SpaceEx: Scalable Veriﬁcation of Hybrid...

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.

SpaceEx

Introduction

SpaceEx

Hybrid Systems

Example

CMACSParameterEstimation inSpaceEx

Reachability Analysis

Support Function

SpaceEx


SpaceEx is a software platform for reachability and safetyverification of hybrid systems developed at Verimag.

http://spaceex.imag.fr/

Hybrid Systems

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


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)

Example


HEAT

dTdt

= fHeat(T )

OFF

dTdt

= fOff(T )

T > 22C

T < 18C

Example


HEAT

dTdt

∈ FHeat(T )

OFF

dTdt

∈ FOff(T )

T > 22C

T < 18C

CMACS

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


CMACS

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


Parameter Estimationwith RoVerGeNe.

CMACS

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


Parameter Estimationwith RoVerGeNe.

Based on abstractionsby discrete automata.

Parameter Estimation in SpaceEx

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


SpaceEx, Reachability for:

LHA HA with linear dynamicsx ∈ P x ∈ Ax+ u | u ∈ U

No parameter estimation.

Parameter Estimation in SpaceEx

Introduction

SpaceEx

Hybrid Systems

Example



Support Function

SpaceEx


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


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx


Continuous Dynamics: x ∈ Ax⊕ U and x(t) ∈ I Hyperplanar guards Affine Reset Maps


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx



Postc : Continuous evolution


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx



Postc : Continuous evolutionPostd : Discrete transition

Reachability for Linear Time Invariant systems

Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx


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


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx




Analytical solution for a given input function u:

x(t) = etAx(0) +

∫ t

0e(t−s)Au(s) ds


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx





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)


Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx





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

Guards

Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx


Representing Sets

Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx


Scalable Computation by Transformation

Introduction



LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx


Support Function

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


Definition

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


The support function of a compact convex set S ⊆ Rd, denoted

ρS , is defined as:

ρS : Rd → R

ℓ 7→ maxx∈S ℓ · x

Definition

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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

Examples

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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

Representing sets

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


If S is convex, then:

S =⋂

ℓ∈Rd

x : x · ℓ ≤ ρS(ℓ)

Properties

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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))

ρΩk(ℓ)

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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)

Ω[0,δk](X0,U)

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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).

ρΩ[0,δk](ℓ)

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx



⊕(

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,δk](ℓ)

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx



⊕(

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))

Time step adaptation with error bounds

Introduction


Support Function

Definition

Examples

Representing sets

Properties

ρΩk(ℓ)

Ω[0,δk ](X0, U)

ρΩ[0,δk](ℓ)

Time Step

SpaceEx


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

SpaceEx Platform - Architecture

Introduction


Support Function

SpaceEx

Architecture


Thank you

Introduction


Support Function

SpaceEx


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