Finite-time attitude tracking control of spacecraft with application to attitude synchronization

好期刊，SCI

2711

Finite-TimeAttitudeTrackingControlofSpacecraft

WithApplicationtoAttitudeSynchronization

HaiboDu,ShihuaLi,andChunjiangQian

Abstract—Thisnoteinvestigatesthefinite-timeattitudecontrolproblemsforasinglespacecraftandmultiplespacecraft.Firstofall,afinite-timecontrollerisdesignedtosolvefinite-timeattitudetrackingproblemforasinglespacecraft.Rigorousproofshowsthatthedesiredattitudecanbetrackedinfinitetimeintheabsenceofdisturbances.Inthepresenceofdis-turbances,thetrackingerrorscanreacharegionaroundtheorigininfinitetime.Then,basedontheneighborrule,adistributedfinite-timeattitudecontrollawisproposedforagroupofspacecraftwithaleader-followerar-chitecture.Underthefinite-timecontrollaw,theattitudesynchronizationcanbeachievedinfinitetime.

IndexTerms—Attitudecontrol,attitudesynchronization,finite-timecon-trol,spacecraft.

以往的

properties[3],[14].Sofar,tothebestofauthors’knowledge,there

有限时isnofinite-timecontrolresultaboutattitudetrackingcontrolfora

singlespacecraftexceptin[15].In[15],thestandardterminalsliding间没有modecontroltechniquewasemployed.Unfortunately,thecontrol针对姿lawof[15]hasasingularityproblemasin[16],[17].Inaddition,

态跟踪fortheattitudecoordinationcontrolofmultiplespacecraft,itmaybe

问题desirabletoachieveattitudesynchronizationinfinitetime.

Inthistechnicalnote,wefocusondesigningafinite-timeattitude

trackingcontrollawforasinglespacecraftandapplyingittosolveco-operativeattitudeproblemformultiplespacecraft.Firstofall,byusingthefinite-timecontrolmethod,anewcontinuousfinite-timeattitudetrackingcontrollawisproposed.Differentfrom[15],thereisnosin-gularityprobleminourcontrollaw,sincehereweuseanotherkindoffinite-timecontroltechnique,i.e.,thetechniqueofaddingapowerintegrator[18].ByconstructingasuitableLyapunovfunction,thefi-nite-timeconvergenceforthetrackingerrorsystemcanbeprovenintheabsenceofdisturbances.Inaddition,arigoroustheoreticalanalysisfordisturbancerejectionperformanceisalsogiven,whichbuildsare-lationshipbetweencontrolparametersandtheboundofsteadytrackingerrors.Basedontheresult,wecangivearigorousexplanationwhyfi-nite-timecontrolsystemscanofferabetterdisturbancerejectionprop-erty.Abovetheseresultsarethemaincontributionsofthistechnicalnote.Secondly,asanapplicationofpreviousdevelopedresults,basedontheneighborrule,adistributedfinite-timecontrollawisproposedforagroupofspacecraftwithaleader-followerarchitecture.Undertheproposedfinite-timecontrollaw,theattitudesynchronizationcanbeachievedinfinitetime.Notethatin[13],basedonslidingmodecontrolmethod,adistributeddiscontinuousfinite-timecontrollawwasalsode-signedforagroupofspacecraftwithadynamicleader.Differentfrom[13],thecooperativefinite-timecontrollawproposedinthistechnicalnoteiscontinuous.

II.PRELIMINARIESANDPROBLEMFORMULATION

A.KinematicsandDynamicsofSpacecraftAttitude

Inthistechnicalnote,theorientationofeachspacecraftwithre-specttotheinertialframeisdescribedintermsoftheModifiedRo-driguezParameters(MRPs)[19].Let󰀀=󰀂tan(󰀃=4)2R3,02󰀆<󰀃<2󰀆,representtheMRPsforthespacecraft,where󰀂istheEuleraxisand󰀃istheEulerangle.Givenavectorv=[v1;v2;v3]T,thesymbols(1)denotesa323skew-symmetricmatrix,thatis,s(v)=[0;v3;0v2;0v3;0;v1;v2;0v1;0].

Remark1:NotethatMRPshaveageometricsingularitywhen󰀃approaches62󰀆.Asin[1],thestabilityanalysisofthistechnicalnoteisreferredtotheattitudesystemdescribedbymodifiedRodriguespa-rameters.Similarly,thecontinuityoftheproposedcontrollawisalsoreferredtotheattitudecontrolsystemdescribedbymodifiedRodriguesparameters.

UsingtheMRPs,theattitudekinematicsanddynamicsofthespace-craftaregivenby[19]

󰀀_=G(󰀀)!;J!_=s(!)J!+󰀎+M(t)

不同于

文献15，本文是没有奇点的理论性知识点不同于文献13，本文是连续的

I.INTRODUCTION

Inrecentyears,theattitudecontrolofarigidspacecraftandatti-tudecoordinationcontrolofmultiplespacecrafthaveattractedagreatdealofinterest.Theinterestismotivatedbyitsmanydifferenttypesofapplications,suchassatellitesurveillance,pointingandslewingofaircraft,formationflying,space-basedinterferometry,etc.

Ononehand,theattitudecontrolproblemofasinglespacecraftcanusuallybeclassifiedintoattitudetrackingproblemandattitudestabilizationproblem[1]–[3].Here,inthistechnicalnote,wefocusonattitudetrackingcontrolproblem.Recently,toimprovetheperfor-mancesoftheclosed-loopsystem,manynon-linearcontrolmethods,suchasslidingmodecontrol[4],optimalcontrol[5],adaptivecontrol[8],outputfeedbackcontrol[6],[7],havebeenemployed.

Ontheotherhand,theattitudecoordinationcontrolofmultiplespacecrafthasgainedincreaseattentionmorerecently.Thecoordi-nationcontrolisoftencatalogedascentralizedcoordinationcontrolanddecentralizedcoordinationcontrol.Usually,thedecentralizedcoordinationcontrolachievesmorebenefits,suchasgreaterefficiency,higherrobustness,andlesscommunicationrequirement[9].Con-sideringthesebenefits,manydistributedcooperativeattitudecontrolalgorithmshavebeenreportedin[9]–[13],tonamejustafew.

相比以Notethatmostoftheexistingattitudecontrollawsforattitude往的控trackingcontrolandattitudecoordinationcontrolareasymptotically

stablecontrollaws,whichmeansthattheconvergencerateisat

制方bestexponentialwithinfinitesettlingtime.Obviously,thecontrol法，有lawswithfinite-timeconvergencearemoredesirable.Besidesfaster限控制convergencerates,theclosed-loopsystemsunderfinite-timecontrol

usuallydemonstratehigheraccuracyandbetterdisturbancerejection

的优点

ManuscriptreceivedJune18,2010;revisedDecember18,2010;acceptedMay23,2011.DateofpublicationJune13,2011;dateofcurrentversionNovember02,2011.ThisworkwassupportedbytheNaturalScienceFoun-dationofChina(61074013),ProgramforNewCenturyExcellentTalentsinUniversity(NCET-10-0328),SpecializedResearchFundfortheDoctoralProgramofHigherEducationofChina(20090092110022),andtheScientificResearchFoundation,GraduateSchoolofSoutheastUniversity.RecommendedbyAssociateEditorP.Tsiotras.

H.DuiswiththeSchoolofAutomation,SoutheastUniversity,Nanjing,210096,China(e-mail:cyy2dhb@yahoo.cn).

S.LiiswiththeSiPaiLou2#,SchoolofAutomation,SoutheastUniversity,Nanjing210096,China(e-mail:lsh@seu.edu.cn).

C.QianiswiththeDepartmentofElectricalandComputerEngineeringTheUniversityofTexasatSanAntonio,SanAntonio,TX78249-0669USA(e-mail:chunjiang.qian@utsa.edu).

DigitalObjectIdentifier10.1109/TAC.2011.2159419

(1)

where!2R3istheangularvelocityofthespacecraftwithrespecttotheinertialframeexpressedinthebodyframe,J2R323and󰀎2R3aretheinertiaandthecontroltorqueofspacecraft,M(t)=[M1(t);M2(t);M3(t)]Tistheboundedexternaldisturbancevector,andG(󰀀)=(1=2)[((10󰀀T󰀀)=2)I30s(󰀀)+󰀀󰀀T]withI3beingthe323identitymatrix.ForthematrixG(󰀀),thefollowingpropertiesareknown[1]:

1+󰀀T󰀀T

󰀀G(󰀀)!=󰀀!;G(󰀀)GT(󰀀)=

4T

1+󰀀T󰀀

42

I3:

0018-9286/$26.00©2011IEEE

2712IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER2011

Forthesakeofstatement,inthefollowing,letd(t)=

T01

[d1(t);d2(t);d3(t)]=JM(t).

Assumption1:Theexternaldisturbancessatisfyjdi(t)j󰀂l<+1,i=1,2,3,wherelisaknownconstant.B.ControlObjectives

Inthistechnicalnote,themaincontrolobjectiveistodesignafinite-timeattitudetrackingcontrollawforasinglespacecraftdescribedbysystem(1).Underthefinite-timecontrollaw,thedesiredattitudecanbetrackedinfinitetimeintheabsenceofdisturbances.Inthepresenceofdisturbances,thetrackingerrorscanreacharegionaroundtheorigininfinitetime.Asanapplicationofpreviousresults,thenextcontrolobjectiveistodesignadistributedfinite-timeattitudecontrollawforagroupspacecraftwithaleader-followerarchitecture.C.SomeLemmas

_=f(x),f(0)=0;x2Rn,Lemma1:[14]:Considersystemx

wheref(1):Rn!Rnisacontinuousfunction.Supposethereexistsacontinuous,positivedefinitefunctionV(x):U!Rdefinedonan

_(x)+c(V(x))󰀃󰀂0openneighborhoodUoftheoriginsuchthatV

onUforsomec>0and󰀐2(0;1).Thentheoriginisafinite-timestableequilibriumofsystemx_=f(x)andthefinitesettlingtimeT

10󰀃

satisfiesT󰀂V(x(0))=c(10󰀐).IfU=RnandVisradiallyunbounded,theoriginisagloballyfinite-timestableequilibrium.Lemma2:[18]:If0Lemma3:[18],[20]:Foranyx2R;y2R,c>0,d>0,jxjcjyjd󰀂c=(c+d)jxjc+d+d=(c+d)jyjc+d.

Lemma4:[20]:Foranyxi2R,i=1;...;n,andarealnumberp2(0;1],(jx1j+111+jxnj)p󰀂jx1jp+111+jxnjp󰀂n10p(jx1j+111+jxnj)p.

A.Finite-TimeAttitudeTrackingControlLawDesignintheAbsenceofDisturbances

Theorem1:Considerthesinglespacecraftsystem(1)withM(t)󰀊0andsupposethereisnosingularityfortheinitialrelativeattitude.Ifthecontroltorque󰀞ischosenas1󰀞=0k11+eTe4p2=p01J(vp+k20s(!)J!e)bb+JRd!_d+Js(v)Rd!d(3)

wherek3>0

p2101=p+3

+k3;

1+p11+p

2101=pk2k1󰀋20pk2󰀋

2101=p+3p2101=p

+k3+3

1+pk2

1attitudecanbetrackedinfinitetime.

Proof:Theproofprocedurecanbedividedintotwosteps.First,vistakenasavirtualinputandisdesignedsuchthatereacheszeroinfinitetime.Then,thecontrollaw󰀞isdesignedsuchthatthevirtualcontrolvcanbetrackedinfinitetime.

Step1:VirtualInputvDesign.

ChooseacandidateLyapunovfunctionasV0=(1=2)eTe.

_0=eTG(e)v=Alongthetrajectoryofsystem(2),wehaveV

((1+eTe)=4)eTv.Takingv=0k2e1=pasthevirtualinputandbyLemma4,oneobtains

_0=0k2V=01+eTe

4k2d

ei󰀂0

4i=13

3i=1d=22

ei

III.FINITE-TIMEATTITUDETRACKINGCONTROL

FORASINGLESPACECRAFT

Inthissection,wefocusonsolvingfinite-timeattitudetrackingproblemforasinglespacecraft.Let󰀖d,!ddenotethedesiredattitudeandthedesiredangularvelocity,respectively.Asin[6],[7],inordertofacilitatethecontrollerdesign,thedesiredtrajectory󰀖d(t)iscon-structedtoavoidthekinematicsingularityassociatedwiththeMRPs_dareassumedtobebounded.Inwhatfollows,tosolvetheat-and!d,!

titudetrackingproblem,asin[4],[19],definee=[e1;e2;e3]T2R3astherelativeattitudeerrorbetweentheactualattitudeandthedesiredattitude,where

01

e=󰀖󰀉󰀖d=

k2d=2d=22V04whered=1+1=p.ByLemma1,weobtainV0(t)reacheszeroinfinitetime.Step2:ControlLaw󰀞Design.ThecandidateLyapunovfunctionisconstructedasV=

3V0+i=1Vi,whereV0isthesameasthatinStep1,andVi=1201p1+p2101=pk2v1=pv3p201=p(sp0vi)ds;󰀖d(󰀖󰀖

T

T

01)+󰀖(10󰀖d󰀖d)+

TTT1+󰀖d󰀖d󰀖󰀖+2󰀖d󰀖2s(󰀖d)󰀖

3vi=0k2ei;i=1;2;3:v

v(4)(sp0

:

ItfollowsfromPropositionsB1andB2in[18]thatb

Definev=[v1;v2;v3]T=!0Rd!d2R3astherelativean-b

istherotationmatrixfromthedesiredgularvelocityerror,whereRd

breferenceframetothebodyreferenceframe.TherotationmatrixRdb

isaproperorthogonalmatrixandisgivenbyRd=R(e),whereTT22

+8s(e)=(1+eTe)2.Then,R(e)=I3+4((10ee)=(1+ee))s(e)

therelativekinematicequationisgivenasin[4]andtherelativedy-namicequationisgivenasin[5],[8]

3p201=pvi)ds,i=1,2,3,aredifferentiable,positivedefiniteandproper.UsingtheresultsinStep1yieldsT_0=1+eeV43i=13eivi1+eTe+43i=13ei(vi0vi):(5)

e_=G(e)v;

bb

!_d0Js(v)Rd!d+M(t)Jv_=s(!)J!+󰀞0JRd

p3p1=p3

ByLemma2,weobtainei(vi0vi)󰀂2101=pjeijjvi0vij,i=1,

p3p

2,3.Forthesakeofbrevity,let󰀠i=vi0vi,i=1,2,3.AccordingtoLemma3,wehave

(2)

01b

,v=!0Rd!d.Therefore,tosolvethefinite-timewheree=󰀖󰀉󰀖d

trackingproblem,weonlyneedtodesignacontrollawsuchthate!0,v!0infinitetime.

3

ei(vi0vi)󰀂2101=pjeijj󰀠ij1=p󰀂2101=p

pjeijdj󰀠ijd

+1+p1+pofthistechnicalnote,ifx=[x;...;x]isavectorandpis

afractionalpower,thenx=[x;...;x].

1Throughout

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER20112713

i=1,2,3.Substitutingthisinequalityinto(5)resultsin

p1+p+2

1+eTe4101=p

3i=1

d

Puttingtogether(6),(9),andnoticingthatjeijd=jeij(1+p)=p=ei,dd

j󰀆ij=󰀆i,weobtain

_0󰀀0k202101=pVedi

3i=1

p2101=p+3_V󰀀0k20

1+pj󰀆ij:

d

1+eTe4101=p

3m=1

jemjd

3m=1

11+p1+eTe4(6)

++

2

101=p

+3p2

+3

1+pk2

1

1p1+eTe4201=p

j󰀆mjd

3

TakingthederivativeofVialongsystem(2)yields_i=0V

1

3p

@(vi)@tvv

3p101=p

(sp0vids)

20

1+p

2101=pk2i=1

󰀆i

󰀍󰀉i:(10)

1+p2101=pk2

+

20

1p201=p

󰀍󰀉󰀆ii;1+p101=p2k2

3vi

1

i=1;2;3(7)

Ifthecontroltorque󰀍ischosenas(3),thenaccordingto(2)andthe

2=p01T

definitionof󰀆i,weobtain󰀍󰀉,i=1,2,i=0k1((1+ee)=4)󰀆i

3.Substitutingthisexpressioninto(10)andbythegainsconditionofTheorem1,wehave

_󰀀0k3V

43i=1

dei

where󰀍󰀉i=@vi=@t.Accordingtothedefinitionof

3p3p3p

@(v1)@(v2)@(v3)

;;@t@t@tT

p@e0k2

and(2),wehave

k3

043i=1

d󰀆i:

(11)

=

@t=

p

G(e)v:0k2

Next,wewillshowthatV(t)willreachzeroinfinitetime.From(4),

bycalculationandLemma2,weget

Vi󰀀󰀀

:

1

2020

1p1+p

2101=pk2

2󰀆i;

3p3p201=p

jvi0vijjvi0vij

BythedefinitionofG(e),wehave

1+e0e0e

21+e0e0e

21+e0e0e

2G(e)v=

12v1+(e1e20e3)v2+(e1e3+e2)v3v2+(e2e30e1)v3+(e1e2+e3)v1v3+(e1e30e2)v1+(e2e3+e1)v2

1

1p1+pk2

i=1;2;3:(12)

222

Noticingthat((1+e10e20e3)=2)v1󰀀

TT

((1+ee)=2)jv1j,(e1e20e3)v2󰀀((1+ee)=2)jv2j,

T

(e1e3+e2)v3󰀀((1+ee)=2)jv3j,thenweobtain

T

[G(e)v]1󰀀((1+ee)=4)(jv1j+jv2j+jv3j),where[G(e)v]idenotestheithelementofvectorG(e)v,i=1,2,3.Byasimilaranal-3pp

ysis,wegetj@(vi)=@tj󰀀k2((1+eTe)=4)(jv1j+jv2j+jv3j),i=1,2,3.Substitutingthisinequalityinto(7)resultsin

3322

Hence,V=V0+3i=1Vi󰀀c1i=1ei+i=1󰀆i,wherec1=1+p

.ByLemma4,wegetmax1=2;1=(201=p)k2

V

d=2

󰀀

d=2c1

3i=1

dei

3

+

i=1

d󰀆i

:(13)

_i󰀀V

12101=pk2

1+eTe3

(jv1j+jv2j+jv3j)jvi0vijj󰀆ij101=p

41201=p

+󰀆i󰀍󰀉(8)i;i=1;2;3:1+p1101=p20p2k2

UsingLemmas2and3,foranyi,m=1,2,3,wehave

3

jvmjjvi0vijj󰀆ij101=p

󰀀2101=pjvmjj󰀆ij1=pj󰀆ij101=p

33󰀀2101=p(j󰀆ijjvm0vmj+j󰀆ijjvmj)

󰀀2101=p(2101=pj󰀆ijj󰀆mj1=p+k2j󰀆ijjemj1=p)󰀀2

101=p

k2(2101=p+k2)pd2101=p

j󰀆ij+j󰀆mjd+jemjd:

1+p1+p1+pDenote󰀓=k3=(8c1)>0.Thenitfollowsfrom(11)and(13)

dd_+󰀓Vd=2󰀀0(k3=8)3qi0(k3=8)3thatVi=1i=1!i󰀀0.

Noticing0Remark2:Notethathereitisrequiredthatthereisnosingularityfortheinitialrelativeattitude.Itmeansthattheresultisnotglobalforboththerelativeattitudeerrordynamicsystemandtheoriginalattitudecontrolsystem.Andthebasinofattractionisforalltheinitialrelativeattitudesexceptforthesingularpoint.Moreover,itshouldbepointedoutthatthesingularitywilldonotoccurfortherelativeattitudeaslongastheinitialrelativeattitudehasnosingularitysinceV(t)󰀀V(0)willbesatisfiedforanyt󰀆0undertheproposedcontrollaw.

Remark3:Ifthefractionpowerp=1,thefinite-timecontrollaw(3)willreducetothefollowingcontrollaw:󰀍=0k1

1+eTe4J(v+k2e)0s(!)J!

bb

+JRd!_d+Js(v)Rd!d

d=2

Substitutingthisinequalityinto(8)yields1+eTe_Vi󰀀

41+

1+p2101=p2101=p+k23pd

j󰀆ij+

k21+p(1+p)k2

3m=1

3m=1

(14)

j󰀆mj

201=p

d

jemjd

+

1

20

1p1+p

2101=pk2

󰀆i

󰀍󰀉i;

i=1;2;3:(9)

2

(1+1=k2+k3).Thiscon-wherek3>0,k2󰀆1+k3;k1󰀆k2

trollawcanbeconsideredasakindofconventionalbacksteppingcon-trollaw.Underthiscontrollaw,thedesiredattitudecanbeasymptot-icallytrackedandtheconvergencerateforthetrackingerrorsystem(2)isexponential,whichcanbeestimatedbythefollowingprocedure.Substitutingp=1intotheproofprocedureofTheorem1,weobtain_+󰀓V󰀀0,whichimpliesthatV(t)󰀀V(t0)e0󰀊(t0t),whereV

2714IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER2011

󰀀=k3=(8c2)>0,c2=max1=2;1=(k2

2).Accordingtothedefi-nitionofV,wehaveeTe1

3V=2+(2k22)(vi+k2ei)2

i=1

3󰀀eT

e+123

2e2i(2k22)vi

0

i=1

i=1

3+3v2i

(8k22)󰀀󰀊1(eTe+vTv)=󰀊1kx(t)k2

(15)

where󰀊1=minf1=3;1=(8k2

2)g,x(t)=[e1;e2;e3;v1;v2;v3]T.Ontheotherhand,by(15),weget

V(t0)󰀆e(t0)T

e(t0)

33

+1(2k22)v2+

2e2i(t0)3v2

i(t0)

ii=1

i=1

3+(t0)

(8k22)󰀆󰀊2kx(t0)k

2

where󰀊2=max5=3;7=(8k2

2).Asaresult

kx(t)k󰀆

V(t)V(t0)e0󰀃(t0t

)

󰀊1󰀆󰀊1

󰀆

󰀊2

kx(t0)ke0(󰀃=2)(t0t)󰀊1

:Comparedwiththisconventionalbacksteppingcontrollaw(14),theclosed-loopsystem(2)underthefinite-timecontrollaw(3)willofferabetterdisturbancerejectionproperty.

B.AnalysisofDisturbanceRejectionPerformanceinthePresenceofDisturbances

Inthissubsection,wewillconsiderthecaseofexistingexternaldis-turbances.

Theorem2:Considerthesinglespacecraftsystem(1)underthefi-nite-timecontrollaw(3)andsupposethereisnosingularityfortheinitialrelativeattitude.IfAssumption1issatisfied,thenthetrackingerrorwillconvergetotheregionQinfinitetime,where

:je4pl2

1=p

p=(20p)

Q=

(eT;vT)ij󰀆c3(2p01)k1+;

2pk3

1=(20p)jvij󰀆c4

4pl21=p

(2p01)k1+i=1;2;3:(16)

2pk3

;andc1=maxf1=2;1=(201=p)k1+2p

g,cp

(2p01)33=(1+p))

p=(1+p)3=2c1(3(1+p)=+

1=2p

c4=k2(2)

1=2p

+2

p0

12k1+2

p1c11

=2p3

(1+p)=(2p01)

+3

3=(1+p)

1=(1+p)

:

Proof:TheanalysisprocedureissimilartothatoftheproofofTheorem1,excepttheappearanceofthedisturbances.jdi(t)j󰀆landsubstitutingthecontrollaw(3)intoV

_Noticing

,followingasimilarprocedurebetween(4)and(11)intheproofofTheorem1,wehave

_333

V

󰀆0k34edi

0k3i=1

4󰀓di

+h

j󰀓ij

201=p

(17)

i=1

i=1

whereh=l=(201=p)2101=pk1+2p

.Thefollowingproofcanbedi-videdintothreesteps.Define

󰀖1=(eT;󰀓T

3

3

):

ed

i

+

󰀓di

i=1

i=1

<3

(1+p)=(2p01)

+3

3=(1+p)

8h(1+p)=(20p)

k3

;

󰀖2=

(eT;󰀓T):V(t)m=c13

(1+p)=(2p01)

+3

3=(1+p)

2=d

8h

2p=(20p)

k3

where󰀓=[󰀓)2=󰀖1;󰀓2;󰀓3]T.First,weprovethefactthat_(xT;vT

V

<0once1.However,thisfactcannotguaranteetheregion󰀖1istheattractiveregion.Thereasonisthatitispossiblethestatesmaycapefrom󰀖1afterenteringit,sincein󰀖1,V

_es-<0isnolongerguaran-teed.However,wecanfindalargerregion󰀖2whichcontains󰀖1,i.e.,󰀖1󰀋󰀖2,suchthat󰀖2istheattractiveregion.Second,weprovethattheregion󰀖2isanattractiveregionforanyinitialstate(eT(0);󰀓T(0))and3canbereachedinfinitetime,whichmeansthereexistsat1,suchthat(eT(t);󰀓T

(t))2󰀖2for8t>t3finitetime

1.Third,wegiveanesti-mationforsteadytrackingerrors.

Step1:Assuming(eT;󰀓T)2=󰀖1,i.e.

33(1+p)=(20p)

edi+

󰀓d

i󰀀

3

(1+p)=(2p01)

+3

3=(1+p)

8h

i=1

i=1

k3

thefollowinganalysisCase1:3idcanbe(1+dividedp)=(2p0into1)

twocases.

(8h=k3)(1+p)=(20p)since=1󰀓i󰀀3

Ononehand,13j󰀓ij

201=p

3

=

(j󰀓ijd)(201=p)=d

i=1

i=1

3(2p01)=(1+p)󰀆3

(20p)=(1+p)

󰀓di

(18)

i=1

whichresultsin3󰀓d

3

i

󰀀3

(p02)=(2p01)

(

j󰀓ij201=p)(1+p)=(2p01):

i=1

i=1

Itfollowsthisinequalitythat:

k333

3

8󰀓di

0h

j󰀓ij

201=p

󰀀

j󰀓ij201=p

i=1

i=1

i=1

2

k(1+p)=(2p01)01

3(p02)=(2p01)

383j󰀓ij201=p

0h:

i=1

(19)

Ontheotherhand,noticing

3

󰀓di󰀀3(1+p)=(2p01)(8h=k3)(1+p)=(20p)

i=1

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER20112715

andbyLemma4,weobtain

3

j󰀀ij

201=p

3

=

(󰀀di)(201=p)=di=1

i=1

3

(201=p)=d󰀂

󰀀di

󰀂3

8h(2p01)=(20p)

i=1

k3

:

Substitutingthisinequalityinto(19)leadstokd

h3i=1j󰀀ij201=pCase2:3󰀂0_d.Itfollows(1+p)=from(2p01)

(17)(8h=kthatV(t)3=<830.i=1󰀀i0

3)(1+p)=(2If3󰀈i=1󰀀di=10p)i<3(1+󰀀ip)<=(23p01)

(8h=k3)(1+p)=(20p)1impliesthat3,then(eT3=(1+(1+p)=(20p);󰀀T)2=by(18),wehavei=13ed

i>3p)201=p

(8h=k󰀅33=(1+3)p)(8h=k3)(2.pIn0addition,

1)=(20p).Substitutingthelasti=1j󰀀ij

twoinequalitiesinto(17)_0(k3=8)nsultsofCasei=1edi0(k3=4)3d

leadsto<0.Hence,bytheVre-󰀅

1andCase2,weknowi=1󰀀Vi

_<0once(eT;󰀀TStep2:Inthesequel,wewillprovethatV_<0once)(2e=T󰀈1.

󰀈;󰀀T)2=

2.Actually,fromStep1,weonlyneedtoprovethat󰀈1isasubsetof󰀈2.Theproofisgivenasfollows.Forany(eT;󰀀T)2󰀈1,by(13)

33

V

d=2

󰀅cd=12

edi

+

󰀀di

i=1

i=1

(1+p)=(20p)

23

(1+p)=(2p01)

+3

3=(1+p)

8hk3

=md=2:

Thatistosay(eT󰀈;󰀀T)2󰀈2,i.e.,󰀈1󰀆󰀈2.Hence,once(eT;󰀀T)2=

2,thenV_<0.Since󰀈2isalevelsetoftheLyapunovfunction,

thenthereexistsafinitetimet31,suchthat(eTT

V(t)t3(t);󰀀(t))2󰀈2,i.e.,

1

.Step3:BythedefinitionofVinTheorem1,whent󰀂t3

1,weconcludethat

jei(t)j󰀅

eT(t)e(t)=2V0(t)󰀅

2V(t)p=(20p)

󰀅p

2m󰀅c3

8hk3

i==11,,2,2,3,3.whereByLemmac3isgiveninTheorem2.Next,letusestimatevi2,js0v3

ij󰀅2101=pjsp0v3ipj1=pi(t),

,thenjspv3ipj󰀂21=p01js0v3i

jp

0

.Ifvi󰀂v3

i,bythepreviousinequalityandthedefinitionofVi,wehaveVi󰀂(2102p=(2p01)k1+p)(vi0v3i)2p

.Inthecaseofv2

Vjv)󰀅V(t)i󰀅i,theproofofthisinequalityi(t8t󰀂t3

issimilar.Since

1i(t)0v3

i(t)j󰀅2[(p01=2)k1+,2pit]

1follows=(2p)m1from=(2p)thisinequalitythat

whent󰀂t31.Com-biningthisinequalityandtheestimationforei(t),itcanbeconcluded

thatwhent󰀂t3

1

jvi(t)j󰀅jv3i(t)j+jvi(t)0v3

i(t)j

1=(20p)

=k2je1i=p

j

+jvi(t)0

v3

i(t)j

󰀅c4

8h

k3

wherec4isgiveninTheorem2.

Remark4:Ifp=1,itcanbeproventhatthetrackingerrorswillconvergetoregionMinfinitetimeundertheconventionalbackstep-pingcontrollaw(14)byasimilarproof,whereM=

(eT;vT):jeij󰀅c5

8l

(k22k3);jvij󰀅c6

8l

(k22k3);i=1;2;3:(20)

andc2k2=maxf1=2;1=(k2

2)g,c5=

2(3+33=2)c2,c6=22(3+33=2)c2.Thefollowingremarkwillshowthatthefi-nite-timecontrollaw(3)hasabetterdisturbancerejectionpropertythantheconventionalbacksteppingcontrollaw(14).

Remark5:OnemayarguethatthroughadjustingcontrolgainskQ2or/andkandM,3tobelargeenough,bothconvergenceregions,i.e.,canberenderedtobeassmallasdesired.Neverthe-less,duetocontrolsaturationconstraint,wecannotselectk2andk3tobesufficientlylarge.Inthiscase,fortheproposedmethodhere,anadditionalparameter,i.e.,thefractionalpowerp,canbeadjustedtoenhancethedisturbancerejectionperfor-mancewithoutincreasingk2andk3tobesufficientlylarge.For

example,supposing0<4pl21=p=(2p01)k1+2p

k3<1issatisfied,

i.e.,selectingk1=(1+p)

2suchthatk2>4pl21=p=(2p01)k3

,wecanselectptoapproximateto2suchthatthepowerexpres-sion1=(20p)inQisbiggerenoughthan1,whichcanmake

4pl21=p=(2p01)k1+p

1=(20p)

3k2󰀋8l=k3k2

2,i.e.,Qismuch

smallerthanM.

IV.FINITE-TIMEATTITUDESYNCHRONIZATION

FORMULTIPLESPACECRAFT

Withoutlossofgenerality,agroupofnspacecraftareconsidered.Let0=f1;2;...;ng.Theattitudekinematicanddynamicequationsoftheithspacecraftaregivenas

󰀖_i=G(󰀖i)!i;Ji!_i=s(!i)Ji!i+󰀚i;i20

(21)

where!i2R3istheangularvelocityoftheithspacecraftwithrespect

totheinertialframeexpressedinthebodyframeoftheithspacecraft,Ji2R323and󰀚i2R3aretheinertiaandcontroltorqueoftheithspacecraft,respectively.Here,thecontroltorque󰀚ifortheithspace-craftistobedesignedbasedonthelocalinformationfromitselfanditsneighbors.

AssumethatthecommunicationtopologyamongnspacecraftismodeledbyadirectedgraphG(A)=fV;E;Ag.V=fv1A;...=;[nag]isi;i=2theRsetn2nofnodes,E󰀌V2VisthesetofedgesandijistheweightedadjacencymatrixofthegraphG(A)withnon-negativeadjacencyelementsaij.Ifthereisanavailableinformationchannelfromspacecraftjtospacecrafti,thena=0.Themagnitudeofanonzeroaij>0,otherwiseaijijcanrepresents/de-terminesthestrengthoftheconnectionbetweenspacecraft.Moreover,weassumethataii=0foralli20.ThesetofneighborsofnodeiisdenotedbyNi=fj:aij>0g.Here,asinTheorem5of[21],thefollowingassumptionaboutgraphG(A)isgiven.

Assumption2:Thecommunicationtopologygraphhasahierar-chicalstructure.Assumev1istheleaderandallnodesofdirectedgraphGfVv(A)canbeclassifiedintothefollowingsubsets:V0=fv1g;V1=j:2vV:vjonlyreceivesinformationfromv1g;...;Vqjonlyreceivesinformationfromnodesin[q01

=fvj2

isnon-emptyforalll=1;...;q.Moreover,[q

l=0Vlg,whereVl

01g;...;0Vl=0Vl=V0=fq=fj:vj2qg.Obviously[q.DenoteTheorem3:Considerthemultiplespacecraftsysteml=00l=0.

(21)underAs-sumption2.Ifthecontroltorque󰀚iischosenas

󰀚i=0s(!i)Ji!i+󰀢i

aijJiRij!_j+Jis(!ij)Ri

j!j

j2N

0k11+󰀖T

ij󰀖ijJi!p2=p01

4ij

+kp2󰀖ij;i=2;...;n(22)

2716IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER2011

Fig.1.Responsecurvesunderfinite-timecontrollaw(3).

Fig.2.Responsecurvesunderconventionalbacksteppingcontrollaw(14).

0;ifj2Naij=0;01

󰀅=󰀅i󰀂󰀅j,

1=j2Naij;ifj2Naij=0,ijii

!ij=!i0Rj!j;Rj=R(󰀅ij),theparametersk1;k2;parethesameasthatofTheorem1,thentheattitudesynchronizationcanbeachievedinfinitetime.

Proof:Theproofismainlybasedonarecursiveargument.We

01

firstconsiderallthenodesinV1.Define󰀅i1=󰀅i󰀂󰀅1,!i1=

i

!i0R1!1;8i201.BasedonTheorem1,itcanbeeasilyproventhat󰀅i!󰀅1,!i!!1;8i201,infinitetimeundertheproposed

3

suchthat󰀅i(t)=󰀅1(t),controllaw.Hence,thereexistsatimeT1

33

!i(t)=!1(t);8i201;8t󰀇T1.AfterT1,asimilarproofleadsto󰀅i!󰀅1,!i!!1;8i202,infinitetime,whichimpliesthatthere

333+T1>T1suchthat󰀅i(t)=󰀅1(t),!i(t)=isatimeinstantT2

33

!1(t);8i202;8t󰀇T2+T1.Byinduction,weconcludethattheattitudesofallspacecraftachievesynchronizationinfinitetime.

Remark6:Obviously,thecontrollaw(22)isdistributedbecauseitiscomposedonlyofthespacecraft’sowninformationandthatofitsneighbors.Inaddition,thefinitesettlingtimeT3forachievingattitudesynchronizationdependsonthenumberoflevelsinthehierarchicalstructureandthemaximumfinitesettingtimeforeachlevel.AssumethemaximumfinitesettingtimeforlthlevelisTl3;l=1;2;...;q,

where󰀀i=

33

thenT3=T1,whereTl3=maxfTi;i20lgandTiis+111+Tq

thetrackingtimeforithspacecraft.

Example1:Considerattitudetrackingproblemforasinglespacecraft(1).Theinertiamatrixisgivenas[2]:J=[100;00:630;000:85]kg1m2.Theinitialstatesaresetas:󰀅(0)=[0:5;0;0:2]T;!(0)=[0;0;0]Trad=s.Thedesired

d

)=󰀓d(t)tan(󰀔d(t)=4),withattitudetrajectoryisselectedas󰀅p(tT

󰀓d(t)=[cos(0:4󰀕t);sin(0:6󰀕t);2]and󰀔d(t)=󰀕rad.

Tohaveafaircomparisonofthedynamicperformancesbetweenthefinite-timecontrollaw(FTCL)(3)andtheconventionalbacksteppingcontrollaw(CBCL)(14),thecontroltorquesarelimitednottoexceed10N1m.FortheFTCL(3),selectp=7=5,k1=14,k2=2:3.FortheCBCL(14),thegainsofarechosenask1=15,k2=2:2.

Intheabsenceofdisturbances,theconvergencetimecomparisonisgiveninTableI.Whenthereexistexternaldisturbances,thesteadytrackingerrorscomparisonisalsoshowninTableIandtheresponsecurvesarepresentedinFigs.1–2,wherethefollowingexternaldistur-bancesM(t)=[0:3sin(t);0:4cos(1:5t);0:5sin(2t+1)]Tareadded.Obviously,theclosed-loopsystem(1)underthefinite-timecontrollaw(3)demonstratesafasterconvergenceandabetterdisturbancerejec-tionproperty.

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.56,NO.11,NOVEMBER20112717

TABLEI

THECOMPARISONSOFCONVERGENCETIME,DISTURBANCEREJECTION

PROPERTYBETWEENTHEFTCL(3)ANDTHECBCL(14)

Convergencetime:thetimeafterwhichjej<103,alwaysholds.

,jvj<10,i=1,2,

[16]Z.Man,A.P.Paplinski,andH.R.Wu,“ArobustMIMOterminal

slidingmodecontrolschemeforrigidroboticmanipulators,”IEEETrans.Autom.Control,vol.39,no.12,pp.2464–2469,Dec.1994.[17]K.B.ParkandJ.J.Lee,“Commentson“ArobustMIMOterminal

slidingmodecontrolschemeforrigidroboticmanipulators”,”IEEETrans.Autom.Control,vol.41,no.5,pp.761–762,May1996.

[18]C.QianandW.Lin,“Acontinuousfeedbackapproachtoglobalstrong

stabilizationofnonlinearsystems,”IEEETrans.Autom.Control,vol.46,no.7,pp.1061–1079,Jul.2001.

[19]M.D.Shuster,“Asurveyofattituderepresentations,”J.Astron.Sci.,

vol.41,no.4,pp.439–517,1993.

[20]G.Hardy,J.Littlewood,andG.Polya,Inequalities.Cambridge,

U.K.:CambridgeUniv.Press,1952.

[21]Y.SunandL.Wang,“Consensusofmulti-agentsystemsindirected

networkswithnonuniformtime-varyingdelays,”IEEETrans.Autom.Control,vol.54,no.7,pp.1607–1613,Jul.2009.

V.CONCLUSION

Basedonthecontinuousfinite-timecontroltechnique,afinite-timeattitudetrackingcontrollawhasbeendesignedforasinglespace-craftandadistributedfinite-timeattitudesynchronizationalgorithmhasalsobeendevelopedforagroupspacecraft.Futureworkincludesextendingtheresultsinthistechnicalnotetocaseswhentheangularve-locityisunmeasurableandthereexistcommunicationdelaysbetweenspacecraft.

PadéDiscretizationforLinearSystemsWithPolyhedralLyapunovFunctions

FrancescoRossi,PatrizioColaneri,andRobertShorten

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attitudeusingfinite-timecontroltechnique,”Int.J.Control,vol.82,no.5,pp.822–836,2009.

[4]G.XingandS.A.Parvez,“Nonlinearattitudestatetrackingcontrol

forspacecraft,”J.Guid.,ControlDynam.,vol.24,no.3,pp.624–626,2001.

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forattitudetrackingofspacecraft,”IEEETrans.Autom.Control,vol.50,no.11,pp.1639–1654,Nov.2005.

[6]H.Wong,M.S.deQueiroz,andV.Kapila,“Adaptivetrackingcontrol

usingsynthesizedvelocityfromattitudemeasurements,”Automatica,vol.37,no.6,pp.947–953,2001.

[7]M.R.Akella,“Rigidbodyattitudetrackingwithoutangularvelocity

feedback,”Syst.ControlLett.,vol.42,no.4,pp.321–326,2001.

[8]Z.ChenandJ.Huang,“Attitudetrackinganddisturbancerejectionof

rigidspacecraftbyadaptivecontrol,”IEEETrans.Autom.Control,vol.54,no.3,pp.600–605,Mar.2009.

[9]W.RenandR.W.Beard,DistributedConsensusinMultivehicle

CooperativeControl:TheoryandApplications.Berlin,Germany:Springer,2007.

[10]J.R.LawtonandR.W.Beard,“Synchronizedmultiplespacecraftro-tations,”Automatica,vol.38,no.8,pp.1359–1364,2002.

[11]A.AbdessameudandA.Tayebi,“Attitudesynchronizationofagroup

ofspacecraftwithoutvelocitymeasurements,”IEEETrans.Autom.Control,vol.54,no.11,pp.2642–2648,Nov.2009.

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Abstract—ThistechnicalnotehasbeenmotivatedbytheneedtoassessthepreservationofpolyhedralLyapunovfunctionsforstablecontinuous-timelinearsystemsundernumericaldiscretizationofthetransitionmatrix.Thisproblemariseswhendiscretizinglinearsystemsinsuchamannerastopreserveacertaintypeofstabilityofthediscretetimeapproximation.Ourmaincontributionistoshowthatacontinuous-timesystemanditsPadédiscretization(ofanyorderandsampling)alwaysshareatleastonecommonpiecewiselinear(polyhedral)Lyapunovfunction.

IndexTerms—Discretization,Lyapunovfunction,stabilityoflinearsystems.

I.INTRODUCTION

Theinvestigationofthepropertiesoflinearsystemswhenpassingfromthecontinuous-timeanalysistothediscrete-timeonehasbeensubjectofparticularattentionintheliteratureofcontroltheory.Forlineartime-invariant(LTI)systems,thisprocedureisalmostcompletelyunderstood,andformsafundamentalbasis,bothforthedesignofcon-trolsystems,andfornumericalsimulation.

Recently,inthecontextofthestudyofswitchedlinearsystems,sev-eraltechnicalnoteshaveconsideredtheproblemofdiscreteapproxi-mationstocontinuous-timeswitchedlinearsystems[1]–[4].Thetheoryofswitchedlinearsystemsisarelativelynewfieldofresearchwheretheknowledgeofthesharedpropertiesbetweencontinuous-timeand

ManuscriptreceivedJune14,2010;revisedDecember01,2010;acceptedJune22,2011.DateofpublicationJune30,2011;dateofcurrentversionNovember02,2011.ThisworkwassupportedinpartbytheItalianNationalResearchCouncil(CNR)andbyScienceFoundationIrelandundergrantPIAward07/IN.1/1901.RecommendedbyAssociateEditorH.Ito.

F.RossiiswiththeLabLSIS,UniversitéPaulCézanne,Marseille,France(e-mail:francesco.rossi@lsis.org).

P.ColaneriiswiththePolitecnicodiMilano,DipartimentodiElettronicaeInformazione,Milan,Italy(e-mail:colaneri@elet.polimi.it).

R.ShorteniswiththeNationalUniversityofIreland,HamiltonInstitute,Maynooth,Co.Kildare,Ireland(e-mail:robert.shorten@nuim.ie).

Colorversionsofoneormoreofthefiguresinthistechnicalnoteareavailableonlineathttp://ieeexplore.ieee.org.

DigitalObjectIdentifier10.1109/TAC.2011.2161028

0018-9286/$26.00©2011IEEE