Linearization and LyapunovFunctions Video 5.1 Segment 1.2 ... · Robo4x 1.2.3.1.a 1 edXRobo4 Mini...

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


Robo4x 1.2.3.1.a 3

MultivariableTaylorSeries• Taylor’sTheorem

§ an“analytic”function§ hasaglobalTaylorexpansion

(aroundaspecifiedpoint)

§ whosefirstorderterm§ iscalleditslinearization


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.


Robo4x 1.2.3.1.a 5

0th TaylorApproximationofFlow• Example:dampedfallingunitmass

§ ConstantapproximationofVFaroundv0=0

§ Flowofconstantapproximation(proposed0th orderapprox.ofFlow)

§ ConstantapproximationofFlowaroundv0=0

• Conclude:nottoobadforverysmallt andd


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


Robo4x 1.2.3.1.a 7

ExpandCCNotion• ChangeofCoordinates(CC)

§ continuous§ continuouslyinvertible

(one-to-one&onto)

• Examples§ Smooth(CCD):differentiablebothways§ Linear(CCL):similarity(withmatrixrep)


Robo4x 1.2.3.1.a 8

GeneralChangeofCoordinatesFormulae

• maps(e.g.,flows)

• VF


Robo4x 1.2.3.1.a 9

ScalarNonlinearChangeofCoordinates• returntodampedfallingunitmass

• introduceproposedCC

§ toassuregoodCCD§ assumeIC“far”fromFP,ve :=b/g

• getconjugacy§ sobehavior§ awayfromFP§ is“identical”§ uptoCC


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]


Robo4x 1.2.3.1.a 11



Video5.2

Segment1.2.3.1.bDynamicalSystemsTheory



July,2017


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


Robo4x 1.2.3.1.a 15

Hyperbolicity ofFP

• SaythataVFishyperbolic ataFP• Ifitslinearizationhasnopurelyimaginaryeigenvalues

• Examples§ scalardampedmass:§ dampedpendulum:

• FPwithzerovelocity• atqe=np• again,neednonzerob


Robo4x 1.2.3.1.a 16

NormalFormNearaFP• IfaFPofaVF• ishyperbolic

• thenthelinearizeddynamics

• islocallyconjugate• viasomeCCdefinedinaneighborhoodoftheFP• Reference:J.Guckenheimer andP.Holmes.NonlinearOscillations,DynamicalSystems,andBifurcationsofVectorFields.Springer,1983.


Robo4x 1.2.3.1.a 17

Pendulum(red)andLinearized(gray)Orbits

NearbottomFP,qb VerynearbottomFP, qb

red:

gray:


Robo4x 1.2.3.1.a 18

Pendulum(red)andLinearized(gray)Orbits

NeartopFP,qt VeryneartopFP, qt

red:

gray:


Robo4x 1.2.3.1.a 19

UnstablePend.(red)andLnrzd.(gray)Orbs.

NearbottomFP,qb VerynearbottomFP, qb

red:

gray:


Robo4x 1.2.3.1.a 20

ImplicationsofFPHyperbolicity

• LinearizationofHyperbolicFPpredictslocalnonlinearbehavior§ numerically:becauselineartermdominatesTaylor

expansion§ formally:becauseCCpreservesqualitativeproperties

• Mostimportantqualitativeproperty:stability


Robo4x 1.2.3.1.a 21



Video5.3

Segment1.2.3.2.aLyapunov Functions



July,2017


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


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:


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”


Robo4x 1.2.3.1.a 25

PDQuadraticFormUnderCCL


Robo4x 1.2.3.1.a 26



Video5.4

Segment1.2.3.2.bLyapunov Functions



July,2017


Robo4x 1.2.3.1.a 27

Stability• Asetisinvariant

§ iftheorbitthroughanyelement

§ remainswithinitforalltime§ itispositive invariant

• iftrajectoriesthroughitselements

• remainwithinitforallfuturetime

• AFPisstable undertheflowofaVF§ ifsufficientlysmall

neighborhoods§ arepositiveinvariant

UndampedPendulumnear“bottom”FP


Robo4x 1.2.3.1.a 28

AsymptoticStability• AFP isasymptoticallystableundertheflowofaVF§ ifitisstableand§ sufficientlysmall

neighborhoods§ approachtheFP

asymptotically§ infuturetime

• inwhichcaseitisanattractor

• whosebasin§ isthesetofICs§ whichasymptotically

approachtheFP

DampedPendulumnear“bottom”FP


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


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.


Robo4x 1.2.3.1.a 31

PDandNSDQuadraticForms(Seg.1.2.1.3)


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


Linearization and LyapunovFunctions Video 5.1 Segment 1.2 ... · Robo4x 1.2.3.1.a 1 edXRobo4 Mini...

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