机械电气工程外文翻译外文文献英文文献原文Word格式文档下载.docx
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机械电气工程外文翻译外文文献英文文献原文Word格式文档下载.docx
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stabilityanalysisoflimitcycles.ConclusionsandextensionsarepresentedinSection5.
2Model
Deterministichybridsystemscanberepresentedbyamodelthatisadaptedfromadifferential-algebraic(DAE)structure.Eventsareincorporatedvia
impulsiveactionandswitchingofalgebraicequations,givingtheImpulsiveSwitched(DAIS)model
■E
去三f他耐+工以九訂(阳(哲0一龙)(!
)
0=灯(冬讪三谚卜
where
x二Raredynamicstatesandy二Rarealgebraicstates;
、(.)istheDiracdelta;
u(.)istheunit-stepfunction;
n-Imn
f,hj:
R>
R;
(0)(i)nmn(.)
g,g:
r>
r;
someelementsofeachgwillusuallybeidentically
zero,butnoelementsofthecompositegshouldbeidenticallyzero;
the
(iJ
g
aredefinedwiththesameformasgin
(2),resultinginarecursivestructureforg;
y,yareselectedelementsofythattriggeralgebraicswitchingand
state
reset(impulsive)eventsrespectively;
y
andymaysharecommon
de
elements.
Theimpulseandunit-steptermsoftheDAISmodelcanbeexpressedinalternativeforms:
Eachimpulsetermofthesummationin
(1)canbeexpressedinthestateresetform
=加(:
fi,旷)whfrn=0⑶
wherethenotationxdenotesthevalueofxjustaftertheresetevent,whilst
xandyrefertothevaluesofxandyjustpriortotheevent.
(i土
Thecontributionofeachg
in
(2)canbeexpressedas
with
(2)becoming
0=三/毋(拓讪+力9"
"
(工,10”⑷
i=l
Thisformisoftenmoreintuitivethan
(2).
Itcanbeconvenienttoestablishthepartitions
■E
鱼
If]
r靈1
X=
z
/=
ihj=
A
■■
c
入-
Xarethecontinuousdynamicstates,forexamplegeneratorangles,velocitiesandfluxes;
zarediscretedynamicstates,suchastransformertappositionsandprotectionrelaylogicstates;
areparameterssuchasgeneratorreactances,controllergainsandswitchingtimes.
Thepartitioningofthedifferentialequationsfensuresthatawayfrom
constant.Similarly,thepartitioningoftheresetequationshjensuresthat
xandremainconstantatresetevents,butthedynamicstateszarereset
Themodelcancapturecomplexbehaviour,fromhysteresisandnon-winduplimitsthroughtorule-basedsystems[I].Amoreextensivepresentationof
thismodelisgivenin[9].
Awayfromevents,systemdynamicsevolvesmoothlyaccordingtothefamiliardifferential-algebraicmodel
金=/(x,y)*⑹
wheregiscomposedof
(o)
togetherwithappropriatechoicesof
gor
0-y)⑺
(i)
dependingonthesignsofthecorrespondingelementsofyd.Atswitchingevents
(2),somecomponentequationsofgchange.Tosatisfythe
newg=0equation,algebraicvariablesymayundergoastepchange.Resetevents(3)forceadiscretechangeinelementsofx.Algebraicvariablesmayalsostepatareseteventtoensureg=0issatisfiedwiththealteredvaluesofx.
Theflowsofandyaredefinedrespectivelyas
工何=如(工饷土)(8)
=©
iXm。
⑼
alongwithinitialconditions,
(10)
(11)
wherex(t)andy(t)satisfy
(1),
(2),站)刃斑如(细S})■0.
3'
IkajectorySensitivities
Sensitivityoftheflows©
xandtoinitialconditionsy0areobtainedby
linearizing(8),(9)aboutthenominaltrajectory,
△述)二
=空芈凹g(12)
3(0=
partialderivativematricesgivenin(12),(13)areknownas
Thetime-varyingtrajectorysensitiuities,andcanbeexpressedinthealternativeforms
(14)
=工牝(f)=Vx(TQlt)
(⑸
Thisgives
(16)
f三f/:
x,andlikewise
X
fortheotherJacobian
matrices.
Notethat
(17)
f,fy,gx,gyareevaluatedalongthetrajectory,andhencearetimevaryingmatrices.Itisshownin19,101thatthenumericalsolutionofthis(potentiallyhighorder)DAEsystemcanbeobtainedasaby-productofnumericallyintegratingtheoriginalDAEsystem(6),(7).Theextracomputationalcostisminimal.
Initialconditionsforareobtainedfrom(10)as
X0
event,sothemodel
(1),
(2)reduces(effectively)totheform
x=f匕讥
(1®
n_f厂&
再)丸叭切<0
I石劝>
o
卢y)—0.
(20)
Let(x(),y(■))bethepointwherethetrajectoryencountersthetriggeringhypersurfaces(x,y)=0,i.e.,thepointwhereaneventisinitiated.Thispointiscalledthejunctionpointandristhejunctiontime.Itisassumedtheencounteristransversal.
Justpriortoeventtriggering,attime一,wehave
8一=盂(广)=氣(%广)
Itisshownin[9]thatthejumpconditionsforthesensitivities
Xxo
aregiven
by
工昭严)=忙f(厂}-(厂-町广)抵(21)
r之舖-旳(9门"
砖儿・冬乂广)
'
(归7茁肩广阪)]严广
r=f任(厂hm
厂=/{®
(r+)iir+(T+))-
Theassumptionthatthetrajectoryandtriggeringhypersurfacemeettransversallyensuresanon-zerodenominatorforThesensitivities
”Xo
y.immediatelyaftertheeventaregivenby
畑b+)w一(ify(T+))*1flf(r+>
a6(r+)^
Followingtheevent,i.e.,fort.,calculationofthesensitivitiesproceedsaccordingto(16),(17)untilthenexteventisencountered.Thejumpconditionsprovidetheinitialconditionsforthepost-eventcalculations.
4LimitCycleAnalysis
StabilityoflimitcyclescanbedeterminedusingPoincaremaps[11,12].Thissectionprovidesabriefreviewoftheseconcepts,andestablishestheconnectionwithtrajectorysensitivities.
APoincarkmapeffectivelysamplestheflowofaperiodicsystemonceeveryperiod.The
conceptisillustratedinFigure1.Ifthelimitcycleisstable,oscillationsapproachthelimitcycleovertime.ThesamplesprovidedbythecorrespondingPoincaremapapproacha
fixedpoint.Anon-stablelimitcycleresultsindivergentoscillations.Forsuchacasethe
samplesofthePoincaremapdiverge.
TodefineaPoincaremap,considerthelimitcycle-showninFigure1.Let\bea
**
encounter3at
afterTseconds,whereTistheminimumperiod
ofthelimitcycle.
hyperplanetransversalto-atx.Thetrajectoryemanatingfromxwillagain
Duetothecontinuityoftheflowxwithrespecttoinitialconditions,trajectories
startingon1inaneighbourhoodof
**
x.will,inapproximatelyTseconds,intersect-inthevicinityofx•Hencex
and1defineamapping
where“Xk”Tisthetimetakenforthetrajectorytoreturnto
二.Completedetails
canhefoundin[11,12].StabilityofthePaincaremap(22)isdeterminedbylinearizingP
atthefixedpointx,i.e.,
FromthedefinitionofP(z)givenby(22),itfollowsthatDP(
X)iscloselyrelatedtothe
trajectorysensitivities
:
X(X,T)_*
*(,T).Infact,itisshownin[11]that:
x\
where-isavectornormaltoZ.
Thematrix
x(X,T)isexactlythetrajectorysensitivitymatrixafteroneperiodofthe
limitcycle,
i.e.,startingfromxandreturningtox•Thismatrixiscalledthe
Monodromy
matrix.Itisshownin[11]thatforanautonomoussystem,oneeigenvalue
ofx(X,T)isalways1,andthecorrespondingeigenvectorliesalong
f(X,y)The
remainingeigenvaluesx(X,T)ofcoincidewiththeeigenvaluesofDP(x),andare
knownasthecharacteristicmultipliersmioftheperiodicsolution.Thecharacteristicmultipliersareindependentofthechoiceofcross-section匕.Therefore,forh
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