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Robo4x 1.2.3.1.a 1
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit3LinearizationandLyapunov Functions
Video5.1
Segment1.2.3.1.aDynamicalSystemsTheory
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 2
AgendaforthisSegment• AnalyticVF
§ have“simple”Taylorapproximations§ yield“simple”closedformflowexpressions
• Hope§ “simple”closedformflowoftheapproximateVF§ mayapproximatethe“complicated”flowoftheVF
• Actuality§ for“typical”situationsthehopeisborneout§ inthosesituations,getmorethan“approximation”
• turnsoutthat“behavior”isindistinguishable• uptochangeofcoordinates(ingeneral,nonlinear)
• Crucialissue:recognize&dealwith“atypical”cases
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 3
MultivariableTaylorSeries• Taylor’sTheorem
§ an“analytic”function§ hasaglobalTaylorexpansion
(aroundaspecifiedpoint)
§ whosefirstorderterm§ iscalleditslinearization
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 4
TaylorSeriesApproximation• Constantterm
§ exactatspecifiedpoint§ “close”approximation
• in“verysmall”• neighborhoodofpoint
• Linearization§ approximatesina“small”§ neighborhoodofthepoint§ nogeneralcriterionfor“small”
• …higherorderterms…better…• References
§ Taylor’sTheoremRichardCourant.Differentialandintegralcalculus,volume2.JohnWiley&Sons,2011.
§ Calculusvia“longpolynomials”R.Ghrist.FunnyLittleCalculusText.R.W.Ghrist,2012.
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 5
0th TaylorApproximationofFlow• Example:dampedfallingunitmass
§ ConstantapproximationofVFaroundv0=0
§ Flowofconstantapproximation(proposed0th orderapprox.ofFlow)
§ ConstantapproximationofFlowaroundv0=0
• Conclude:nottoobadforverysmallt andd
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 6
0th TaylorApproximationFailure
• Ingeneral,for“typical”v0§ flowoftheconstantapproximationVF
§ is“close”to§ constantapproximationoftheflowoftheactual§ forverysmallt andd
• Whatabout“atypical”v0 ?§ majorexception:failsbadlyatFPv0 =ve :=b/g§ since
• We’llneed1st TaylorApproximationfor“typical”FP• Firstshowthat“behaviorissame”intypicalcase
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 7
ExpandCCNotion• ChangeofCoordinates(CC)
§ continuous§ continuouslyinvertible
(one-to-one&onto)
• Examples§ Smooth(CCD):differentiablebothways§ Linear(CCL):similarity(withmatrixrep)
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 8
GeneralChangeofCoordinatesFormulae
• maps(e.g.,flows)
• VF
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 9
ScalarNonlinearChangeofCoordinates• returntodampedfallingunitmass
• introduceproposedCC
§ toassuregoodCCD§ assumeIC“far”fromFP,ve :=b/g
• getconjugacy§ sobehavior§ awayfromFP§ is“identical”§ uptoCC
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 10
NormalForm
• Previousresult§ conjugacybetweenVF&0th orderVFapproximant§ onneighborhoodsawayfromFP
• Leadstonotionof“normalform”§ thelowestdegreepolynomialapproximant§ thatstilladmitslocalconjugacytotheVF
• GeneralizestoanalyticVFinarbitrarydimensions§ Flowbox Theorem:
• thenormalformforaVFintheneighborhoodofanonFP• istheconstantVF,e.g.,fcons(x) := [1, 0, ..,0]T
• [V.I.Arnold.OrdinaryDifferentialEquations.MITPress,1973]
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 11
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit3LinearizationandLyapunov Functions
Video5.2
Segment1.2.3.1.bDynamicalSystemsTheory
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 12
AgendaforthisSegment• NormalForm(introducedinsegmentjustprevious)
§ Flowbox theoremtellsus• awayfromFP• VFisconjugatetoconstantvelocity
§ nowseeknormalformintheneighborhoodofaFP
• TaylorapproximationnearFP
§ seemsdominatedby§ goodnewswhentrulyanormalform§ sinceweunderstandLTIsystemsverywell
• NowexplorewhenthisintuitionholdsPropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 13
ConditionsforFPNormalForm:Scalars
• ForVFtohaveLTInormalformatscalarFPxe
§ musthave§ else
§ inwhichcaselinearizedVFcannotbeconjugate
§ because,e.g.,
• soh growswithoutboundalongflowofquadraticfield• whereasitisconstantalongflowofthelinearfield
§ noCCcanconjugateanunboundedtoaboundedfnc.PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 14
ConditionsforFPNormalForm:Vectors• ButforVFtohaveLTInormalformatvectorFPxe
• needmorethansimply• e.g.
§ hencelinearizedVFcannotbeconjugate
§ because,e.g.,
vs• soh growswithoutboundalongflowoffNLRC
• whereasitisconstantalongflowoffL
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 15
Hyperbolicity ofFP
• SaythataVFishyperbolic ataFP• Ifitslinearizationhasnopurelyimaginaryeigenvalues
• Examples§ scalardampedmass:§ dampedpendulum:
• FPwithzerovelocity• atqe=np• again,neednonzerob
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 16
NormalFormNearaFP• IfaFPofaVF• ishyperbolic
• thenthelinearizeddynamics
• islocallyconjugate• viasomeCCdefinedinaneighborhoodoftheFP• Reference:J.Guckenheimer andP.Holmes.NonlinearOscillations,DynamicalSystems,andBifurcationsofVectorFields.Springer,1983.
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 17
Pendulum(red)andLinearized(gray)Orbits
NearbottomFP,qb VerynearbottomFP, qb
red:
gray:
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 18
Pendulum(red)andLinearized(gray)Orbits
NeartopFP,qt VeryneartopFP, qt
red:
gray:
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 19
UnstablePend.(red)andLnrzd.(gray)Orbs.
NearbottomFP,qb VerynearbottomFP, qb
red:
gray:
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 20
ImplicationsofFPHyperbolicity
• LinearizationofHyperbolicFPpredictslocalnonlinearbehavior§ numerically:becauselineartermdominatesTaylor
expansion§ formally:becauseCCpreservesqualitativeproperties
• Mostimportantqualitativeproperty:stability
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 21
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit3LinearizationandLyapunov Functions
Video5.3
Segment1.2.3.2.aLyapunov Functions
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 22
AgendaforthisSegment• Lyapunov functions:generalizephysicalenergy
§ FundamentalTheoremofDynamicalSystems• allsystemshavesuchageneralized“energy”• defines“basinsofattraction”around“attractors”(generalizedsteadystateconditions– perhapsvery complicated)
§ Ouruseinthiscourse• inferthemfromhyperbolicattractors• findthemwhenhyperbolicity fails• usethemtofind(conservativeapproximationsof)basins
• Longtermaim:programmingwork§ attractorbasinsassymbols§ energylandscapesasprograms§ seekcompositionalmethodsfordesigninglandscapes
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 23
PositiveDefiniteFunctions• Pindownnotionof“norm-like”
§ crucial:levelsurfacesencloseneighborhoods§ guaranteedatlocalminimumofcontinuousfunctions§ preferdifferentiabilityaswell(wanttocomputepower)
• Acontinuous,scalarvaluedfunction,V,isPositiveDefinite(PD)atxe§ ifitisnonnegativeonaneighborhoodandvanishesonlyatxe
§ oftenwritten(sloppily)as“V >0”• SayPositiveSemi-Definite(PSD)whentheremightbeotherzerovalues• SayNegativeDefinite(ND)orNegativeSemi-Definite(NSD)whenthesign
reverses• e.g.NSDproperty:
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 24
PositiveDefiniteMatrices• Aquadraticform isascalar-valuedpolynomialofdegree2
§ e.g.,norm-squared||x||2 =x Tx =x12 +…+xn
2
§ e.g.,totalenergy,hHO=k +fS, foraHooke’slawspringpotential,fS§ moregenerally,canrepresentanyquadraticformwithamatrix,
• Aquadraticform,V =x T P x,representedbythematrixP, isPDat0§ ifandonlyifthereisaCCL,y=P-1/2x, suchthat
§ inwhichcaseP iscalledaPDmatrixwritten(sloppily)as“P >0”
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 25
PDQuadraticFormUnderCCL
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 26
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit3LinearizationandLyapunov Functions
Video5.4
Segment1.2.3.2.bLyapunov Functions
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 27
Stability• Asetisinvariant
§ iftheorbitthroughanyelement
§ remainswithinitforalltime§ itispositive invariant
• iftrajectoriesthroughitselements
• remainwithinitforallfuturetime
• AFPisstable undertheflowofaVF§ ifsufficientlysmall
neighborhoods§ arepositiveinvariant
UndampedPendulumnear“bottom”FP
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 28
AsymptoticStability• AFP isasymptoticallystableundertheflowofaVF§ ifitisstableand§ sufficientlysmall
neighborhoods§ approachtheFP
asymptotically§ infuturetime
• inwhichcaseitisanattractor
• whosebasin§ isthesetofICs§ whichasymptotically
approachtheFP
DampedPendulumnear“bottom”FP
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 29
Instability• AFPisunstable undertheflowofaVF§ ifitfailstobestable§ i.e.,everyneighborhood§ hasICswhosetrajectories§ leaveatsomefuturetime
• anditisarepeller§ ifitisasymptoticallystable
inreversetime§ i.e.,everyICinsmallenough
neighborhoods§ hastrajectoriesthatleavein
futuretime
DampedPendulumnear“top”FP
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 30
Lyapunov Theory• PDV,isaLyapunov function(LF)foraVFataFP
• ifitspowerfunctionisNSD:• itisstrict ifthepowerfunctionisND:• Lyapunov’s Theorem:
§ LFforVFatFPimpliesstabilityofFP§ strictLFforVFatFPimpliesasymptoticstabilityofFP
• ConverseTheorem:§ ifFPofVFisstablethenithasaLF§ ifasymptoticallystablethenithasastrictLF
• Reference:V.I.Arnold.OrdinaryDifferentialEquations.MITPress,1973.
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 31
PDandNSDQuadraticForms(Seg.1.2.1.3)
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.3.1.a 32
MovingAhead• LocalLyapunov Theory
§ necessaryandsufficientconditionsforsteadystatestability§ notconstructive(buttypically“energy-like”)
• LinearizedDynamics§ numericallycloseandbehaviorallyexactaccount§ offlowintheneighborhoodofFP§ algorithmicconstructionofLF§ conservativeestimateofbasin;ofrobustness
• GlobalLyapunov Theory§ notdevelopedinthiscourse§ fundamentaltheoremofdynamicalsystems§ idea:generalizedenergylandscapesforprogrammingwork
PropertyofPennEngineeringandDanielE.Koditschek