spaceex: scalable verification of hybrid...
TRANSCRIPT
![Page 1: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/1.jpg)
Colas Le Guernic CMACS meeting – 1 / 22
SpaceEx:
Scalable Verification of Hybrid Systems
Colas Le Guernic
April 29, 2011
joint work with:Goran Frehse, Alexandre Donze, Scott Cotton, Rajarshi Ray, Olivier Lebeltel,
Rodolfo Ripado, Antoine Girard, Thao Dang, and Oded Maler.
![Page 2: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/2.jpg)
SpaceEx
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 2 / 22
SpaceEx is a software platform for reachability and safetyverification of hybrid systems developed at Verimag.
http://spaceex.imag.fr/
![Page 3: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/3.jpg)
Hybrid Systems
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 3 / 22
x ∈ f1(x) x ∈ f2(x)
x ∈ f3(x)
x ∈ f4(x)
x ∈ G1,2
x← R1,2(x)
x ∈ G2,4
x← R2,4(x)
x ∈ G2,3
x← R2,3(x)
x ∈ G3,2
x← R3,2(x)
x ∈ G3,1
x← R3,1(x)
![Page 4: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/4.jpg)
Example
Colas Le Guernic CMACS meeting – 4 / 22
HEAT
dTdt
= fHeat(T )
OFF
dTdt
= fOff(T )
T > 22C
T < 18C
![Page 5: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/5.jpg)
Example
Colas Le Guernic CMACS meeting – 4 / 22
HEAT
dTdt
= fHeat(T )
OFF
dTdt
= fOff(T )
T > 22C
T < 18C
![Page 6: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/6.jpg)
Example
Colas Le Guernic CMACS meeting – 4 / 22
HEAT
dTdt
= fHeat(T )
OFF
dTdt
= fOff(T )
T > 22C
T < 18C
![Page 7: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/7.jpg)
Example
Colas Le Guernic CMACS meeting – 4 / 22
HEAT
dTdt
= fHeat(T )
OFF
dTdt
= fOff(T )
T > 22C
T < 18C
![Page 8: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/8.jpg)
Example
Colas Le Guernic CMACS meeting – 4 / 22
HEAT
dTdt
∈ FHeat(T )
OFF
dTdt
∈ FOff(T )
T > 22C
T < 18C
![Page 9: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/9.jpg)
CMACS
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 5 / 22
![Page 10: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/10.jpg)
CMACS
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 5 / 22
Parameter Estimationwith RoVerGeNe.
![Page 11: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/11.jpg)
CMACS
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 5 / 22
Parameter Estimationwith RoVerGeNe.
Based on abstractionsby discrete automata.
![Page 12: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/12.jpg)
Parameter Estimation in SpaceEx
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 6 / 22
SpaceEx, Reachability for:
LHA HA with linear dynamicsx ∈ P x ∈ Ax+ u | u ∈ U
No parameter estimation.
![Page 13: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/13.jpg)
Parameter Estimation in SpaceEx
Introduction
SpaceEx
Hybrid Systems
Example
CMACSParameterEstimation inSpaceEx
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 6 / 22
SpaceEx, Reachability for:
LHA HA with linear dynamicsx ∈ P x ∈ Ax+ u | u ∈ U
Parameters as variables with 0 derivative.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x1
x2
initial states
final states
reachable
final states
reachable
states R1
![Page 14: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/14.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
![Page 15: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/15.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
Postc : Continuous evolution
![Page 16: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/16.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
Postc : Continuous evolutionPostd : Discrete transition
![Page 17: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/17.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
Postc : Continuous evolutionPostd : Discrete transition
![Page 18: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/18.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
Postc : Continuous evolutionPostd : Discrete transition
![Page 19: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/19.jpg)
Reachability Analysis
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 7 / 22
Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps
Postc : Continuous evolutionPostd : Discrete transition
![Page 20: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/20.jpg)
Reachability for Linear Time Invariant systems
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 8 / 22
Describe all x(t) ∈ Rd for any t in [0;T ] such that :
x(t) = Ax(t) + u(t) with x(0) ∈ X0 and u(t) ∈ U
![Page 21: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/21.jpg)
Reachability for Linear Time Invariant systems
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 8 / 22
Describe all x(t) ∈ Rd for any t in [0;T ] such that :
x(t) = Ax(t) + u(t) with x(0) ∈ X0 and u(t) ∈ U
Analytical solution for a given input function u:
x(t) = etAx(0) +
∫ t
0e(t−s)Au(s) ds
![Page 22: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/22.jpg)
Reachability for Linear Time Invariant systems
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 8 / 22
Describe all x(t) ∈ Rd for any t in [0;T ] such that :
x(t) = Ax(t) + u(t) with x(0) ∈ X0 and u(t) ∈ U
Analytical solution for a given input function u:
x(t) = etAx(0) +
∫ t
0e(t−s)Au(s) ds
Temporal discretization:
Reach[tk,tk+δk](X0) = eAtkReach[0,δk](X0) ⊕ Reach[tk,tk](0)
![Page 23: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/23.jpg)
Reachability for Linear Time Invariant systems
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 8 / 22
Describe all x(t) ∈ Rd for any t in [0;T ] such that :
x(t) = Ax(t) + u(t) with x(0) ∈ X0 and u(t) ∈ U
Analytical solution for a given input function u:
x(t) = etAx(0) +
∫ t
0e(t−s)Au(s) ds
Temporal discretization:
Ψk+1 = Ψk ⊕ eAtkΨδk(U)
Ωk = eAtkΩ[0,δk](X0,U) ⊕ Ψk
![Page 24: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/24.jpg)
Guards
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 9 / 22
![Page 25: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/25.jpg)
Guards
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 9 / 22
![Page 26: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/26.jpg)
Guards
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 9 / 22
![Page 27: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/27.jpg)
Guards
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 9 / 22
![Page 28: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/28.jpg)
Representing Sets
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 10 / 22
![Page 29: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/29.jpg)
Scalable Computation by Transformation
Introduction
Reachability Analysis
Reachability Analysis
LTI
Guards
Representing Sets
Scalable
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 11 / 22
![Page 30: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/30.jpg)
Support Function
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 12 / 22
![Page 31: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/31.jpg)
Definition
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 13 / 22
The support function of a compact convex set S ⊆ Rd, denoted
ρS , is defined as:
ρS : Rd → R
ℓ 7→ maxx∈S ℓ · x
![Page 32: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/32.jpg)
Definition
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 13 / 22
The support function of a compact convex set S ⊆ Rd, denoted
ρS , is defined as:
ρS : Rd → R
ℓ 7→ maxx∈S ℓ · x
−2
0
2
4
6
8
ℓ
![Page 33: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/33.jpg)
Examples
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 14 / 22
support function of the unit cube B∞:
ρB∞(ℓ) = ‖ℓ‖1 =
d−1∑
i=0
|ℓi|
support function of a ball S of center c and radius r:
ρS(ℓ) = c · ℓ+ r‖ℓ‖2
support function of a polytope P = x : Ax ≤ b: any LPalgorithm solving:
maxx · ℓAx ≤ b
![Page 34: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/34.jpg)
Representing sets
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 15 / 22
If S is convex, then:
S =⋂
ℓ∈Rd
x : x · ℓ ≤ ρS(ℓ)
![Page 35: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/35.jpg)
Properties
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 16 / 22
Linear transformation:
ρAS(ℓ) = ρS(A⊤ℓ)
Minkowski sum:
X ⊕ Y = x+ y : x ∈ X and y ∈ Y
ρX⊕Y(ℓ) = ρX (ℓ) + ρY(ℓ)
Convex union:
ρCH(X∪Y)(ℓ) = max(ρX (ℓ), ρY(ℓ))
![Page 36: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/36.jpg)
ρΩk(ℓ)
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 17 / 22
Ψk+1 = Ψk ⊕ eAtkΨδk(U)
Ωk = eAtkΩ[0,δk](X0,U) ⊕ Ψk
For any direction ℓ:
ψk+1(ℓ) = ψk(ℓ) + ρΨδk((eAtk)⊤ℓ)
ρΩk(ℓ) = ρΩ[0,δk]
((eAtk)⊤ℓ) ⊕ ψk(ℓ)
![Page 37: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/37.jpg)
Ω[0,δk](X0,U)
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 18 / 22
Let λ ∈ [0, 1], and Ωλ(X0,U , δ) be the convex set defined by :
Ωλ(X0,U , δ) = (1 − λ)X0 ⊕ λeδAX0 ⊕ λδU
⊕(
λE+Ω (X0, δ) ∩ (1 − λ)E−
Ω (X0, δ))
⊕ λ2EΨ(U , δ)
where E+Ω (X0, δ) = ⊡
(
Φ2(|A|, δ) ⊡(
A2X0
))
and E−
Ω (X0, δ) = ⊡(
Φ2(|A|, δ) ⊡(
A2eδAX0
))
and EΨ(U , δ) = ⊡ (Φ2(|A|, δ) ⊡ (AU)) .
Then Reachλδ,λδ(X0,U) ⊆ Ωλ(X0,U , δ). If we define Ω[0,δ](X0,U) as:
Ω[0,δ](X0,U) = CH(
⋃
λ∈[0,1]
Ωλ(X0,U , δ))
,
then Reach0,δ(X0) ⊆ Ω[0,δ](X0,U).
![Page 38: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/38.jpg)
ρΩ[0,δk](ℓ)
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 19 / 22
Ωλ(X0,U , δ) = (1 − λ)X0 ⊕ λeδAX0 ⊕ λδU
⊕(
λE+Ω (X0, δ) ∩ (1 − λ)E−
Ω (X0, δ))
⊕ λ2EΨ(U , δ)
Ω[0,δ](X0,U) = CH(
⋃
λ∈[0,1]
Ωλ(X0,U , δ))
,
![Page 39: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/39.jpg)
ρΩ[0,δk](ℓ)
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 19 / 22
Ωλ(X0,U , δ) = (1 − λ)X0 ⊕ λeδAX0 ⊕ λδU
⊕(
λE+Ω (X0, δ) ∩ (1 − λ)E−
Ω (X0, δ))
⊕ λ2EΨ(U , δ)
Ω[0,δ](X0,U) = CH(
⋃
λ∈[0,1]
Ωλ(X0,U , δ))
,
ρΩ[0,δ](ℓ) = max
λ∈[0,1]
(
(1 − λ)ρX0(ℓ) + λρX0
((eδA)⊤ℓ) + λδρU (ℓ)
+ ρλE+Ω∩(1−λ)E−
Ω(ℓ) + λ2ρEΨ
(ℓ))
![Page 40: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/40.jpg)
Time step adaptation with error bounds
Introduction
Reachability Analysis
Support Function
Definition
Examples
Representing sets
Properties
ρΩk(ℓ)
Ω[0,δk ](X0, U)
ρΩ[0,δk](ℓ)
Time Step
SpaceEx
Colas Le Guernic CMACS meeting – 20 / 22
A user defined time step is arbitrary Time step guided by requested quality of approximation:
ǫΩk(ℓ) = ρ (ℓ,Ωk) − ρ
(
ℓ,Reachtk,tk+1(X0)
))
linear accumulation of errors
ǫΨk(ℓ) + ǫΨ
δΨk
(U)(eAtΨk
⊤ℓ) ≤
tΨk + δΨkT
ǫΨ
computed independently for each direction
![Page 41: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/41.jpg)
SpaceEx Platform - Architecture
Introduction
Reachability Analysis
Support Function
SpaceEx
Architecture
Colas Le Guernic CMACS meeting – 21 / 22
![Page 42: SpaceEx: Scalable Verification of Hybrid Systemscmacs.cs.cmu.edu/presentations/UMD-slides/LeGuernic-CMACS.pdfColas Le Guernic CMACS meeting – 1 / 22 SpaceEx: Scalable Verification](https://reader033.vdocument.in/reader033/viewer/2022060808/608d36c5f929f4754d06d6d9/html5/thumbnails/42.jpg)
Thank you
Introduction
Reachability Analysis
Support Function
SpaceEx
Colas Le Guernic CMACS meeting – 22 / 22