EECE 396-1 Hybrid and Embedded Systems: Computation

EECE 396-1Hybrid and Embedded Systems: Computation

T. John Koo, Ph.D.Institute for Software Integrated Systems

Department of Electrical Engineering and Computer ScienceVanderbilt University

300 Featheringill HallApril 6-15, 2004

john.koo@vanderbilt.eduhttp://www.vuse.vanderbilt.edu/~kootj

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Analysis: Timed Automata

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Topics Bisimular Transition Systems

Transition Systems Forward and Backward Reachability Algorithms Simulation and Bisimulation Relations Transition Systems and Quotient Transition Systems Bisimulation between Transition Systems Bisimulation Algorithm

Bisimulations of Timed Automata Executions of Timed Automata Region Equivalent Relation and Partition

Ref: [1] R. Alur, T.A. Henzinger, and P.-H. Ho.

Automatic Symbolic Verification of Embedded Systems. IEEE Transactions on Software Engineering 22:181-201, 1996.

[2] J. Lygeros, Lecture Notes on Hybrid Systems, Cambridge, 2003. [3] J. Lygeros, C. Tomlin, and S. Sastry, The Art of Hybrid Systems, July 2001.

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Analysis: Timed AutomataBisimular Transition Systems

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Transition System

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Transition System

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Transition System

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Transition System

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Forward and Backward Reachability Algorithms

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Forward and Backward Reachability Algorithms

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Forward and Backward Reachability Algorithms

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Forward and Backward Reachability Algorithms

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Simulation and Bisimulation Relations

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Bisimulation Algorithm

How?

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Bisimulation Algorithm

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Analysis: Timed AutomataBisimulations of Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Timed Automata

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Hybrid Automaton

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Hybrid Automaton

i

012

tfinite

i

012

tinfinite

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Hybrid Automaton

i

012

tfinite

i

012

tZeno

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Hybrid Automaton Non-Determinism

Multiple Executions for the same initial condition Sources of non-determinism

Non-Lipschitz continuous vectorfields, f Multiple discrete transition destinations, E & G Choice between discrete transition and continuous evolution, D & G Non-unique continuous state assignment, R

Definition: A hybrid automaton H is deterministic if for all initial conditions there exists a unique maximal sequence

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Hybrid Automaton Blocking

No Infinite executions for some initial states Source of blocking

Cannot continue in domain due to reaching the boundary of the domain where no guard is defined

Have no place to make discrete transition to

Definition: A hybrid automaton H is non-blocking if for every initial condition there exists at least one infinite execution

?

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Hybrid Automaton Zeno Executions

Infinite execution defined over finite time Infinite number of transitions in finite time Transition times converge

Definition: A hybrid automaton H is zeno if there exists an initial condition for which all infinite executions are Zeno

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Transition System

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Transition System

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Region Equivalence Relation

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Region Equivalence Relation

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Timed Automata

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Timed Automata

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Timed Automata

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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Region Equivalence Relation

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End