Modeling and Analysis of Hybrid Systems Erika brahm RWTH Aachen - - PowerPoint PPT Presentation

▶
modeling and analysis of hybrid systems
SMART_READER_LITE
LIVE PREVIEW

Modeling and Analysis of Hybrid Systems Erika brahm RWTH Aachen - - PowerPoint PPT Presentation

Aug 13, 2022 166 likes •1.06k views

Modeling and Analysis of Hybrid Systems Erika brahm RWTH Aachen University, Germany Beijing, September 2013 Contents 1 Modeling 2 Reachability analysis 3 Counterexample generation Erika brahm - Modeling and Analysis of Hybrid Systems 1

slide-1
SLIDE 1

Modeling and Analysis of Hybrid Systems

Erika Ábrahám

RWTH Aachen University, Germany

Beijing, September 2013

slide-2
SLIDE 2

Contents

1 Modeling 2 Reachability analysis 3 Counterexample generation

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 1 / 28

slide-3
SLIDE 3

Contents

1 Modeling 2 Reachability analysis 3 Counterexample generation

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 2 / 28

slide-4
SLIDE 4

Modeling with hybrid automata

Thermostat example

ℓon ˙ x = K(h − x) x ≤ 23 ℓoff ˙ x = −Kx x ≥ 17 x = 20 x ≥ 22 x ≤ 18

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 3 / 28

slide-5
SLIDE 5

Some interesting subclasses of hybrid automata

subclass derivatives conditions bounded unbounded reachability reachability timed automata ˙ x = 1 x ∼ c decidable decidable initialized ˙ x ∈ [c1, c2] x ∼ [c1, c2] decidable decidable rectangular automata reset by derivative change linear hybrid automata I ˙ x = c x ∼ glinear( x) decidable undecidable linear hybrid automata II ˙ x = flinear( x) x ∼ glinear( x) undecidable undecidable general hybrid automata ˙ x = f( x) x ∼ g( x) undecidable undecidable [Henzinger et al., 1998]

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 4 / 28

slide-6
SLIDE 6

Contents

1 Modeling 2 Reachability analysis 3 Counterexample generation

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 5 / 28

slide-7
SLIDE 7

Some tools

Uppaal [Behrmann et al., 2004] HyTech [Henzinger et al., 1997] PHAVer [Frehse, 2005] SpaceEx [Frehse et al., 2011] d/dt [Asarin et al., 2002] Ellipsoidal toolbox [Kurzhanski et al., 2006] MATISSE [Girard et al., 2007] Multi-Parametric Toolbox [Kvasnica et al., 2004] Flow∗ [Chen et al., 2012]

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 6 / 28

slide-8
SLIDE 8

The two most popular techniques for reachability analysis

Given: hybrid automaton + set of unsafe states Abstraction Iterative forward/backward search

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 7 / 28

slide-9
SLIDE 9

Iterative forward search

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

slide-10
SLIDE 10

Iterative forward search

We need a (possibly over-approximative) state set representation and

  • perations on them like intersection, union, linear transformation and

Minkowski sum.

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

slide-11
SLIDE 11

Iterative forward search

We need a (possibly over-approximative) state set representation and

  • perations on them like intersection, union, linear transformation and

Minkowski sum. The representation is crucial for the representation size, efficiency and accuracy.

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

slide-12
SLIDE 12

Iterative forward search

We need a (possibly over-approximative) state set representation and

  • perations on them like intersection, union, linear transformation and

Minkowski sum. The representation is crucial for the representation size, efficiency and accuracy.

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

slide-13
SLIDE 13

Minkowski sum

x1 x2 1 2 3 1 2 3 P

⊕

x1 x2 1 2 3 1 2 3 Q

=

x1 x2 1 2 3 1 2 3 P ⊕ Q

P ⊕ Q = {p + q | p ∈ P and q ∈ Q}

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 9 / 28

slide-14
SLIDE 14

Most well-known state set representations

Geometric objects: hyperrectangles [Moore et al., 2009]

  • riented rectangular hulls [Stursberg et al., 2003]

convex polyhedra [Ziegler, 1995] [Chen at el, 2011]

  • rthogonal polyhedra [Bournez et al., 1999]

template polyhedra [Sankaranarayanan et al., 2008] ellipsoids [Kurzhanski et al., 2000] zonotopes [Girard, 2005]) Other symbolic representations: support functions [Le Guernic et al., 2009] Taylor models [Berz and Makino, 1998, 2009] [Chen et al., 2012]

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 10 / 28

slide-15
SLIDE 15

Example: Polytopes

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-16
SLIDE 16

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-17
SLIDE 17

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z

l1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-18
SLIDE 18

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces

l1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-19
SLIDE 19

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces

l1 l2 l3 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-20
SLIDE 20

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron

l1 l2 l3 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-21
SLIDE 21

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron

l1 l2 l3 l4 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-22
SLIDE 22

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-23
SLIDE 23

Example: Polytopes

Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron representation union intersection Minkowski sum V-representation by vertices easy hard easy H-representation by facets hard easy hard

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

slide-24
SLIDE 24

Linear hybrid automata I: Time evolution

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-25
SLIDE 25

Linear hybrid automata I: Time evolution

x1 x2 P

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-26
SLIDE 26

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 Q

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-27
SLIDE 27

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-28
SLIDE 28

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-29
SLIDE 29

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-30
SLIDE 30

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2 P ⊕ cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-31
SLIDE 31

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2 P ⊕ cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-32
SLIDE 32

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2 P ⊕ cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-33
SLIDE 33

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2 P ⊕ cone(Q)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-34
SLIDE 34

Linear hybrid automata I: Time evolution

x1 x2 P ˙ x1 ˙ x2 cone(Q) x1 x2 (P ⊕ cone(Q)) ∩ Inv(ℓ)

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

slide-35
SLIDE 35

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-36
SLIDE 36

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-37
SLIDE 37

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4 ℓ′ x1 x2

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-38
SLIDE 38

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4 ℓ′ x1 x2

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-39
SLIDE 39

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4 ℓ′ x1 x2 2 4

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-40
SLIDE 40

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4 ℓ′ x1 x2 2 4

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-41
SLIDE 41

Linear hybrid automata I: Discrete steps (jumps)

ℓ x1 x2 5 4 ℓ′ x1 x2 2 4 Computed via projection and Minkowski sum.

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

slide-42
SLIDE 42

Linear hybrid automata II: Time evolution

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-43
SLIDE 43

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-44
SLIDE 44

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu

R[0,δ] R[δ,2δ] R[2δ,3δ]

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-45
SLIDE 45

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

R[0,δ] R[δ,2δ] R[2δ,3δ] Ω0 Ω1 Ω2

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-46
SLIDE 46

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment:

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-47
SLIDE 47

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment:

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-48
SLIDE 48

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-49
SLIDE 49

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-50
SLIDE 50

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-51
SLIDE 51

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-52
SLIDE 52

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-53
SLIDE 53

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-54
SLIDE 54

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-55
SLIDE 55

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-56
SLIDE 56

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-57
SLIDE 57

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

V0 eAδV0

Ω0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-58
SLIDE 58

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: t δ 2δ

Ω0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-59
SLIDE 59

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-60
SLIDE 60

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0 eAδΩ0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-61
SLIDE 61

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0 eAδΩ0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-62
SLIDE 62

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0 eAδΩ0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-63
SLIDE 63

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-64
SLIDE 64

Linear hybrid automata II: Time evolution

Assume ˙ x = Ax + Bu Compute Ω0, Ω1, . . . such that R[iδ,(i+1)δ] ⊆ Ωi

The first flowpipe segment: The remaining ones: t δ 2δ

Ω0 eAδΩ0 ⊕ V

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

slide-65
SLIDE 65

Linear hybrid automata II: Discrete steps (jumps)

Ω0 Ω1 Ω2 Ω3

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-66
SLIDE 66

Linear hybrid automata II: Discrete steps (jumps)

Ω0 Ω1 Ω2 Ω3

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-67
SLIDE 67

Linear hybrid automata II: Discrete steps (jumps)

Ω0 Ω1 Ω2 Ω3

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-68
SLIDE 68

Linear hybrid automata II: Discrete steps (jumps)

Ω′

1

Ω′

2

Ω′

3

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-69
SLIDE 69

Linear hybrid automata II: Discrete steps (jumps)

Π1 Π2 Π3 Ω′

1

Ω′

2

Ω′

3

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-70
SLIDE 70

Linear hybrid automata II: Discrete steps (jumps)

Π1 Π2 Π3 Ω′

1

Ω′

2

Ω′

3

V1

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 15 / 28

slide-71
SLIDE 71

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-72
SLIDE 72

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-73
SLIDE 73

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-74
SLIDE 74

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-75
SLIDE 75

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-76
SLIDE 76

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-77
SLIDE 77

Linear hybrid automata II: The global picture

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 16 / 28

slide-78
SLIDE 78

Our contribution: Taylor model representation of state sets

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 17 / 28

slide-79
SLIDE 79

Fetch the tool

http://systems.cs.colorado.edu/research/cyberphysical/taylormodels/

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 18 / 28

slide-80
SLIDE 80

Contents

1 Modeling 2 Reachability analysis 3 Counterexample generation

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 19 / 28

slide-81
SLIDE 81

Counterexamples for hybrid automata

Only a few approaches are available, mostly for rectangular automata Possible techniques?

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 20 / 28

slide-82
SLIDE 82

Simulation-based approach

Identify timed traces as possible counterexamples l0

[t0,t′

0],e0

− → l1

[t1,t′

1],e1

− → . . . Validate them using simulation

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 21 / 28

slide-83
SLIDE 83

Extracting the path length

  • n

. . . ∧ ˙ tr=0 t≤22

  • ff

. . . ∧ ˙ tr=0 t≥18 t≥21 ∧ tr ≤maxtr−1 ∧ tr′=tr+1 t≤19 ∧ tr ≤maxtr−1 ∧ tr′=tr+1 t=20∧ tr=0

  • n

. . . ∧ ˙ tr=0 t≤22 ∧ tr ≤maxtr

  • ff

. . . ∧ ˙ tr=0 t≥18 ∧ tr ≤maxtr t≥21 ∧ tr′=tr+1 t≤19 ∧ tr′=tr+1 t=20∧ tr=0 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 22 / 28

slide-84
SLIDE 84

Naive trace encoding

  • n0

. . . ∧

i∈{1,2,3} ˙

tri = 0 t ≤ 22

  • ff1

. . . ∧

i∈{1,2,3} ˙

tri = 0 t ≥ 18

  • n2

. . . ∧

i∈{1,2,3} ˙

tri = 0 t ≤ 22

  • ff3

. . . ∧

i∈{1,2,3} ˙

tri = 0 t ≥ 18 t ≥ 21 ∧ tr′

1 = 0

t ≤ 1 9 ∧ t r

′ 2

= 1 t ≥ 21 ∧ tr′

3 = 0

t = 20

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 23 / 28

slide-85
SLIDE 85

Getting time information

  • n0

t ≤ 22 ˙ t = 1 ∧ ˙ timer = 1∧

  • i∈{1,...,3}

˙ tstampi = 0

  • ff1

t ≥ 18 ˙ t = 1 ∧ ˙ timer = 1∧

  • i∈{1,...,3}

˙ tstampi = 0

  • n2

t ≤ 22 ˙ t = 1 ∧ ˙ timer = 1∧

  • i∈{1,...,3}

˙ tstampi = 0

  • ff3

t ≥ 18 ˙ t = 1 ∧ ˙ timer = 1∧

  • i∈{1,...,n}

˙ tstampi = 0 t ≥ 21 ∧ tstamp′

1 = timer

∧timer′ = 0 t ≤ 1 9 ∧ t s t a m p′

2 = timer

∧ t i m e r′ = t ≥ 21 ∧ tstamp′

3 = timer

∧timer′ = 0 t = 20 ∧ timer = 0∧

  • i∈{1,...,n} tstampi = 0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 24 / 28

slide-86
SLIDE 86

The search tree

(l0, P0) 3 (l1, P1) 3 (l2, P2) 2 (l4, P4) 0 (l3, P3) 0 (l5, P5) 1 (l6, P6) 0 (l7, P7) 0

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 25 / 28

slide-87
SLIDE 87

Simulation-based validation

Domain contraction Linear ODEs: time successor computations closed to exact Non-liner ODEs: numerical methods Problems: invariants, hit equality guards Search heuristics Validity metrics

100 200 300 400 500 600 700 800 900 1000 10 20 30 40 50 60 70 80 Temperatures of Coffee and Room

Time (s) Temperature (C)

Coffee - Exact Room - Exact Coffee - Euler Room - Euler Coffee - RK2 Room - RK2 Coffee - RK4 Room - RK4

100 200 300 400 500 600 700 800 900 1000 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Approximation Error of Temperatures Time (s) Relative Approximation Error (%)

Coffee - Euler Room - Euler Coffee - RK2 Room - RK2 Coffee - RK4 Room - RK4

284 285 286 287 288 289 290 31.6 31.65 31.7 31.75 31.8 31.85 31.9 Time (s) Temperature (C) 161.8 161.85 161.9 161.95 162 162.05 162.1 162.15 162.2 20.645 20.65 20.655 20.66 20.665 20.67 20.675 20.68 Time (s) Temperature (C)

Euler RK2, RK4 Coffee Room Coffee, Room Euler RK2, RK4 Exact RK2, RK4 Euler Exact Coffee Room

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 26 / 28

slide-88
SLIDE 88

Under-approximative reachability analysis

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 27 / 28

slide-89
SLIDE 89

Provably correct counterexamples using Ariadne

Erika Ábrahám - Modeling and Analysis of Hybrid Systems 28 / 28