Verification and Qualitative analysis of hybrid systems: dimension 2 - PowerPoint PPT Presentation

Verification and Qualitative analysis of hybrid systems: dimension 2 Joint work with Gerardo Schneider, Sergio Yovine and Gordon Pace Eugene Asarin VERIMAG AS - Paris - 27/06/02 – p.1/38

Outline • Motivation and Context • SPDI - decidable 2d systems • The model • Reachability is decidable • Beyond verification : algorithmic phase portrait construction. • SPeeDI the tool AS - Paris - 27/06/02 – p.2/38

Outline • Motivation and Context • SPDI - decidable 2d systems • More complex systems: between decidable and undecidable • The reference model : 1d PAM • 2d Linear Hybrid Automata ≡ PAM • PCD on 2d manifolds ≡ iPAM AS - Paris - 27/06/02 – p.2/38

Outline • Motivation and Context • SPDI - decidable 2d systems • More complex systems: between decidable and undecidable • Undecidable 2d systems • LHA + counter AS - Paris - 27/06/02 – p.2/38

Outline • Motivation and Context • SPDI - decidable 2d systems • More complex systems: between decidable and undecidable • Undecidable 2d systems • Discussion AS - Paris - 27/06/02 – p.2/38

Motivation and Context AS - Paris - 27/06/02 – p.3/38

The Problem • Explore decidability of reachability for classes of 2d hybrid systems. • Trace the boundary between decidable and undecidable. • Find good algorithms for decidable problems. AS - Paris - 27/06/02 – p.4/38

Why? • Why hybrid systems? • What kind of hybrid systems? • Why reachability? • Why 2d? AS - Paris - 27/06/02 – p.5/38

Hybrid systems Discrete+continuous ⇒ interesting and useful Our basic model: PCD (simple dynamics, no jumps) c 1 P 1 y x x = c i for x ∈ P i ˙ t AS - Paris - 27/06/02 – p.6/38

Around reachability • Reach ( x, y ) ⇔ ∃ a trajectory from x to y Also Reach ( A, B ) : set-to-set reachability. AS - Paris - 27/06/02 – p.7/38

Around reachability • Reach ( x, y ) ⇔ ∃ a trajectory from x to y Also Reach ( A, B ) : set-to-set reachability. • Key to safety verification: x is safe ⇔ ¬ Reach ( x, Bad ) AS - Paris - 27/06/02 – p.7/38

Around reachability • Reach ( x, y ) ⇔ ∃ a trajectory from x to y Also Reach ( A, B ) : set-to-set reachability. • Key to safety verification: x is safe ⇔ ¬ Reach ( x, Bad ) • MP94: Reach is decidable for 2d PCD. AS - Paris - 27/06/02 – p.7/38

Around reachability • Reach ( x, y ) ⇔ ∃ a trajectory from x to y Also Reach ( A, B ) : set-to-set reachability. • Key to safety verification: x is safe ⇔ ¬ Reach ( x, Bad ) • MP94: Reach is decidable for 2d PCD. • AM95: Reach is undecidable for 2d PCD. AS - Paris - 27/06/02 – p.7/38

Where is the boundary? The boundary between decidable and undecidable lies somewhere in dimension 2. Let us explore more general 2d systems: • SPDI = Non-deterministic PCD • PCD on surfaces • Linear Hybrid Automata = PCD + jumps • LHA+1 counter • . . . AS - Paris - 27/06/02 – p.8/38

SPDI - a new class of decidable systems AS - Paris - 27/06/02 – p.9/38

SPDI Simple Polygonal Differential Inclusion = the non-deterministic version of PCD= • A partition of the plane into polygonal regions • A constant differential inclusion for each region x ∈ ∠ b a if x ∈ R i ˙ AS - Paris - 27/06/02 – p.10/38

SPDI Simple Polygonal Differential Inclusion = R 3 e 3 R 4 R 2 e 2 x e 1 y e 4 R 5 R 1 e 8 e 5 e 7 e 6 R 6 R 8 R 7 AS - Paris - 27/06/02 – p.10/38

Difficulties Too many trajectories ( even locally ) e 3 e 2 e 4 e 1 e 5 e 8 e 6 e 7 AS - Paris - 27/06/02 – p.11/38

Difficulties Too many signatures e 2 e 3 e 4 e 1 e 9 e 12 e 10 e 11 e 8 e 5 e 6 e 7 AS - Paris - 27/06/02 – p.11/38

Difficulties Self-crossing trajectories e 2 e 3 e 4 e 1 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 AS - Paris - 27/06/02 – p.11/38

Plan of solution • Simplify trajectories • Enumerate types of signatures • Test reachability for each type using accelerations AS - Paris - 27/06/02 – p.12/38

Simplification 1: Straightening x ′ R i b a x AS - Paris - 27/06/02 – p.13/38

Simplification 2: Removing self-crossings x f x ′ y ′ e ′ e ′ 2 1 e 2 b a y x e 1 x 0 x f y ′ e ′ e ′ 1 2 e 2 b a x e 1 x 0 Bottom line: Reach ( x, y ) ⇔ ∃ a simple piecewise straight trajectory from x to y AS - Paris - 27/06/02 – p.14/38

Signatures of simplified trajectories • Representation Theorem: Any edge signature can be represented as σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 • Properties • r i is a seq. of pairwise different edges; • s i is a simple cycle; • r i and r j are disjoint • s i and s j are different Proof based on Jordan’s theorem (MP94) AS - Paris - 27/06/02 – p.15/38

Classification of signatures Any edge signature belongs to a type r 1 ( s 1 ) ∗ r 2 ( s 2 ) ∗ . . . r n ( s n ) ∗ r n +1 s 1 s 2 s n r 1 r 2 r 3 r n r n +1 There are finitely many types! AS - Paris - 27/06/02 – p.16/38

How to explore one type? s 1 s 2 s n r 1 r 2 r 3 r n r n +1 Recipe: compute successors and accelerate cycles. AS - Paris - 27/06/02 – p.17/38

Successors (by σ ) One step ( σ = e 1 e 2 ) R 3 e 3 e 2 R 2 R 4 e 4 e 1 e 9 R 5 e 12 e 13 e 10 R 1 e 11 e 5 e 8 e 7 e 6 R 7 R 8 R 6 I ′ = Succ e 1 e 2 ( x ) = [ f b ( x ) , f a ( x )] = F ( x ) AS - Paris - 27/06/02 – p.18/38

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