Locally Optimal Reach Set Over-approximation for Nonlinear ...cfan10.web.engr.illinois.edu/wp-content/uploads/2018/08/EMSOFT2016talk.pdf · Locally Optimal Reach Set Over-approximation

LocallyOptimalReachSetOver-approximationfor

NonlinearSystems

EMSOFT 2016

ChuchuFan Sayan Mitra JimKapinski XiaoqingJin

EMSOFT2016⋅ Locallyoptimalreachability⋅ Chuchu Fan⋅ UIUC

Howtochecksafetyofanautonomousmaneuver?

2

𝜔 𝑠$

reachthreshold

switchtoleft

overtake

switchtoright

gainthreshold

abort

Givencontrollerandseparationrequirement,checksafetywithrespecttorangesofinitialrelativepositions,speeds,roadconditions.


certificate

model,simulator,

requirements

bugtraceVerificationAlgorithms

Verificationchallenge

Bugdiscovery→ fasterdevelopment

Certificate→ evidenceforDO178C,ISO26262,etc.

Challenge:modelsofcomplexcontrolsystemsoftendonothave

analyticalsolutions→ Simulation⇒ proofs?3


SafetyverificationproblemConsidernonlinearODE�� = 𝑓 𝑥 ,𝑥 ∈ ℝ-

‒ Trajectory𝜉 𝑥/, 𝑡 : stateattime𝑡 frominitialstate𝑥/

‒ Reachtube 𝜉(𝐵(𝑥/, 𝛿), 𝑇):allstatesreachablefrominitialset𝐵(𝑥/, 𝛿) ⊆ ℝ- uptotime𝑇

Safetyverificationproblem:giveninitialset𝐵(𝑥/, 𝛿), unsafesetU,timebound𝑇, decide𝜉 𝐵(𝑥/, 𝛿), 𝑇 ∩ U = ∅?

4

Unsafe

𝜉 𝑑/, 𝑡

time

Relativedistance

𝑑/𝐵(𝑑/, 𝛿) 𝜉(𝐵(𝑥/, 𝛿), 𝑇)


5

Simulation-drivenverificationstrategyGivenstartandunsafeComputefinitecoverofinitialsetSimulatefromthecenter𝑥/ ofeachcoverGeneralize simulationtoreachtube sothatreachtube containsalltrajectoriesfromthecoverCheckintersection/containmentwith𝑈RefineUnion=over-approximationofreachset

Θ 𝑈

Keystep:𝜉 𝑥/, 𝑡 ->𝜉 𝐵 𝑥/, 𝛿 , 𝑇

𝜉 𝑑/, 𝑡

time

Relativedistance

𝑑/𝐵(𝑑/, 𝛿) 𝜉(𝐵(𝑥/, 𝛿), 𝑇)

Greytube:UnknownGreentube:Safe

FeedbackFridayPresentationEMSOFT2016⋅ Locallyoptimalreachability⋅ Chuchu Fan⋅ UIUC

6

Main problem: How to quantifygeneralization?

Discrepancy formalizes generalization:

Discrepancy is a continuousfunction𝛽 thatboundsthedistancebetweenneighboringtrajectories

𝜉 𝑥B, 𝑡 − 𝜉(𝑥D, 𝑡) ≤ 𝛽 𝑥B − 𝑥D , 𝑡 ,

From a single simulation of 𝜉(𝑥B, 𝑡)anddiscrepancy 𝛽 we can over-approximatethereachtube

𝑥B

𝑥D𝜉 𝑥D, 𝑡

𝜉 𝑥B, 𝑡𝛽(‖𝑥B − 𝑥D‖, 𝑡)


7

Asimpleexampleofdiscrepancyfunction

If𝑓(𝑥)hasaLipschitzconstant𝐿:

∀𝑥, 𝑦 ∈ ℝ-, 𝑓 𝑥 − 𝑓 𝑦 ≤ 𝐿 𝑥 − 𝑦

Example:�� = −2𝑥,Lipschitzconstant𝐿 = 2

thena(bad)discrepancyfunctionis

𝜉 𝑥B, 𝑡 − 𝜉(𝑥D, 𝑡) ≤ 𝑥B − 𝑥D 𝑒MN = 𝛽 𝑥B − 𝑥D , 𝑡

𝑥B




8

Asimpleexampleofdiscrepancyfunction

�� = −2𝑥,Lipschitzconstant𝐿 = 2, 𝛿 = 1

𝑥B




𝛽(‖𝑥B − 𝑥D‖, 𝑡)𝑥B


𝜉 𝑥B, 𝑡

9

Whatisagooddiscrepancy?

General:Appliestogeneralnonlinear𝑓

Accurate:Smallerrorin𝛽

Effective:Computing𝛽isfast(inpractice)


Matrixmeasures can givetightdiscrepancy

Theorem [Sontag10]: For any 𝒟 ⊆ ℝ-,if all trajectoriesstarting from the line betweenanytwoinitialstates𝑥B and𝑥Dremainsin𝒟then: 𝜉 𝑥B, 𝑡 − 𝜉 𝑥D, 𝑡 ≤ 𝑥B − 𝑥D 𝑒QN,

where c = max$∈𝒟

𝜇 𝐽 𝑥 and

𝜇 𝐽 𝑥 is amatrix measure of Jacobian

𝐽 𝑥 = XYZ $X$[

istheJacobianmatrixoff

This𝑐 can be <0, usually <<Lipschitz constant

10

𝒟

𝑥B

𝑥D 𝜉 𝑥D, 𝑡

𝜉 𝑥B, 𝑡

Example: �� = 𝑣D + 𝑤D

−𝑣

Jacobian:𝐽 𝑣𝑤 = 2𝑣 2𝑤

−1 0


Matrixmeasurefor 𝐴 ∈ ℝ-×-

Matrixmeasure[Dahlquist 59]:

𝜇 𝐴 = limN→/f

𝐼 + 𝑡𝐴 − 𝐼𝑡

2-norm:𝜇(𝐴) = 𝜆ij$klkm

D

11

Matrixnorm

𝐴 = max$n/

𝐴𝑥𝑥

𝐴 D = 𝜆ij$(𝐴o𝐴)�


Matrixmeasure[Desoer 72]:

𝜇 𝐴 = limN→/f

𝐼 + 𝑡𝐴 − 𝐼𝑡

2-norm:𝜇(𝐴) = 𝜆ij$klkm

D

Definitionofmatrixmeasures

12

Foranymatrix𝐴 ∈ ℝ-×-

Matrixnorm

𝐴 = max$n/

𝐴𝑥𝑥

𝐴 D = max 𝜆ij$(𝐴o𝐴)�

𝑐 = max$∈𝒟

𝜇 𝐽 𝑥 ①

≡ 𝑐 = max$∈𝒟

limN→/f

𝐼 + 𝑡𝐽 𝑥 − 𝐼𝑡 ②

min 𝑐

s.t. ∀𝐴 ∈ 𝒜 𝒟, 𝐽 , 𝑀𝐴 + 𝐴o𝑀 ≼ 2𝑐𝐼

𝑀 ≻ 0

…FromoriginalproblemtoanSDPprobleminthenextslides


Baselinealgorithmwith2-norm[FanandMitraATVA15]

Choosingordinarymatrix2-norm,𝜇 𝐽 𝑥 becomes:

𝜆ij$𝐽 𝑥 + 𝐽o 𝑥

2

[ATVA15]useseigenvalueofcenterJacobianmatrixandperturbationboundtomaximizethisquantityover𝒟

[CAV15]applicationtoPowertrainverificationproblem[Jin 16]

[CAV16]toolC2E2implementingthisalgorithm

13


14

Coordinatetransformationmakesreachtubetighter

Under2-norm,approximationsarerepresentedbyspheres

Usinglinearcoordinatetransformationsofstate,wecangettighterover-approximationswithellipsoids

Undercoordinatetransformation𝑃:matrixmeasureis𝜇| 𝐴 = 𝜇(𝑃𝐴𝑃}B)

𝑥B




15

Coordinatetransformationmakesreachtubetighter

Under2-normapproximationsarerepresentedbyspheres

Usinglinearcoordinatetransformationsofstate,wecangettighterover-approximationswithellipsoids

Undercoordinatetransformation𝑃:matrixmeasureis𝜇| 𝐴 = 𝜇(𝑃𝐴𝑃}B)

𝑥B



𝑐 = max$∈𝒟

𝜇 𝐽 𝑥 ①


limN→/f

𝐼 + 𝑡𝐽 𝑥 − 𝐼𝑡

②


𝜆ij$𝑃𝐽 𝑥 𝑃}B + (𝑃}B)o𝐽 𝑥 𝑃o

2 ③

Plugindefinition[Originalproblem]

[Usingcoordinatetransformation]


𝒟

𝑥B


𝜉 𝑥B, 𝑡

ApproximatingJ(x)withanintervalmatrix

𝒟 isacompactset

Each𝐽��: 𝒟 → ℝ iscontinuousandhasupper(𝑢��) andlowerbounds(𝑙��)

Computeintervalmatrix𝒜(𝒟, 𝐽) =

[∗,∗] ⋯ [∗,∗]⋮ [𝑙��, 𝑢��] ⋮

[∗,∗] ⋯ [∗,∗]

Forall𝑥 ∈ 𝒟, 𝐽 𝑥 ∈ 𝒜(𝒟, 𝐽) 𝐽(𝑥)

16

𝑐 = max$∈𝒟

𝜇 𝐽 𝑥


limN→/f




2


𝒟

𝑥B


𝜉 𝑥B, 𝑡

ApproximatingJ(x)withanintervalmatrix

𝒟 isacompact

Each𝐽��: 𝒟 → ℝ iscontinuousandthereforehasupper(𝑢��) andlowerbounds(𝑙��) over𝒟

𝒜(𝒟, 𝐽) = [∗,∗] ⋯ [∗,∗]⋮ [𝑙��, 𝑢��] ⋮

[∗,∗] ⋯ [∗,∗]

𝐽(𝑥)

17

𝑐 = max$∈𝒟

𝜇 𝐽 𝑥 ①


limN→/f


②



2 ③

⇐ maxk∈𝒜 𝒟,�

𝜆ij$𝑃𝐴𝑃}B + (𝑃}B)o𝐴𝑃o

2④

[Originalproblem]

[Usingcoordinatetransformation]

[Bound𝐽(𝑥) withintervalmatrix]


Makeitasemi-definiteproblem

18

maxk∈𝒜 𝒟,�

𝜆ij$|k|��l(|��)mk|m

D

≡ min 𝑐

s.t. ∀𝐴 ∈ 𝒜 𝒟, 𝐽 𝑃𝐴𝑃}B + (𝑃}B)o𝐴𝑃o ≼ 2𝑐𝐼

≡ min 𝑐

s.t. ∀𝐴 ∈ 𝒜 𝒟, 𝐽 , 𝑀𝐴 + 𝐴o𝑀 ≼ 2𝑐𝐼

𝑃o 𝑃 𝑃o 𝑃 𝑃o𝑃𝑃o𝑃𝐴 +𝐴𝑃o𝑃 ≼ 2𝑐𝐼{𝑀

{

𝑀

𝑐 = max$∈𝒟

𝜇 𝐽 𝑥


limN→/f




2

⇐ maxk∈𝒜 𝒟,�

𝜆ij$𝑃𝐴𝑃}B + (𝑃}B)o𝐴𝑃o

2

𝒟

𝑥B


𝜉 𝑥B, 𝑡

𝑥B − 𝑥D �𝑒QN


BoundthematrixmeasurebysolvingSDPproblem

OPT1:min𝑐s. t. 𝑀𝐴 + 𝐴o𝑀 ≼ 2𝑐𝑀, ∀𝐴 ∈ 𝒜(𝒟, 𝐽)

𝑀 ≻ 0

Theorem. Thesolution𝑐 ofOPT1giveslocally

optimal discrepancy 𝑥B − 𝑥D �𝑒QN.

Givessmallest𝑐 foranychoiceofMoverD

NotanordinarySDP,infinitenumberofconstraints!

19

𝒟

𝑥B


𝜉 𝑥B, 𝑡

𝑥B − 𝑥D �𝑒QN


Vertexmatrixalgorithm

20

𝒜(𝒟, 𝐽) =[∗,∗] ⋯ [∗,∗]⋮ ⋱ ⋮

[∗,∗] ⋯ [∗,∗]=interval 𝐵, 𝐶

where

𝐵 = ∗ ⋯ ∗⋮ ⋱ ⋮∗ ⋯ ∗

,C = ∗ ⋯ ∗⋮ ⋱ ⋮∗ ⋯ ∗

Foranyintervalmatrix𝒜(𝒟, 𝐽)=interval 𝐵, 𝐶 , itsvertexmatricesare:

𝒱 = 𝑉 ∈ ℝ-×- 𝑣�� = 𝑏�� ∨ 𝑣�� = 𝑐��}

Theorem.OPT1≡

OPT2:min𝑐

s. t. ∀𝑉 ∈ 𝒱, 𝑀𝑉 + 𝑉o𝑀 ≼ 2𝑐𝑀

𝑀 ≻ 0

Potentially2-¢ ofinequalities


𝑀 ≻ 0


Centermatrixalgorithm

21

Foranyintervalmatrix𝒜(𝒟, 𝐽)=interval 𝐵, 𝐶 ,itscentermatrixisCT 𝒜 𝒟, 𝐽 = ¤l¥

D

[∗,∗] ⋯ [∗,∗]⋮ ⋱ ⋮

[∗,∗] ⋯ [∗,∗]

∗l∗D

⋯ ∗l∗D

⋮ ⋱ ⋮∗l∗D

⋯ ∗l∗D

centermatrix

SolvetheoptimizationproblemOPT3:min𝑐’

s. t. 𝑀CT 𝒜 𝒟, 𝐽 + CT 𝒜 𝒟, 𝐽 o𝑀 ≼ 2𝑐′𝑀𝑀 ≻ 0

Computeerrorbound𝛿 ≥ 𝐸o𝑀 +𝑀𝐸 D, ∀𝐸 ∈ 𝒜 − CT(𝒜)

𝑐 = 𝑐ª +𝛿

𝜆«¬(𝑀)

Theorem.Theabove𝑐 isanupperboundofthesolutionofOPT1


𝑀 ≻ 0

Canbeachievedconservativelyinlineartime


Howtocomputetheerrorbound

Computeerrorbound 𝛿 ≥ 𝐸o𝑀 +𝑀𝐸 D, ∀𝐸 ∈ 𝒜 − CT(𝒜)

isequivalentto𝛿 ≥ ℰ D,whereℰ = 𝒜 − CT 𝒜 o𝑀 +𝑀 𝒜 − CT 𝒜 isalsoanintervalmatrix

Intervalmatrixnorm: 𝒜 = supk∈𝒜

𝐴

Theorem:foranyintervalmatrix𝒜 = interval 𝐵, 𝐶 ,for𝑝 = 1,∞

𝒜 ± =¤l¥D

+ ¥}¤D ±

22


Puttingitalltogether

Upper-bounding withasinglecforentiretimehorizoncan be tooconservative

Compute piece-wiseor local upper-bounds

Thatis,M¬, 𝑐� foreachtimeinterval𝑡�, 𝑡�lB inT

𝑥B = −𝑥D;𝑥D = 𝑥BD − 1 𝑥D + 𝑥B;

23


𝑥/

Puttingitalltogether

upper-bounding matrix measure forall𝑡 can betooconservative

Compute piece-wiseor local upper-bounds on thematrix measure

Divide 0, 𝑇 into𝑁 consecutivetimeintervals,and

ComputeexponentofdiscrepancyM¬, 𝑐� foreachtimeinterval 𝑡�, 𝑡�lB

24

𝜉 𝑥/, 𝑡

𝑡B𝑡/

𝑴𝟎, 𝒄𝟎

𝑡D

𝑴𝟏, 𝒄𝟏 …


Locallyoptimalalgorithms:accuracy

25

(Arbitraryprecision)Approximationerror→ 0whensizeoftheinitialset𝛿 → 0

(Asymptoticconvergence)Approximationerror→0 as𝑡 → ∞ forcontractivenonlinearsystemandstablelinearsystems


𝒟

𝑥B

𝑥D

𝜉 𝑥D, 𝑡

𝜉 𝑥B, 𝑡

Algorithmusing2-norm(withouttransformation)

Matrixperturbationtheorem[Teschl,99]:If𝐴 and𝐸 are𝑛×𝑛 symmetricmatrices,then

𝜆º 𝐴 + 𝐸 − 𝜆º 𝐴 ≤ 𝐸 D

Method[Fan15]:

- Findthecenterpoint𝑑/ of 𝒟,compute 𝐽Q = 𝐽(𝑑/)

- Computethelargesteigenvalue 𝜆 of 𝑆𝐽Q = (𝐽Qo + 𝐽Q)/2

- Computeerrorbound 𝑒 ≥ 𝑆𝐽 𝑥 − 𝑆𝐽Q D, ∀𝑥 ∈ 𝒟

- 𝑐 = 𝜆 + 𝑒

26

𝑑/

𝜇 𝐽 𝑥 ≤ 𝑐

min𝑐s. t. 𝑀𝐴 + 𝐴o𝑀 ≼ 2𝑐𝑀, ∀𝐴 ∈ 𝒜 𝒟, 𝐽

𝑀 ≻ 0

Let𝑀 = 𝐼,𝑐 canbecomputedwithoutsolvingtheoptimizationproblem


Summary:Locallyoptimaldiscrepancy

Methods

Baselinealgorithm Locallyoptimalalgorithms

Largesteigenvalueofcentermatrixandperturbationbound

Vertexmatrix Centermatrix

#optimizationproblems 01convexproblem with

upto2-¢ + 1constraints

1convexproblem withupto2 constraints

Tightness ofthediscrepancy

Nolocaloptimalityguarantee Locallyoptimal Locally optimalforthe

centermatrix

27


Runningtimecomparison

0.1

1

10

100

1000

10000

Flow*

LocallyoptimalAlgorithm

BaselineAlgorithm

Seconds

28

2 28Dimension


Accuracycomparison

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

1E+09 1E+10

0.1

100000000

1E+17

1E+26

1E+35

1E+44

1E+53

1E+62

Laub-Loomis BiologyModelASPolynomialHelicopter(L)

Flow*

LocallyoptimalAlgorithm

BaselineAlgorithm

29


Futuredirections:Applicationsinautomotivesystems

30

sx (blue):relativedistancealongroaddirectionsy (green):relativedistanceorthogonaltosx


Debuggingsystemswithhigh-fidelitymodels

31


Summaryandfuturedirections

Simulation+discrepancyanalysis⇒ proofs(reachtube)

Discrepancyanalysisinfluencesefficiencyandconservativenessofverification

Matrixmeasuresenableautomaticlocallyoptimalreachabilityanalysis

Future:methodsforsystemswithpartiallyknownmodels

32


LinksandreferencesPictureslinks:https://images.google.com/

References:

[Dahlquist 59]G.DAHLQUIST,Stabilityanderrorboundsinthenumericalintegrationsofordinarydifferentialequations,Trans.Roy.Inst.Tech.Stockholm130(1959).[Jin 16]Jin,Xiaoqing,etal."Powertraincontrolverificationbenchmark."Proceedingsofthe17thinternationalconferenceonHybridsystems:computationandcontrol.ACM,2014.[Sontag10]E.D.Sontag,“Contractivesystemswithinputs,”inPerspectivesinMathematicalSystemTheory,Control,andSignalProcessing.Berlin,Germany:Springer-Verlag,2010,pp.217–228.[Fan15]Fan,Chuchu,andSayan Mitra."Boundedverificationwithon-the-flydiscrepancycomputation."InternationalSymposiumonAutomatedTechnologyforVerificationandAnalysis.SpringerInternationalPublishing,2015.[Fan16]Fan,Chuchu,etal."AutomaticReachabilityAnalysisforNonlinearHybridModelswithC2E2."InternationalConferenceonComputerAidedVerification.SpringerInternationalPublishing,2016.

33

Thank you

for your precioustimeandattention

Documents

Locally Optimal Reach Set Over-approximation for Nonlinear ...cfan10.web.engr.illinois.edu/wp-content/uploads/2018/08/EMSOFT2016talk.pdf · Locally Optimal Reach Set Over-approximation