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 20 , 2004 [email protected]http://www.vuse.vanderbilt.edu/~kootj
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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 20 , 2004 [email protected] - PowerPoint PPT Presentation
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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
Verification Temporal Logic Model Checking Time Automata
q1q2 q3
u
xç = f (x) + g(x)u
x
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Hybrid Automaton
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Hybrid Automaton Hybrid Automaton (Lygeros, 2003)
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Hybrid Automaton
Q
X
Execution
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Examples: Thermostat
t
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Examples: Bouncing Ball
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Motivating Examples:Two Tanks
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Hybrid Automaton
t
i
012
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Hybrid Automaton
i
012
34
t
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Hybrid Automaton
i
012
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t
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Hybrid Automaton
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Examples: Bouncing Ball
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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 Zeno of Elea, 490BC
Ancient Greek philosopher The race of Achilles and the turtle
Achilles, a renowned runner, was challenged by the turtle to a race. Being a fair sportsman, Achilles decided to give the turtle a 10 meter head-start. To overtake the turtle, Achilles will have to first cover half the distance separating them. To cover the remaining distance, he will have to cover half that distance, and so on.
No matter how fast Achilles is, he can never overtake the turtle. Why???
Ans: Covering each one of the segments in this series requires a non zero amount of time. Since there is an infinite number of segments, Achilles will never overtake the turtle.
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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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Exercise
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Examples: Bouncing Ball
Is this model: Deterministics? Non-Blocking? Zeno?
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Examples: Bouncing Ball
Is this model: Deterministics?
Yes, the Guard and Domain contains only one element. Reset maps from one point to exactly another point. Also, the vector field is Lipschitz continuous.
Non-Blocking? Zeno?
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Examples: Bouncing Ball
Is this model: Deterministics? Non-Blocking?
Yes, the guard is always reachable from any initial condition within the domain and also the reset makes the state start within the domain.
Zeno?
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Examples: Bouncing Ball
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Examples: Bouncing Ball
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Examples: Bouncing Ball
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Examples: Bouncing Ball
Is this model: Deterministics? Non-Blocking? Zeno?
Yes, it is Zeno since the time sequence converges.
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Thermostat
Is this model: Deterministics? Non-Blocking? Zeno?
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Thermostat
Is this model: Deterministics? No. Non-Blocking? Yes. Zeno? No.
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Two Tanks
Is this model: Deterministics? Yes. Non-Blocking? Yes. Zeno? Yes.
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Zeno—infinitely many jumps in finite time
If
Water Tank Automaton
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Timed Automata
Is this model: Deterministics? Non-Blocking? Zeno?
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Timed Automata
Is this model: Deterministics? No. Non-Blocking? Yes. Zeno? No.