Modeling and Analysis of Hybrid Systems Erika brahm RWTH Aachen - PowerPoint PPT Presentation

Modeling and Analysis of Hybrid Systems Erika Ábrahám RWTH Aachen University, Germany Beijing, September 2013

Contents 1 Modeling 2 Reachability analysis 3 Counterexample generation Erika Ábrahám - Modeling and Analysis of Hybrid Systems 1 / 28

Contents 1 Modeling 2 Reachability analysis 3 Counterexample generation Erika Ábrahám - Modeling and Analysis of Hybrid Systems 2 / 28

Modeling with hybrid automata Thermostat example x ≥ 22 ℓ on ℓ off x = 20 x = K ( h − x ) ˙ x = − Kx ˙ x ≤ 23 x ≥ 17 x ≤ 18 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 3 / 28

Some interesting subclasses of hybrid automata subclass derivatives conditions bounded unbounded reachability reachability timed automata x = 1 ˙ decidable decidable x ∼ c initialized x ∈ [ c 1 , c 2 ] ˙ x ∼ [ c 1 , c 2 ] decidable decidable rectangular automata reset by derivative change linear hybrid automata I x = c ˙ x ∼ g linear ( � x ) decidable undecidable linear hybrid automata II x = f linear ( � ˙ x ) x ∼ g linear ( � x ) undecidable undecidable general hybrid automata x = f ( � ˙ x ) x ∼ g ( � x ) undecidable undecidable [Henzinger et al., 1998] Erika Ábrahám - Modeling and Analysis of Hybrid Systems 4 / 28

Contents 1 Modeling 2 Reachability analysis 3 Counterexample generation Erika Ábrahám - Modeling and Analysis of Hybrid Systems 5 / 28

Some tools Uppaal [Behrmann et al., 2004] HyTech [Henzinger et al., 1997] PHAVer [Frehse, 2005] SpaceEx [Frehse et al., 2011] d/dt [Asarin et al., 2002] Ellipsoidal toolbox [Kurzhanski et al., 2006] MATISSE [Girard et al., 2007] Multi-Parametric Toolbox [Kvasnica et al., 2004] Flow ∗ [Chen et al., 2012] Erika Ábrahám - Modeling and Analysis of Hybrid Systems 6 / 28

The two most popular techniques for reachability analysis Given: hybrid automaton + set of unsafe states Iterative forward/backward search Abstraction Erika Ábrahám - Modeling and Analysis of Hybrid Systems 7 / 28

Iterative forward search Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

Iterative forward search We need a (possibly over-approximative) state set representation and operations on them like intersection, union, linear transformation and Minkowski sum. Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

Iterative forward search We need a (possibly over-approximative) state set representation and operations on them like intersection, union, linear transformation and Minkowski sum. The representation is crucial for the representation size, efficiency and accuracy. Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

Iterative forward search We need a (possibly over-approximative) state set representation and operations on them like intersection, union, linear transformation and Minkowski sum. The representation is crucial for the representation size, efficiency and accuracy. Erika Ábrahám - Modeling and Analysis of Hybrid Systems 8 / 28

Minkowski sum x 2 x 2 x 2 P ⊕ Q 3 3 3 P 2 ⊕ 2 2 = Q 1 1 1 0 0 0 x 1 x 1 x 1 1 2 3 1 2 3 1 2 3 P ⊕ Q = { p + q | p ∈ P and q ∈ Q } Erika Ábrahám - Modeling and Analysis of Hybrid Systems 9 / 28

Most well-known state set representations Geometric objects: hyperrectangles [Moore et al., 2009] oriented rectangular hulls [Stursberg et al., 2003] convex polyhedra [Ziegler, 1995] [Chen at el, 2011] orthogonal polyhedra [Bournez et al., 1999] template polyhedra [Sankaranarayanan et al., 2008] ellipsoids [Kurzhanski et al., 2000] zonotopes [Girard, 2005]) Other symbolic representations: support functions [Le Guernic et al., 2009] Taylor models [Berz and Makino, 1998, 2009] [Chen et al., 2012] Erika Ábrahám - Modeling and Analysis of Hybrid Systems 10 / 28

Example: Polytopes Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z l 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces l 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces l 3 l 2 l 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron l 3 l 2 l 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron l 3 l 2 l 4 l 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Example: Polytopes Halfspace: set of points satisfying l · x ≤ z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron representation union intersection Minkowski sum V -representation by vertices easy hard easy H -representation by facets hard easy hard Erika Ábrahám - Modeling and Analysis of Hybrid Systems 11 / 28

Linear hybrid automata I: Time evolution Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution x 2 P 0 x 1 Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution x 2 P 0 x 1 x 2 ˙ Q 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution x 2 P 0 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution x 2 P 0 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution P ⊕ cone ( Q ) x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution P ⊕ cone ( Q ) x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution P ⊕ cone ( Q ) x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution P ⊕ cone ( Q ) x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Time evolution ( P ⊕ cone ( Q )) ∩ Inv ( ℓ ) x 2 x 2 P 0 0 x 1 x 1 x 2 ˙ cone ( Q ) 0 x 1 ˙ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 12 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 0 x 1 ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 5 4 0 x 1 ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 x 2 5 4 x 1 0 x 1 0 ℓ ′ ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 x 2 5 4 x 1 0 x 1 0 ℓ ′ ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 x 2 5 4 4 2 x 1 0 x 1 0 ℓ ′ ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 x 2 5 4 4 2 x 1 0 x 1 0 ℓ ′ ℓ Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata I: Discrete steps (jumps) x 2 x 2 5 4 4 2 x 1 0 x 1 0 ℓ ′ ℓ Computed via projection and Minkowski sum. Erika Ábrahám - Modeling and Analysis of Hybrid Systems 13 / 28

Linear hybrid automata II: Time evolution Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

Linear hybrid automata II: Time evolution Assume ˙ x = Ax + Bu Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

Linear hybrid automata II: Time evolution Assume ˙ x = Ax + Bu R [0 ,δ ] R [ δ, 2 δ ] R [2 δ, 3 δ ] Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28

Linear hybrid automata II: Time evolution Assume ˙ x = Ax + Bu Compute Ω 0 , Ω 1 , . . . such that R [ iδ, ( i +1) δ ] ⊆ Ω i Ω 0 R [0 ,δ ] Ω 1 R [ δ, 2 δ ] Ω 2 R [2 δ, 3 δ ] Erika Ábrahám - Modeling and Analysis of Hybrid Systems 14 / 28