respectively,andv(n01)0(tj)6=v(n01)+(tj),j=1;.
.
.
;k,maybesatised.
FromtheconjectureinFig.
1,weobtainTheorem1asfollows.
Theorem1:If(2)issatisedwithAssumptions1–3,signalv(t)issatisedwithAssumption4.
Thenforsystemdx1dt=x2111dxn01dt=xn"ndxndt=fx10v(t);"x2;.
.
.
;"n01xn(5)thereexist>0(where>n)and0>0suchthatxi0v(i01)(t)=O("0i+1)(6)fortj>ttj01+"0(4(")e+(tj01)),j=1;.
.
.
;k+1,(lett0=0andtk+1=1)i=1;.
.
.
;nwithxn(tj)0v(n01)0(tj)=O("0n+1);j=1;.
.
.
;k(7)where">0istheperturbationparameterandO("0i+1)de-notestheapproximationof"0i+1order[13]betweenxiandv(i01)(t);and=(10)=,2(0;minf=(+n);1=2g),n2.
ei=xi0v(i01),i=1;.
.
.
;n,e=[e1.
.
.
en]T,e+(tj01)=e1(tj01).
.
.
en01(tj01)e+n(tj01)T,e+n(tj01)=xi(tj01)0v(i01)+(tj01),and4(")=diagf1;n01g.
Proof:Throughouttheproof,wewillassume"0istheup-boundnessofjdiv=dtij,i=1;.
.
.
;n01;andhn>0istheup-boundnessofthen-orderderivativeofv(t)(includingthen-ordergeneralizedderivativeatt=tj01).
Let=minfiigand=an01i=1hi+hn.
WehaveD+(Vz)()_V+"Man01i=1hi+hn=_V+"M:(15)Itisrequiredthattheperturbationin(13)iscontinuous,andthisrequirementissatisedoneachtimedomain[tj01;tj),j=1;.
.
.
;k+1.
Therefore,from[15,Th.
5.
2andProp.
2.
4],thereexists0>0(0>Tf)suchthatkz()k(V(z()))10rc(10)2"Mc(10)=rc(10)(16)fortj=">tj01="+0(z(tj01=")),j=1;.
.
.
;k+1.
Tomake2"M=c1,let"3=minf[c=(2M)]1=;1gand"2(0;"3),l=1=(rc(10))>0;and=2M=c,=(10)=.
Fromthecoordinatetransform(12),weget[e1"e2.
.
.
"n01en]T"l(17)fortj>ttj01+"0(4(")e(tj01)),j=1;.
.
.
;k+1.
Therefore,wehavejeij"0i+1l;i=1;.
.
.
;n(18)fortj>ttj01+"0(4(")e(tj01)),j=1;.
.
.
;k+1,i.
e.
,xi0v(i01)(t)=O("0i+1);i=1;.
.
.
;n(19)fortj>ttj01+"0(4(")e(tj01)),j=1;.
.
.
;k+1.
Moreover,becausev(n01)+(tj01)andv(n01)0(tj)arerightandleftcontinuous,respectively,wehavexn(tj)0v(n01)0(tj)=O("0n+1);j=1;.
.
.
;k:(20)Especially,whenv(t)isacontinuousn-orderderivablesignal,wehaveaconclusionthatxi0v(i01)(t)=O("0i+1),i=1;.
.
.
;n,fort"0(4(")e(0)).
From[15,Th.
5.
2],welet20;min+n;12;n2:(21)Importantly,from[15,Th.
4.
3],canbechosentobearbitrarilysmall.
Hencetherequirementthatliein2(0;minf=(+n);1=2g)isnotrestrictive.
Accordingly,dueto0.
Therefore,0i+1>1fori=1;.
.
.
;n.
Thisconcludestheproof.
Remark2:If(2)issatisedwithAssumptions1–4andjzi(2)jjzi(1)jwhen1ttj01+"0(4(")e+(tj01)),j=1;.
.
.
;k+1,i=n,respectively,withxn(tj)0v(n01)0(tj)=O("0n+1),j=1;.
.
.
;k.
Fromtheanalysisabove,theconvergenttimeofthepresenteddifferentiatorisdecidedbythetime0andtheperturbationparameter".
Inaddition,canbechosentobearbitrarilysmall;thenisarbitrarilylarge.
From(16),wechoose"2(0;"3);therefore,jeijisarbitrarilysmall.
Thesmaller"is,theshortertheconvergenttimeisandsmallerthetrackingerrorsare.
Forthisdifferentiator,becausetheconvergenttimeissufcientlyshortandtrackingerrorissufcientlysmallbychoosing"2(0;"3),thegeneralizedderivativesofsomeclassesofcontinuousandpiecewisederivablesignals(forexample,triangularwave)canbealsoobtainedrapidlybesidethoseoftheusualcontinuousderivablesignals.
D+(Vz)()=@V@z(z)z0fz1;z2+dvd;.
.
.
;zn+dn01vdn010dnvdnT=@V@z(z)[z0f(z1;.
.
.
;zn)]T+@V@z(z)z0fz1;z2+dvd;.
.
.
;zn+dn01vdn010dnvdnT0[z0f(z1;.
.
.
;zn)]T_V+Man01i=1divdi+Mdnvdn=_V+Man01i=1"idivdti+"ndnvdtn_V+Man01i=1hi"i+hn"n(14)1734IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
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Finite-Time-ConvergentSecond-OrderDifferentiatorAnite-time-convergentsecond-orderdifferentiatorisdesignedasdx1dt=x2"2dx2dt=uy=x2(22)u=0sat"sign((x10v(t);"x2)2j((x10v(t);"x2)j=(20)0sat"fsign(x2)j"x2jg(23)where(x10v(t);"x2)=x10v(t)+sign(x2)j"x2j2020sat"(x)=x;jxj0(where>2and=minf;=(20)g==(20)),suchthatxi0v(i01)(t)=O("0i+1)(25)fort"0(4(")e(0)),i=1,andtj>ttj01+"0(4(")e+(tj01)),j=1;.
.
.
;k+1,i=2,respectively,withx2(tj)0v00(tj)=O("01),j=1;.
.
.
;k,where2(0;1).
Infact,weknowthat[16]_x1=x2_x2=0sat"sign((x1;x2)j((x1;x2)j=(20)0sat"fsign(x2)jx2jg(26)isnite-time-convergentwithrespecttotheoriginwithina-nitetimeTf,anditissatisedwithAssumptions1–4.
From[16]and[18,Ex.
3.
2],wehavethatjxi(2)jjxi(1)jwhen10k>0;i=1;.
.
.
;n(30)0=r2;(i+1)ri+1(i01+1)ri>0;i=1;.
.
.
;n02;n01>0(31)constantli>0fori=1;.
.
.
;n,andsig(x)=jxjsign(x)(32)foracontinuousandpiecewisen-orderderivablesignalv(t),thereex-ists>0[where>nand=n01i=0(ri+1+k)=(ri+1i)],suchthatxi0v(i01)(t)=O("0i+1)(33)fortj>ttj01+"0(4(")e+(tj01)),j=1;.
.
.
;k+1,i=1;.
.
.
;n,withxn(tj)0v(n01)0(tj)=O("0n+1),j=1;.
.
.
;k.
Infact,weknowthat[17],[18](34)and(35),asshownatthebottomofthepage,arenite-time-convergentwithrespecttotheoriginwithinanitetimeTf,andtheyaresatisedwithAssumptions1–4(letmi=1dx1dt=x2111dxn01dt=xn"ndxndt=un(28)u0=0ui+1(x10v(t);.
.
.
;"ixi+1)=0li+1sigsig"ixi+10siguix10v(t);.
.
.
;"i01xi(r+k)=(r);i=0;.
.
.
;n01;n2(29)_x1=x2111_xn01=xn_xn=un(34)u0=0ui+1(x1;.
.
.
;xi+1)=0li+1sigsig(xi+1)0sig(ui(x1;.
.
.
;xi))(r+k)=(r);i=0;.
.
.
;n01;n2(35)IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
52,NO.
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FromTheorem1,weobtained(33)forthedifferentiator(28)and(29).
III.
SIMULATIONSInthefollowingsimulations,weselectthefunctionsofsintandtriangularwave,respectively,asthesignalv(t).
Moreover,usinganIntelCeleronMProcessor1.
50-GHz760-MBcomputer,weconsiderthetimecomputationsforthedifferentiatorsanddifferentvaluesof",andtheinputsignal"sint"isillustrated.
Theinitialconditionsarex1(0)=0,x2(0)=0,andx3(0)=0.
1)Two-slidingalgorithmdifferentiator[5]dx1dt=x201jx10v(t)j(1=2)sign(x10v(t))dx2dt=02sign(x10v(t))y=x2where1=1:6=0:04and2=5=0:04.
2)Linearhigh-gaindifferentiator[4]dx1dt=x20(x10v(t))0:04dx2dt=02(x10v(t))0:04y=x2:3)Finite-time-convergentsecond-orderdifferentiatordx1dt=x2"2dx2dt=0sat1x10v(t)+35("x2)(5=3)(1=5)0sat1f("x2)(1=3)gy=x2sat1(x)=x;jxj<1sign(x);jxj1:4)Thenite-time-convergentthird-orderdifferentiatordx1dt=x2dx2dt=x30:043dx3dt=023=54x10v(t)+(0:04x2)9=71=3040:042x33=5y1=x2y2=x3:Fromthesimulationsabove,wendthatthechatteringphenom-enonhappensinthetwo-slidingdifferentiator,andsteadyerrorap-pearsobviously[seeFig.
2(b)and(c)].
In[5],selecting1=6and2=8,steadyerrordoesnotappear;however,slowconvergenceandthechatteringphenomenonexist.
Theintensivevibrationhappensinthehigh-gaindifferentiator[seeFig.
3(b)and(c)].
Rapidandhigh-pre-cisiontrackingcanbeguaranteedforthenite-time-convergentdif-ferentiator[seeFig.
4(a)–(d)].
Bydecreasingtheperturbationparam-eter",thecomputedderivativebecomesbetterandbetter[seeFig.
4(b)(where"=0:04and"=0:1,respectively)],andthequalityindexofcomputedderivative(i.
e.
,themaximumabsolutevalue)becomesbiggerandbigger.
Whentheblockofsimulinkisadoptedtocarryoutthederivativeofthetriangularwaveor"sint,"slowconvergencehap-pensobviously[seeFig.
4(b)].
Fig.
5showsthesimulationcurvesofthepresentedthird-orderdiffer-entiator.
Itmeansthatx2andx3tracktheidealrst-orderandsecond-Fig.
2.
Two-slidingdifferentiator.
(a)Tracking"sint"wave(computationtime:23s)(b)Themagniedgureof(a).
(c)Tracking"triangular"wave.
orderderivativesoftheinput(sint),respectively.
Notimelaggingphe-nomenonhappensobviously.
Inaddition,thenonlineardifferentialequationsbecomecomplicatedcomparedwithlineardifferentialequations,andsonumericalproblemsforthesimulationcomeout.
However,thedevelopmentofhigh-per-formanceprocessorsandthealgorithmsofnumericalsolutionsofnon-1736IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
52,NO.
9,SEPTEMBER2007Fig.
3.
Linearhigh-gaindifferentiator.
(a)Tracking"sint"wave(computationtime:1s)(b)Themagniedgureof(a).
(c)Tracking"triangular"wave.
lineardifferentialequationsprovideusanappropriatebasementofthealgorithmsofthenonlineardifferentiators.
Somesimulationmethodsareexploited,whichhaveasatisfyingexecutiontime;forexample,thefunctionof'ode45'inMATLABboxisquitetforthenonlineardiffer-entialequations.
Moreover,forthispresentednonlineardifferentiator,Fig.
4.
Finite-time-convergentsecond-orderdifferentiator.
(a)Tracking"sint"wave("=0:04(computationtime:3s);"=0:1(computationtime:1.
5s).
(b)Themagniedgureof(a).
(c)Tracking"triangular"wave("=0:04).
(d)Themagniedgureof(c).
somesimplepowerfunctionsareused,whichareeasytocomputeinsimulationorinoating-pointdigitalsignalprocessors.
IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.
52,NO.
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5.
Finite-time-convergentthird-orderdifferentiator.
IV.
CONCLUSIONSInthispaper,wepresentanalgorithmofnite-time-convergentdifferentiatorbasedonsingularperturbationtechnique.
Becausethetheoryoftimeoptimizationandthesingularperturbationareintroducedintothepresenteddifferentiator,itcankeeprapid-conver-gentvelocityandcanbeadaptedtothesystems,whichneedrapidconvergence.
Withtheintroductionofsaturatingvariousterms,thechatteringphenomenoncanbeavoidedandthenite-time-stabilitycanbestillguaranteed.
Besidesthederivativesofthederivablesignals,thegeneralizedderivativesofsomeclassesofsignalscanbeobtainedthroughthistypeofdifferentiator.
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SolvingRank-ConstrainedLMIProblemsWithApplicationtoReduced-OrderOutputFeedbackStabilizationSeog-JooKim,Young-HyunMoon,andSoonmanKwonAbstract—Thispaperpresentsaniterativepenaltyfunctionmethodforsolvingrank-constrainedlinearmatrixinequality(LMI)problemsandil-lustratesitsapplicationtoreduced-orderoutputfeedbackstabilization.
Weproposeapenalizedobjectivefunctiontoreplacetherankcondition,sothatasolutiontotheoriginalnonconvexLMIfeasibilityproblemcanbeobtainedbysolvingaseriesofconvexLMIoptimizationsubproblems.
Nu-mericalexperimentswereperformedtodemonstratetheproposedmethod.
IndexTerms—Linearmatrixinequality(LMI),penaltyfunctionmethod,rankcondition,reduced-orderoutputfeedback.
I.
INTRODUCTIONReduced-orderoutputfeedback(ROF)stabilizationoflineartime-invariant(LTI)systemsisoneofthemostchallengingproblemsincon-trolengineering(seesurveypaper[1]).
ROFstabilizationischaracter-izedasarank-constrainedlinearmatrixinequality(LMI)problemintheconventionalLMIframeworkbythecelebratedeliminationlemma(e.
g.
,[2]),wheretwoLyapunovmatricesarecoupledwitharankcon-dition[3],[4].
Theproblembecomesnonconvexanddifculttosolveduetothiscouplingrankcondition.
Overthelastdecade,manyiterativetechniqueshavebeenpresentedtosolverank-constrainedLMIproblems.
Theseincludealternatingprojectionmethods[5]–[7],alinearizationmethod[4],andaugmentedLagrangianmethods[8]–[10].
Recently,anotherlocalalgorithmbasedManuscriptreceivedDecember7,2004;revisedMarch6,2006,November15,2006,andMarch5,2007.
RecommendedbyAssociateEditorC.
Beck.
S.
-J.
KimandS.
KwonarewithI&CGroup,KoreaElectrotechnologyResearchInstitute,Kyungnam641-120,Korea(e-mail:sjkim@keri.
re.
kr;smkwon@keri.
re.
kr).
Y.
-H.
MooniswiththeDepartmentofElectricalandElectronicEngineering,YonseiUniversity,Seoul120-749,Korea(e-mail:moon@yonsei.
ac.
kr).
DigitalObjectIdentier10.
1109/TAC.
2007.
9042930018-9286/$25.
002007IEEE