EECE 396 1 Hybrid and Embedded Systems Computation

EECE 396 -1 Hybrid and Embedded Systems: Computation T. John Koo, Ph. D. Institute for Software Integrated Systems Department of Electrical Engineering and Computer Science Vanderbilt University 300 Featheringill Hall April 6 -15, 2004 john. koo@vanderbilt. edu http: //www. vuse. vanderbilt. edu/~kootj

Analysis: Timed Automata

Topics n Bisimular Transition Systems n Forward and Backward Reachability Algorithms n Simulation and Bisimulation Relations n Transition Systems and Quotient Transition Systems n Bisimulation between Transition Systems n Bisimulation Algorithm n Bisimulations of Timed Automata n Executions of Timed Automata n Region Equivalent Relation and Partition n Ref: n [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. n [2] J. Lygeros, Lecture Notes on Hybrid Systems, Cambridge, 2003. n [3] J. Lygeros, C. Tomlin, and S. Sastry, The Art of Hybrid Systems, July 2001.

Analysis: Timed Automata Bisimular Transition Systems

Transition System

Forward and Backward Reachability Algorithms

Simulation and Bisimulation Relations

Bisimulation Algorithm How?

Bisimulation Algorithm

Analysis: Timed Automata Bisimulations of Timed Automata

Timed Automata

Hybrid Automaton

Hybrid Automaton i i 2 2 1 1 0 0 finite t infinite t

Hybrid Automaton i i 2 2 1 1 0 0 finite t Zeno t

Hybrid Automaton n Non-Determinism n Multiple Executions for the same initial condition n Sources of non-determinism n n 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

Hybrid Automaton n Blocking n No Infinite executions for some initial states n Source of blocking n n 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 ?

Hybrid Automaton n Zeno Executions n Infinite execution defined over finite time n n 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

Transition System

Region Equivalence Relation

Timed Automata

Region Equivalence Relation

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