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1 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) Reducing the Wrapping Effect.

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Presentation on theme: "1 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) Reducing the Wrapping Effect."— Presentation transcript:

1 1 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) Reducing the Wrapping Effect in Flowpipe Construction using Pseudo-Invariants Stanley Bak United States Air Force Research Lab Information Directorate – Rome, NY

2 2 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 2 Cyber-Physical Systems Include computational (discrete) components, and physical-world (continuous) aspects Discrete components are typically modeled using finite state machines with switching rules. The physical world is typically described by differential equations. Autonomous Cars Air Traffic Control Fault-Tolerant Power Distribution

3 3 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 3 Outline Hybrid Automata as Models for Cyber-Physical Systems Flowpipe Construction Wrapping-Effect Error Pseudo-Invariants Improvement Example

4 4 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 4 Hybrid Automaton u continuous dynamics invariant: hybrid automaton may remain in u as long as x  I(u) location (discrete state) u’ reset condition guard condition edge initial condition Fusion of discrete dynamics with continuous dynamics From: “Hybrid System Verification Using Discrete Model Approximations,” Chutinan

5 5 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 5 Hybrid Automaton Example Simple example: on/off heater in a room (x = temperature)

6 6 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 6 Hybrid Automaton Execution Executions of a hybrid automaton capture one valid trajectory in the model. There can be unaccountably many.

7 7 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 7 Hybrid Automaton Reachability Reachability captures all possible behaviors, and is therefore useful for verification.

8 8 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 8 Outline Hybrid Automata as Models for Cyber-Physical Systems Flowpipe Construction Wrapping-Effect Error Pseudo-Invariants Improvement Example

9 9 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 9 Reachability Algorithm One way to (overapproximate) reachability is to iteratively compute discrete and continuous successors

10 10 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 10 Continuous Successors Many ways to compute continuous successors (abstraction, symbolic reasoning, flowpipe construction) Flowpipe construction methods compute the set of states at snapshots in time T=0.0 T=0.1 T=0.2 Between 0.0 and 0.1 Between 0.1 and 0.2

11 11 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 11 Discrete Successors Flowpipe methods typically aggregate states across discrete transitions Otherwise, a single tracked set can spawn multiple flowpipes in subsequent modes Image from: “Safety Analysis of Hybrid Systems with SpaceEx,” Frehse et al., http://cmacs.cs.cmu.edu/seminars/slides/frehse.pdf

12 12 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 12 Demonstration (buck_open.hyc) Aggregation

13 13 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 13 No Aggregation

14 14 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 14 With Aggregation

15 15 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 15 Outline Hybrid Automata as Models for Cyber-Physical Systems Flowpipe Construction Wrapping-Effect Error Pseudo-Invariants Improvement Example

16 16 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 16 Flowpipe Construction Issues Iterative flowpipe construction leads to overapproximation error Depends on representation Image from: “Perspectives on Enclosure Methods”, Kulisch et al.

17 17 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 17 Outline Hybrid Automata as Models for Cyber-Physical Systems Flowpipe Construction Wrapping-Effect Error Pseudo-Invariants Improvement Example

18 18 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 18 Pseudo-Invariants Key Idea of the Talk: Use discrete successor aggregation to reduce wrapping-effect error How? Force discrete-successor aggregation to occur by introducing an artificial invariant (called the pseudo- invariant) and associated transition – Basically, split one mode into two with identical dynamics

19 19 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 19 Splitting a Single Mode The new automaton is a bisimulation of the original one

20 20 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 20 Effect on Reachability

21 21 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 21 Outline Hybrid Automata as Models for Cyber-Physical Systems Flowpipe Construction Wrapping-Effect Error Pseudo-Invariants Improvement Example

22 22 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 22 Van der Pol Dynamics (Demonstration, ShowReachability.jar)

23 23 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 23 Van der Pol Simulations

24 24 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 24 Computing Reachability Using Flow*, a reachability tool that uses Taylor Models as a state-set representation, we ran the following two models (initial state: x = 1.0, y=[-0.5, 0.5]): poly ode 1 { x'= y y'= y - x - x^2 * y } first { poly ode 1 { x'= y y'= y - x - x^2 * y } inv { x in [0.75, 999] } } second { poly ode 1 { x'= y y'= y - x - x^2 * y } inv {}} jumps { first -> second guard { x in [-999,0.75] } reset { } parallelotope aggregation {}} Error gets too large at t=1.54, tool exits Completes Successfully!

25 25 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 25 Flow* Reachability

26 26 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 26 What Happens Around t=1.54?

27 27 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 27 With Pseudo-Invariant

28 28 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 28 Conclusion Key Idea of the Talk: Use discrete successor aggregation to reduce wrapping-effect error Concerns: – What if the intersections are not accurate? – How do you come up with pseudo-invariants? – What if you choose a “poor” pseudo-invariant?

29 29 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 29 Bonus How do you come up with pseudo-invariants? Idea: during reachability, when the wrapping errors appears to be getting large, generate a pseudo-invariant on-the-fly based on the currently-tracked set (demonstration, van_der_pol.hyc)

30 30 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 30 Without Pseudo-Invariants

31 31 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 31 Using Pseudo-Invariants

32 32 DISTRIBUTION A. Approved for public release; Distribution unlimited. (Approval AFRL PA # 88ABW-2014-1552, 09 April 2014) 32 Using Pseudo-Invariants (2)

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