Techniques and Tools for Hybrid Systems Reachability Analysis
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- Ambrose Thomas
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1 which is funded by the German Research Council (DFG). Techniques and Tools for Hybrid Systems Reachability Analysis Stefan Schupp Johanna Nellen Erika Ábrahám RiSE4CPS, Heidelberg, Germany April 23, 2017 This work is part of the project
2 Hybrid systems in computer science discrete continuous f(t) + t Erika Ábrahám: Hybrid Systems Reachability Analysis 1/45
3 Hybrid systems in computer science discrete continuous f(t) + t Example: Physical systems controlled by programmable logic controllers (PLCs) Plant Physical quantities V cont Actuators V act Sensors V sen PLC read write Input V in Output V out Programs Computation V loc Erika Ábrahám: Hybrid Systems Reachability Analysis 1/45
4 Hybrid systems in computer science discrete continuous f(t) + t Example: Physical systems controlled by programmable logic controllers (PLCs) Plant Physical quantities V cont Actuators V act Sensors V sen PLC read write Input V in Output V out Programs Computation V loc Such systems are typically complex and safety critical. Erika Ábrahám: Hybrid Systems Reachability Analysis 1/45
5 Formal verification System Requirements Formal model Formal specification Formal verification engine Satisfied Violated Unknown Erika Ábrahám: Hybrid Systems Reachability Analysis 2/45
6 Formal verification System Hybrid system Requirements Safety Formal model Hybrid automata Formal specification Reachability property Formal verification engine Flowpipe construction Satisfied Violated Unknown Erika Ábrahám: Hybrid Systems Reachability Analysis 2/45
7 A hybrid automaton model of a steering controller Source: Möhlmann et al.: Verifying a PI Controller using SoapBox and Stabhyli. ARCH 2016 Erika Ábrahám: Hybrid Systems Reachability Analysis 3/45
8 Some example requirements for the steering controller The distance of the current state (dist cur, β ori ) to the optimal state (0, 0) is always bounded: G(dist cur [ 10,10] β ori [ 5,5 ]) The car s orientation changes smoothly: G( β ori [ 0.3,0.3]) Source: Möhlmann et al.: Verifying a PI Controller using SoapBox and Stabhyli. ARCH 2016 Erika Ábrahám: Hybrid Systems Reachability Analysis 4/45
9 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
10 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Initial states Target states Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
11 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Initial states Target states Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
12 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Initial states Target states Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
13 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Initial states Target states Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
14 The reachability problem for hybrid systems The reachability problem for hybrid systems poses the question, whether a given hybrid system can reach a certain set of target states from its initial states. Initial states Target states The reachability problem for hybrid systems is in general undecidable. Despite this fact, there are different (incomplete but practically useful) algorithms and tools for hybrid systems reachability analysis. Erika Ábrahám: Hybrid Systems Reachability Analysis 5/45
15 Impressive tool development in the last decade (incomplete list) HSolver [Ratschan et al., HSCC 2005] isat-ode [Eggers et al., ATVA 2008] KeYmaera (X) [Platzer et al., IJCAR 2008] PowerDEVS [Bergero et al., Simulation 2011] SpaceEx [Frehse et al., CAV 2011] S-TaLiRo [Annapureddy et al., TACAS 2011] Ariadne [Collins et al., ADHS 2012] HySon [Bouissou et al., RSP 2012] Flow* [Chen et al., CAV 2013] HyCreate [Bak et al., HSCC 2013] HyEQ [Sanfelice et al., HSCC 2013] NLTOOLBOX [Testylier et al., ATVA 2013] SoapBox [Hagemann et al., ARCH 2014] Acumen [Taha et al., IoT 2015] C2E2 [Duggirala et al., TACAS 2015] Cora [Althoff et al., ARCH 2015] dreach [Kong et al, TACAS 2015] Isabelle/HOL [Immler, TACAS 2015] HyLAA [Bak et al., HSCC 2017] HyPro/HyDRA [Schupp et al., NFM 2017] Erika Ábrahám: Hybrid Systems Reachability Analysis 6/45
16 Verification techniques/tools for hybrid systems (Rigorous/verified) simulation: Besides simulation for testing, rigorous/verified simulation can be used for (bounded) reachability analysis. Some tools: Acumen, C2E2, HyEQ, HyLAA, HySon, S-TaLiRo, PowerDEVS Source: Erika Ábrahám: Hybrid Systems Reachability Analysis 7/45
17 Verification techniques/tools for hybrid systems Deduction: Finding and showing invariants using theorem proving. Some tools: Ariadne, Isabelle/HOL, KeYmaera Source: Erika Ábrahám: Hybrid Systems Reachability Analysis 8/45
18 Verification techniques/tools for hybrid systems Bounded model checking / interval arithmetic: System executions and requirements are encoded by logical formulas; satisfiability checking tools (SMT solvers) are used for (bounded) reachability analysis. Some tools: dreach, HSolver, isat-ode Source: Erika Ábrahám: Hybrid Systems Reachability Analysis 9/45
19 Verification techniques/tools for hybrid systems Over-approximating flowpipe construction: Iterative (bounded) forward reachability analysis based on some over-approximative symbolic state set representations. Some tools: Cora, Flow*, HyCreate, HyPro/HyDRA, NLTOOLBOX, SoapBox, SpaceEx P Exact behavior P Over-approximation Source: [Bournez et al., HSCC 1999] [Stursberg et al., HSCC 2003] Erika Ábrahám: Hybrid Systems Reachability Analysis 10/45
20 Forward reachability analysis Input: Hybrid automaton H, initial states X 0, target states T. Output: (Bounded) reachability of T from X 0 in H. Algorithm: Queue := {X 0 }; R := {X 0 }; while (Queue = and not break cond){ Take a set P from Queue; Compute successor sets Reach(P); Add successor sets to Queue and R; }; if ( P R P) T = ) return no else return yes ; Erika Ábrahám: Hybrid Systems Reachability Analysis 11/45
21 Forward reachability analysis Input: Hybrid automaton H, initial states X 0, target states T. Output: (Bounded) reachability of T from X 0 in H. Algorithm: Problems: Queue := {X 0 }; R := {X 0 }; while (Queue = and not break cond){ Take a set P from Queue; Compute successor sets Reach(P); Add successor sets to Queue and R; }; if ( P R P) T = ) return no else return yes ; How to represent state sets? How to compute set operations on them? How to compute Reach( )? Erika Ábrahám: Hybrid Systems Reachability Analysis 11/45
22 Most well-known state set representations Geometric objects: boxes (hyper-rectangles) [Moore et al., 2009] oriented rectangular hulls [Stursberg et al., 2003] convex polyhedra [Ziegler, 1995] [Chen at el, 2011] template polyhedra [Sankaranarayanan et al., 2008] orthogonal polyhedra [Bournez et al., 1999] zonotopes [Girard, 2005]) ellipsoids [Kurzhanski et al., 2000] Other symbolic representations: support functions [Le Guernic et al., 2009] Taylor models [Berz and Makino, 1998, 2009] [Chen et al., 2012] Erika Ábrahám: Hybrid Systems Reachability Analysis 12/45
23 Set operations Some needed set operations: intersection union linear transformation Minkowski sum projection test for membership test for emptiness Erika Ábrahám: Hybrid Systems Reachability Analysis 13/45
24 Set operations Some needed set operations: intersection union linear transformation Minkowski sum projection test for membership test for emptiness Reminder: Minkowski sum x 2 3 P 2 1 x Q = x P Q x x x 1 P Q = {p + q p P and q Q} Erika Ábrahám: Hybrid Systems Reachability Analysis 13/45
25 Example state set representation: Polytopes Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
26 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
27 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z l 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
28 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Polyhedron: an intersection of finitely many halfspaces l 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
29 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Polyhedron: an intersection of finitely many halfspaces l 3 l 2 l 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
30 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron l 3 l 2 l 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
31 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron l 3 l 4 l 2 l 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
32 Example state set representation: Polytopes Halfspace: set of points x satisfying l x z Polyhedron: an intersection of finitely many halfspaces Polytope: a bounded polyhedron Erika Ábrahám: Hybrid Systems Reachability Analysis 14/45
33 Example state set representation: Polytopes Halfspace: set of points x 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: Hybrid Systems Reachability Analysis 14/45
34 Linear hybrid automata I and II Linear hybrid automata I: derivatives: boxes conditions: convex linear sets resets: linear functions Linear hybrid automata II: derivatives: linear differential equations conditions: convex linear sets resets: linear functions Erika Ábrahám: Hybrid Systems Reachability Analysis 15/45
35 Linear hybrid automata I: Time evolution Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
36 Linear hybrid automata I: Time evolution x 2 initial state set X 0 0 x 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
37 Linear hybrid automata I: Time evolution x 2 initial state set X 0 0 x 1 ẋ 2 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
38 Linear hybrid automata I: Time evolution x 2 initial state set X 0 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
39 Linear hybrid automata I: Time evolution x 2 initial state set X 0 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
40 Linear hybrid automata I: Time evolution x 2 x 2 initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
41 Linear hybrid automata I: Time evolution x 2 x 2 X 0 cone(q) initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
42 Linear hybrid automata I: Time evolution x 2 x 2 X 0 cone(q) initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
43 Linear hybrid automata I: Time evolution x 2 x 2 X 0 cone(q) initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
44 Linear hybrid automata I: Time evolution x 2 x 2 X 0 cone(q) initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
45 Linear hybrid automata I: Time evolution x 2 (X 0 cone(q)) Inv(l) x 2 initial state set X 0 0 x 1 0 x 1 ẋ 2 cone(q) 0 derivatives Q ẋ 1 Erika Ábrahám: Hybrid Systems Reachability Analysis 16/45
46 Linear hybrid automata I: Discrete steps (jumps) x 2 Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] P 0 x 1 l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
47 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] P 0 x 1 l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
48 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 P G P 0 x 1 0 x 1 l l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
49 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 (P G) x1 P 0 x 1 0 x 1 l l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
50 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 ((P G) x1 ) R P 0 x 1 0 x 1 l l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
51 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 0 x 1 0 x 1 l l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
52 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 0 x 1 0 x 1 l l Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
53 Linear hybrid automata I: Discrete steps (jumps) x Example jump: ( l, 4 x 2 5, }{{} x 2 : [2,4], }{{} l ) Edge G R [4,5] R R [2,4] x 2 4 (((P G) x1 ) R) R P 2 0 x 1 0 x 1 l l Additionally: intersect the result with the invariant of l. Computed via projection, Minkowski sum and intersection. Erika Ábrahám: Hybrid Systems Reachability Analysis 17/45
54 Linear hybrid automata II: Time evolution Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
55 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
56 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu time [0,δ] time [δ,2δ] time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
57 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i P 0 time [0,δ] time [δ,2δ] P 1 P 2 time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
58 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
59 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: 0 δ 2δ t X 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
60 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
61 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 conv(x 0, e Aδ X 0 ) 0 δ 2δ t X 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
62 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
63 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
64 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! conv(x 0, e Aδ X 0 B 1 ) e Aδ X 0 0 δ 2δ t X 0 over-approximates flowpipe for time [0,δ] under dynamics ẋ = Ax Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
65 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 disturbance! Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
66 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 disturbance! Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
67 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! e Aδ X 0 0 δ 2δ t X 0 disturbance! Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
68 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! P 0 = conv(x 0, e Aδ X 0 B 1 B 2 ) e Aδ X 0 0 δ 2δ t X 0 over-approximates flowpipe for time [0,δ] under dynamics ẋ = Ax+Bu Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
69 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! 0 δ 2δ t P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
70 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
71 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 e Aδ P 0 over-approximates flowpipe for time [δ,2δ] under dynamics ẋ = Ax Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
72 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 e Aδ P 0 disturbance! Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
73 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 e Aδ P 0 disturbance! Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
74 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
75 Linear hybrid automata II: Time evolution Assume initial set X 0 and flow ẋ = Ax+Bu Over-approximate flowpipe segment for time [iδ,(i + 1)δ] by P i The first flowpipe segment: Reminder matrix exponential: e X = k=0 Xk k! The remaining ones: 0 δ 2δ t P 0 P 1 = e Aδ P 0 B 2 over-approximates flowpipe for time [δ,2δ] under dynamics ẋ = Ax+Bu Erika Ábrahám: Hybrid Systems Reachability Analysis 18/45
76 Linear hybrid automata II: Discrete steps (jumps) P 3 P 2 P 1 P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
77 Linear hybrid automata II: Discrete steps (jumps) P 3 P 2 P 1 P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
78 Linear hybrid automata II: Discrete steps (jumps) P 3 P 2 P 1 P 0 Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
79 Linear hybrid automata II: Discrete steps (jumps) Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
80 Linear hybrid automata II: Discrete steps (jumps) Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
81 Linear hybrid automata II: Discrete steps (jumps) Erika Ábrahám: Hybrid Systems Reachability Analysis 19/45
82 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
83 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
84 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
85 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
86 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
87 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
88 Linear hybrid automata II: The global picture Erika Ábrahám: Hybrid Systems Reachability Analysis 20/45
89 Reachability analysis search tree Erika Ábrahám: Hybrid Systems Reachability Analysis 21/45
90 Reachability analysis search tree Erika Ábrahám: Hybrid Systems Reachability Analysis 21/45
91 Flow* [Chen et al., CAV 2013] A verification tool for cyber-physical systems Available at Taylor model-based approach non-linear dynamic adaptive refinement methods Image: Xin Chen Has been used in a variety of verification tasks, e.g. biological/medical systems (glucose control, spiking neurons, Lotka Volterra equations), circuits (oscillators, van der Pol circuit), mechanical systems (jet engine model) Erika Ábrahám: Hybrid Systems Reachability Analysis 22/45
92 HyPro/HyDRA [Schupp et al., NFM 2017] A free and open-source C++ library for state set representations for the reachability analysis of hybrid systems Available at state set representations conversion between different representations further datastructures and utility functions (hybrid automata, parser, logging, plotting) templated number type Fast implementation of specialized reachability analysis methods Dimension reduction via sub-space computations [Schupp et al., QAPL 2017] CEGAR-like refinement loops and parallelisation (work in progress) Erika Ábrahám: Hybrid Systems Reachability Analysis 23/45
93 HyPro/HyDRA: Structure algorithms Hybrid automaton Point Halfspace datastructures Box HPolytope VPolytope PPL-Polytope Zonotope SupportFunction Orthogonal polyhedra Taylor model representations GeometricObject <Interface> Parser Reachability analysis Converter Optimizer glpk SMT-RAT Z3 SoPlex Logger Plotter util Erika Ábrahám: Hybrid Systems Reachability Analysis 24/45
94 HyPro: State set representations y y y I y I x x x x Boxes Convex polyhedra (H, V, PPL) y y y g 0 c g 1 x x x Zonotopes Support functions Orthogonal polyhedra Taylor models Source: Xin Chen Erika Ábrahám: Hybrid Systems Reachability Analysis 25/45
95 HyPro/HyDRA: Structure algorithms Hybrid automaton Point Halfspace datastructures Box HPolytope VPolytope PPL-Polytope Zonotope SupportFunction Orthogonal polyhedra Taylor model representations GeometricObject <Interface> Parser Reachability analysis Converter Optimizer glpk SMT-RAT Z3 SoPlex Logger Plotter util Erika Ábrahám: Hybrid Systems Reachability Analysis 26/45
96 HyPro: Linear optimization HyPro offers different number representations: cln::cl RA, mpq class, double Obstacles: inexact linear optimization not suitable exact linear optimization expensive combined application Compute optimal solution glpk solution s Compute optimal solution s s SMT-RAT/SoPlex/Z3 solution s solution s Solution no solution no solution Compute optimal solution SMT-RAT/SoPlex/Z3 no solution No solution Erika Ábrahám: Hybrid Systems Reachability Analysis 27/45
97 HyPro/HyDRA: Structure algorithms Hybrid automaton Point Halfspace datastructures Box HPolytope VPolytope PPL-Polytope Zonotope SupportFunction Orthogonal polyhedra Taylor model representations GeometricObject <Interface> Parser Reachability analysis Converter Optimizer glpk SMT-RAT Z3 SoPlex Logger Plotter util Erika Ábrahám: Hybrid Systems Reachability Analysis 28/45
98 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
99 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y linear transformation Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
100 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y union Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
101 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y Minkowski sum Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
102 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
103 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
104 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x intersection Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
105 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x linear transformation Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
106 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x linear transformation Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
107 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x x 0.25 x intersection Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
108 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x x 0.25 x 0.3 y := 0.9y x := x linear transformation Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
109 HyDRA: Example x [0.5,0.6] y [0.1,0.2] l 0 ẋ = x + 4y ẏ = 4x + y x x 0.25 x 0.3 y := 0.9y x := x linear transformation Erika Ábrahám: Hybrid Systems Reachability Analysis 29/45
110 Other HyPro applications Two examples for HyPro applications: Sub-space computations CEGAR-based reachability analysis and parallelisation Erika Ábrahám: Hybrid Systems Reachability Analysis 30/45
111 HyPro application: Sub-space computations Motivation: PLC-controlled plants High-dimensional models Erika Ábrahám: Hybrid Systems Reachability Analysis 31/45
112 HyPro application: Sub-space computations Motivation: PLC-controlled plants High-dimensional models Relevant number of discrete variables Clocks for cycle synchronisation Erika Ábrahám: Hybrid Systems Reachability Analysis 31/45
113 HyPro application: Sub-space computations Motivation: PLC-controlled plants High-dimensional models Relevant number of discrete variables Clocks for cycle synchronisation Idea: Partition variable set sub-spaces Compute reachability in sub-spaces Synchronise on time time [0,0] time [0,δ] time [δ,2δ] time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 31/45
114 Variable set partitioning time [0,0] time [0,δ] time [δ,2δ] time [2δ,3δ] What assures that we can compute locally in sub-spaces? Erika Ábrahám: Hybrid Systems Reachability Analysis 32/45
115 Variable set partitioning time [0,0] time [0,δ] time [δ,2δ] time [2δ,3δ] What assures that we can compute locally in sub-spaces? All variables x, y in different partitions should be syntactically independent. (ẋ,ẏ) = A (x,y) T + B u Inv ẋ = A x x T + B x u ẏ = A y y T + B y u Inv x Inv y guard reset guard x guard y reset x reset y Erika Ábrahám: Hybrid Systems Reachability Analysis 32/45
116 Variable set partitioning Global space: X 0 P 1 = conv(x 0 e Aδ X 0 V A V B ) P 2 = e Aδ P 1 V B,U P 3 = e Aδ P 2 V B,U time [0,0] time [0,δ] time [δ,2δ] time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 33/45
117 Variable set partitioning Global space: X 0 P 1 = conv(x 0 e Aδ X 0 V A V B ) P 2 = e Aδ P 1 V B,U P 3 = e Aδ P 2 V B,U time [0,0] time [0,δ] time [δ,2δ] time [2δ, 3δ] Sub-space: X 0 X 0,x = X 0 x X 0,y = X 0 y time [0,0] P 1,x = conv(x 0,x e A xδ X 0,x V A,x V B,x )... P 2,x = e A xδ P 1,x V Bx,u... P 3,x = e A xδ P 2,x V Bx,u... time [0,δ] time [δ,2δ] time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 33/45
118 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
119 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
120 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
121 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
122 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
123 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
124 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
125 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
126 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
127 Discrete variables y ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
128 Discrete variables y 2.5 y 2.8 ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
129 Discrete variables y 2.5 y 2.8 ẋ = 0 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 34/45
130 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
131 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
132 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
133 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
134 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
135 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
136 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
137 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
138 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
139 Representation: Boxes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
140 Representation: Boxes y 2.5 y 2.8 ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
141 Representation: Boxes y 2.5 y 2.8 ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 35/45
142 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
143 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
144 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
145 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
146 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
147 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
148 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
149 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
150 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
151 Representation: Polytopes y ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
152 Representation: Polytopes y 2.5 y 2.8 ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
153 Representation: Polytopes y 2.5 y 2.8 ẋ = 1 ẏ = 1 x Erika Ábrahám: Hybrid Systems Reachability Analysis 36/45
154 Reachability computation 1 Partition variable set 2 Decompose initial state sets 3 Compute successors in sub-spaces 4 Discrete variables: no flowpipe, neglect disabled jumps for whole flowpipe global state set time [0,0] disc clock rest time [0,δ] Inv sub-space flowpipe segment sub-space flowpipe segment time [δ,2δ] sub-space flowpipe segment sub-space flowpipe segment time [2δ, 3δ] Erika Ábrahám: Hybrid Systems Reachability Analysis 37/45
155 Some more aspects Erika Ábrahám: Hybrid Systems Reachability Analysis 38/45
156 Some more aspects We can use different state set representations for different sub-spaces. We could even use different reachability analysis methods for different sub-spaces. Erika Ábrahám: Hybrid Systems Reachability Analysis 38/45
157 Some more aspects We can use different state set representations for different sub-spaces. We could even use different reachability analysis methods for different sub-spaces. In our implementation: 3 variable partitions semantically independent discrete variables semantically independent clocks rest For discrete variables we use boxes, for the rest support functions. Erika Ábrahám: Hybrid Systems Reachability Analysis 38/45
158 Results: Leaking tank HyPro clock + rest HyPro original 14 x t Erika A braha m: Hybrid Systems Reachability Analysis /45
159 Results: Leaking tank x HyPro clock + rest HyPro original t Erika Ábrahám: Hybrid Systems Reachability Analysis 40/45
160 Results: Two tanks HyPro full separation HyPro original x t Erika Ábrahám: Hybrid Systems Reachability Analysis 41/45
161 Running times Bench- HyPro SpaceEx mark Rep. Agg orig clock disc disc & clock orig box agg Leaking box none tank sf agg TO TO sf none TO box agg Two box none tanks sf agg TO TO TO sf none TO TO TO box agg Ther- box none mostat sf agg sf none Erika Ábrahám: Hybrid Systems Reachability Analysis 42/45
162 Other HyPro applications Two examples for HyPro applications: Sub-space computations CEGAR-based reachability analysis and parallelisation Erika Ábrahám: Hybrid Systems Reachability Analysis 43/45
163 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
164 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A B C Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
165 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A B C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
166 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A B C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
167 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
168 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
169 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
170 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
171 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
172 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
173 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
174 Other HyPro applications Strategy: S1: box, δ = 0.1 S2: support f., δ = 0.01 S3: polytope, δ = 0.01 Search tree: A E G B F C D Extension: Parallelized search in different branches. Erika Ábrahám: Hybrid Systems Reachability Analysis 44/45
175 Conclusion HyPro: open-source programming library State set representations for the implementation of hybrid systems reachability analysis algorithms Exact as well as inexact number representations Flexibility to deviate from standard methods Examples: sub-space computations CEGAR parallelisation Available at Erika Ábrahám: Hybrid Systems Reachability Analysis 45/45
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