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

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/

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)

Example

Colas Le Guernic CMACS meeting – 4 / 22

HEAT

dTdt

= fHeat(T )

OFF

dTdt

= fOff(T )

T > 22C

T < 18C

Example

Colas Le Guernic CMACS meeting – 4 / 22

HEAT

dTdt

= fHeat(T )

OFF

dTdt

= fOff(T )

T > 22C

T < 18C

Example

Colas Le Guernic CMACS meeting – 4 / 22

HEAT

dTdt

= fHeat(T )

OFF

dTdt

= fOff(T )

T > 22C

T < 18C

Example

Colas Le Guernic CMACS meeting – 4 / 22

HEAT

dTdt

= fHeat(T )

OFF

dTdt

= fOff(T )

T > 22C

T < 18C

Example

Colas Le Guernic CMACS meeting – 4 / 22

HEAT

dTdt

∈ FHeat(T )

OFF

dTdt

∈ FOff(T )

T > 22C

T < 18C

CMACS

Introduction

SpaceEx

Hybrid Systems

Example

CMACSParameterEstimation inSpaceEx

Reachability Analysis

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 5 / 22

CMACS

Introduction

SpaceEx

Hybrid Systems

Example

CMACSParameterEstimation inSpaceEx

Reachability Analysis

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 5 / 22

Parameter Estimationwith RoVerGeNe.

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.

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.

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

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

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

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

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

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

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

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

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

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)

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

Guards

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 9 / 22

Guards

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 9 / 22

Guards

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 9 / 22

Guards

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 9 / 22

Representing Sets

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 10 / 22

Scalable Computation by Transformation

Introduction

Reachability Analysis

Reachability Analysis

LTI

Guards

Representing Sets

Scalable

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 11 / 22

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

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

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

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

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)

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))

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)

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).

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,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))

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

SpaceEx Platform - Architecture

Introduction

Reachability Analysis

Support Function

SpaceEx

Architecture

Colas Le Guernic CMACS meeting – 21 / 22

Thank you

Introduction

Reachability Analysis

Support Function

SpaceEx

Colas Le Guernic CMACS meeting – 22 / 22