Computing Invariance Kernels of Polygonal Hybrid Systems G ERARDO S - PowerPoint PPT Presentation

Computing Invariance Kernels of Polygonal Hybrid Systems G ERARDO S CHNEIDER gerardos@it.uu.se U PPSALA U NIVERSITY D EPARTMENT OF I NFORMATION T ECHNOLOGY U PPSALA , S WEDEN Computing Invariance Kernels of Polygonal Hybrid Systems – p.1/26

Overview of the presentation • Motivation • Introduction: Hybrid System • Polygonal Differential Inclusion System (SPDI) • Successors and Predecessors • Classification of Simple Cycles • Phase Portrait of SPDIs • Invariance Kernels • Conclusions Computing Invariance Kernels of Polygonal Hybrid Systems – p.2/26

Motivation and Related Work • For Hybrid Systems • Verification (reachability, ...): • Qualitative behavior (Phase Portrait, ...) Computing Invariance Kernels of Polygonal Hybrid Systems – p.3/26

Motivation and Related Work • For Hybrid Systems • Verification (reachability, ...): • Qualitative behavior (Phase Portrait, ...) • For a class of non-deterministic systems (SPDI) • Verification (HSCC’01) • Undecidability of some extensions (CONCUR’02) • Phase Portrait (HSCC’02): • Viability Kernel • Controllability Kernel • Invariance Kernels Computing Invariance Kernels of Polygonal Hybrid Systems – p.3/26

Why Invariance Kernels? • Important objects for giving SPDIs • Crucial for proving termination of a BFS reachability algorithm for SPDI Computing Invariance Kernels of Polygonal Hybrid Systems – p.4/26

Overview of the presentation • Motivation • Introduction: Hybrid System • Polygonal Differential Inclusion System (SPDI) • Successors and Predecessors • Classification of Simple Cycles • Phase Portrait of SPDIs • Invariance Kernels • Conclusions Computing Invariance Kernels of Polygonal Hybrid Systems – p.5/26

Hybrid Systems • Hybrid Systems: interaction between discrete and continuous behaviors • Examples: thermostat, automated highway systems, air traffic management systems, robotic systems, chemical plants, etc. Computing Invariance Kernels of Polygonal Hybrid Systems – p.6/26

Hybrid Systems Model: Hybrid Automata label x = M On Off dynamics x = 3 − x x = − x ˙ ˙ x ≤ M x = m /γ x ≥ m invariant reset guard Computing Invariance Kernels of Polygonal Hybrid Systems – p.6/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Hybrid Systems Example: Swimmer in a whirlpool e 2 e 3 e 4 e 1 x 0 e 9 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.7/26

Overview of the presentation • Motivation • Introduction: Hybrid System • Polygonal Differential Inclusion System (SPDI) • Successors and Predecessors • Classification of Simple Cycles • Phase Portrait of SPDIs • Invariance Kernels • Conclusions Computing Invariance Kernels of Polygonal Hybrid Systems – p.8/26

Polygonal Differential Inclusion Systems (SPDIs) • A partition of the plane into convex polygonal regions • A constant differential inclusion for each region x ∈ ∠ b a if x ∈ R i ˙ e 2 e 3 e 4 e 12 e 1 e 9 e 11 e 10 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.9/26

Polygonal Differential Inclusion Systems (SPDIs) • A partition of the plane into convex polygonal regions • A constant differential inclusion for each region x ∈ ∠ b a if x ∈ R i ˙ x ′ ∠ b a b R i b a a x Computing Invariance Kernels of Polygonal Hybrid Systems – p.9/26

Polygonal Differential Inclusion Systems (SPDIs) • The “swimmer” is a hybrid system • Hybrid Automata? Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

Polygonal Differential Inclusion Systems (SPDIs) • The “swimmer” is a hybrid system • Hybrid Automata? e 2 e 3 e 4 e 12 e 1 e 9 e 11 e 10 e 5 e 8 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

Polygonal Differential Inclusion Systems (SPDIs) • The “swimmer” is a hybrid system • Hybrid Automata? e 2 e 3 e 2 e 3 e 4 e 12 e 1 e 9 e 4 e 12 e 1 e 9 e 10 e 11 e 5 e 8 e 11 e 10 e 5 e 8 e 6 e 7 e 7 e 6 Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

Polygonal Differential Inclusion Systems (SPDIs) • The “swimmer” is a hybrid system • Hybrid Automata? R 4 R 2 R 3 x = e 3 x = e 2 x ∈ ∠ b x = a 4 ˙ ˙ x = a 2 ˙ a Inv ℓ 4 Inv ℓ 3 Inv ℓ 2 x = e 4 x = e 9 x = e 1 x = e 12 R 5 R 1 x = a 5 ˙ x = a 1 ˙ Inv ℓ 5 Inv ℓ 1 x = e 10 x = e 11 x = e 5 x = e 8 R 6 R 7 R 8 x = e 6 x = e 7 x = a 6 ˙ x = a 7 ˙ x = a 8 ˙ Inv ℓ 6 Inv ℓ 7 Inv ℓ 8 Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

Polygonal Differential Inclusion Systems (SPDIs) • The “swimmer” is a hybrid system • Hybrid Automata? We will use the “geometric” representation instead of the hybrid automata Computing Invariance Kernels of Polygonal Hybrid Systems – p.10/26

Overview of the presentation • Motivation • Introduction: Hybrid System • Polygonal Differential Inclusion System (SPDI) • Successors and Predecessors • Classification of Simple Cycles • Phase Portrait of SPDIs • Invariance Kernels • Conclusions Computing Invariance Kernels of Polygonal Hybrid Systems – p.11/26

Successor Operators For a signature σ = e 1 . . . e 8 e 1 : R 3 Succ σ ([ l, u ]) R 4 R 2 e 3 e 2 e 4 e 1 u l R 5 R 1 e 8 e 5 Succ σ ([ l, u ]) e 6 e 7 R 6 R 8 R 7 Computing Invariance Kernels of Polygonal Hybrid Systems – p.12/26

Successor Operators • Successors have the form Succ σ ( l , u ) = [ a 1 l + b 1 , a 2 u + b 2 ] ∩ J if [ l, u ] ⊆ S • Fixpoint equations [ a 1 l ∗ + b 1 , a 2 u ∗ + b 2 ] = [ l ∗ , u ∗ ] can be explicitely solved (without iterating). Computing Invariance Kernels of Polygonal Hybrid Systems – p.13/26

Predecessor Operators For σ = e 1 . . . e 8 e 1 : R 3 f Pre σ ([ l, u ])) R 4 R 2 e 3 e 2 e 4 e 1 l u R 5 R 1 e 8 e 5 f Pre σ ([ l, u ])) e 6 e 7 R 6 R 8 R 7 Computing Invariance Kernels of Polygonal Hybrid Systems – p.14/26

Overview of the presentation • Motivation • Introduction: Hybrid System • Polygonal Differential Inclusion System (SPDI) • Successors and Predecessors • Classification of Simple Cycles • Phase Portrait of SPDIs • Invariance Kernels • Conclusions Computing Invariance Kernels of Polygonal Hybrid Systems – p.15/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] L ≤ l ∗ ≤ u ∗ ≤ U STAY: u ∗ < L ∨ l ∗ > U DIE: l ∗ < L ∧ u ∗ > U EXIT-BOTH: l ∗ < L ≤ u ∗ ≤ U EXIT-LEFT: L ≤ l ∗ ≤ U < u ∗ EXIT-RIGHT: Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] STAY: l ∗ u ∗ e 1 L U Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] DIE: l ∗ u ∗ e 1 L U Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] EXIT-BOTH: l ∗ u ∗ e 1 L U Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] EXIT-LEFT: l ∗ u ∗ e 1 L U Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26

Classification of Simple Cycles Given a cyclic signature σ = e 1 . . . e 8 e 1 . Let e 1 = [ L, U ] . Succ ∗ σ = [ l ∗ , u ∗ ] Succ σ ([ l, u ]) ⊆ [ l ∗ , u ∗ ] EXIT-RIGHT: l ∗ u ∗ e 1 L U Computing Invariance Kernels of Polygonal Hybrid Systems – p.16/26