EE363Winter2008-09Lecture11Invariantsets,conservation,anddissipationi - PDF document

EE363Winter2008-09Lecture11Invariantsets,conservation,anddissipationinvariantsetsconservedquantitiesdissipatedquantitiesderivativealongtrajectorydiscrete-timecase11{1 Invariantsetsweconsiderautonomous,time-invariantnonlinearsystem_x=(x)asetCRnisinvariant(w.r.t.system,or)ifforeverytrajectoryx,x()2C=)x()2CforalliftrajectoryentersC,orstartsinC,itstaysinCtrajectoriescancrossintoboundaryofC,butneveroutofC CInvariantsets,conservation,anddissipation11{2 Examplesofinvariantsetsgeneralexamples:fxg,where(x)=0(i.e.,xisanequilibriumpoint)anytrajectoryorunionoftrajectories,e.g.,fx()jx(0)2D;t0;_x=(x)gmorespecicexamples:_x=Ax,C=spanfv;:::;vkg,whereAvi=ivi_x=Ax,C=fzj0wzag,wherewA=w,0Invariantsets,conservation,anddissipation11{3 InvarianceofnonnegativeorthantwhenisnonnegativeorthantRn+invariantfor_x=Ax?(i.e.,whendononnegativetrajectoriesalwaysstaynonnegative?)answer:ifandonlyifAij0fori=jrstassumeAij0fori=j,andx(0)2Rn+;we'llshowthatx()2Rn+for0x()=tAx(0)=limk(I+(t=k)A)kx(0)forklargeenoughthematrixI+(t=k)Ahasallnonnegativeentries,so(I+(t=k)A)kx(0)hasallnonnegativeentrieshencethelimitabove,whichisx(),hasnonnegativeentriesInvariantsets,conservation,anddissipation11{4 nowlet'sassumethatAij0forsomei=j;we'llndtrajectorywithx(0)2Rn+butx()62Rn+forsomet&#x-0.1;儤0let'stakex(0)=j,soforsmallh&#x-0.1;儤0,wehavex(h)j+hAejinparticular,x(h)ihAij0forsmallpositiveh,i.e.,x(h)62Rn+thisshowsthatifAij0forsomei=j,Rn+isn'tinvariantInvariantsets,conservation,anddissipation11{5 Conservedquantitiesscalarvaluedfunction:Rn!Riscalledintegralofthemotion,aconservedquantity,orinvariantfor_x=(x)ifforeverytrajectoryx,(x())isconstantclassicalexamples:totalenergyofalosslessmechanicalsystemtotalangularmomentumaboutanaxisofanisolatedsystemtotal\ruidinaclosedsystemlevelsetorlevelsurfaceof,fz2Rnj(z)=ag,areinvariantsetse.g.,trajectoriesoflosslessmechanicalsystemstayinsurfacesofconstantenergyInvariantsets,conservation,anddissipation11{6 Example:nonlinearlosslessmechanicalsystem mq()Fqmq=F=(q),wherem�0ismass,q()isdisplacement,Fisrestoringforce,isnonlinearspringcharacteristicwith(0)=0withx=(q;_q),wehave_x=_qq=x(1=m)(x)Invariantsets,conservation,anddissipation11{7 potentialenergystoredinspringis (q)=Zq(u)dutotalenergyiskineticpluspotential:E(x)=(m=2)_q+ (q)Eisaconservedquantity:ifxisatrajectory,then E(x())=(m=2) _q+ (q)=m_qq+(q)_q=m_q((1=m)(q))+(q)_q=0i.e.,E(x())isconstantInvariantsets,conservation,anddissipation11{8 Derivativeoffunctionalongtrajectorywehavefunction:Rn!Rand_x=(x)ifxistrajectoryofsystem,then (x())=D(x())dx =r(x())(x)wedene_:Rn!Ras_(z)=r(z)(z)intepretation:_(z)givesd dt(x()),ifx()=ze.g.,if_(z)�0,then(x())isincreasingwhenx()passesthroughzInvariantsets,conservation,anddissipation11{9 ifisconserved,then(x())isconstantalonganytrajectory,so_(z)=r(z)(x)=0forallzthismeansthevectoreld(z)iseverywhereorthogonaltor,whichisnormaltothelevelsurfaceInvariantsets,conservation,anddissipation11{10 Dissipatedquantitieswesaythat:Rn!Risadissipatedquantityforsystem_x=(x)ifforalltrajectories,(x())is(weakly)decreasing,i.e.,(x())(x())forallclassicalexamples:totalenergyofamechanicalsystemwithdampingtotal\ruidinasystemthatleakscondition:_(z)0forallz,i.e.,r(z)(z)0_issometimescalledthedissipationfunctionifisdissipatedquantity,sublevelsetsfzj(z)agareinvariantInvariantsets,conservation,anddissipation11{11 Geometricinterpretation =const.r(z)(z)x()vectoreldpointsintosublevelsetsr(z)(z)0,i.e.,randalwaysmakeanobtuseangletrajectoriescanonly\slipdown"tolowervaluesofInvariantsets,conservation,anddissipation11{12 Examplelinearmechanicalsystemwithdamping:Mq+D_q+Kq=0q()2RnisdisplacementorcongurationM=M�0ismassorinertiamatrixK=K�0isstinessmatrixD=D0isdampingorlossmatrixwe'llusestatex=(q;_q),so_x=_qq=0IMKMDxInvariantsets,conservation,anddissipation11{13 considertotal(potentialpluskinetic)energyE=1 2qKq+1 2_qM_q=1 2xK00Mxwehave_E(z)=rE(z)(z)=zK00M0IMKMDz=z0KKDz=_qD_q0makessense: (totalstoredenergy)=(powerdissipated)Invariantsets,conservation,anddissipation11{14 Trajectorylimitwithdissipatedquantitysuppose:Rn!Risdissipatedquantityfor_x=(x)(x())!as!1,where2R[f1giftrajectoryxisboundedand_iscontinuous,x()convergestothezero-dissipationset:x()!D=fzj_(z)=0gi.e.,dist(x();)!0,as!1(moreonthislater)Invariantsets,conservation,anddissipation11{15 Linearfunctionsandlineardynamicalsystemsweconsiderlinearsystem_x=Axwhenisalinearfunction(z)=czconservedordissipated?_=r(z)(z)=cAz_(z)0forallz()_(z)=0forallz()Ac=0i.e.,isdissipatedifonlyifitisconserved,ifandonlyififAc=0(cislefteigenvectorofAwitheigenvalue0)Invariantsets,conservation,anddissipation11{16 Quadraticfunctionsandlineardynamicalsystemsweconsiderlinearsystem_x=Axwhenisaquadraticform(z)=zPzconservedordissipated?_(z)=r(z)(z)=2zPAz=z(AP+PA)zi.e.,_isalsoaquadraticformisconservedifandonlyifAP+PA=0(whichmeansAandAshareatleastRank(P)eigenvalues)isdissipatedifandonlyifAP+PA0Invariantsets,conservation,anddissipation11{17 Acriterionforinvariancesuppose:Rn!Rsatises(z)=0=)_(z)0thenthesetC=fzj(z)0gisinvariantidea:alltrajectoriesonboundaryofCcutintoC,sononecanleavetoshowthis,supposetrajectoryxsatisesx()2C,x(s)62C,sconsider(dierentiable)function:R!Rgivenby()=(x())satises()0,(s)&#x-0.1;儤0anysuchfunctionmusthaveatleastonepointT2[t;s]where(T)=0,0(T)0(forexample,wecantakeT=minfj()=0g)thismeans(x(T))=0and_(x(T))0,acontradictionInvariantsets,conservation,anddissipation11{18 Discrete-timesystemsweconsidernonlineartime-invariantdiscrete-timesystemorrecursionxt+1=(xt)wesayCRnisinvariant(withrespecttothesystem)ifforeverytrajectoryx,xt2C=)x2Cforalli.e.,trajectoriescanenter,butcannotleavesetCequivalentto:z2C=)(z)2Cexample:whenisnonnegativeorthantRn+invariantforxt+1=Axt?answer:,Aij0fori;j=1;:::;nInvariantsets,conservation,anddissipation11{19 Conservedanddissipatedquantities:Rn!Risconservedunderxt+1=(xt)if(xt)isconstant,i.e.,((z))=(z)forallzisadissipatedquantityif(xt)is(weakly)decreasing,i.e.,((z))(z)forallzwedene
:Rn!Rby
(z)=((z))(z)
(z)giveschangein,overonestep,startingatzisconservedifandonlyif
(z)=0forallzisdissipatedifandonlyif
(z)0forallzInvariantsets,conservation,anddissipation11{20 Quadraticfunctionsandlineardynamicalsystemsweconsiderlinearsystemxt+1=Axtwhenisaquadraticform(z)=zPzconservedordissipated?
(z)=(Az)P(Az)zPz=z(APAP)zi.e.,
isalsoaquadraticformisconservedifandonlyifAPAP=0(whichmeansAandAshareatleastRank(P)eigenvalues,ifAinvertible)isdissipatedifandonlyifAPAP0Invariantsets,conservation,anddissipation11{21