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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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