Transcript Optimization-based Formation Reconfiguration Planning For
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
Summary 2
Hybrid System
A system built from atomic discrete components and continuous components by parallel and serial composition, arbitrarily nested.
The behaviors and interactions of components are governed by models of computation (MOCs).
Discrete Components
Finite State Machine (FSM)
Discrete Event (DE) Synchronous Data Flow (SDF) Continuous Components
Ordinary Differential Equation (ODE) Partial Differential Equation (PDE)
u q 1 q 2 q 3 x xç = f (x) + g(x)u
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Why Hybrid Systems?
Modeling abstraction of
Continuous systems with phased operation (e.g. walking robots, mechanical systems with collisions, circuits with diodes) Continuous systems controlled by discrete inputs (e.g. switches, valves, digital computers) Coordinating processes (multi-agent systems) Important in applications
Hardware verification/CAD, real time software Manufacturing, communication networks, multimedia Large scale, multi-agent systems
Automated Highway Systems (AHS) Air Traffic Management Systems (ATM)
Uninhabited Aerial Vehicles (UAV) Power Networks 4
Topics
Modeling
Finite State Machines Time Automata Ordinary Differential Equations Hybrid Automata Analysis
Reachability - Discrete Reachability - Continuous Reachability - Hybrid Tool
Ptolemy II HyTech Requiem d/dt Checkmate Verification
Temporal Logic Model Checking Time Automata
u q 1 q 2 q 3 x xç = f (x) + g(x)u
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Hybrid Automaton 6
Hybrid Automaton
Hybrid Automaton (Lygeros, 2003) 7
Q
Hybrid Automaton Execution
X
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Examples: Thermostat
t
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Examples: Bouncing Ball 10
Motivating Examples:Two Tanks 11
i 4 3 2 1 0 Hybrid Automaton t 12
Hybrid Automaton i 4 3 2 1 0 t 13
Hybrid Automaton i 4 3 2 1 0 t 14
Hybrid Automaton 15
Examples: Bouncing Ball 16
Hybrid Automaton 2 1 0 i finite t 2 1 0 i infinite t 17
Hybrid Automaton 2 1 0 i finite t 2 1 0 i Zeno t 18
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. 19
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 20
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 22
Exercise 23
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 27
Examples: Bouncing Ball 28
Examples: Bouncing Ball 29
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? 35
Timed Automata
Is this model:
Deterministics? No. Non-Blocking? Yes.
Zeno? No.
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In Summary Deterministic Non-Blocking Thermostat NO Bouncing Ball YES Two Tanks YES Time Automaton NO YES YES YES YES NO YES YES NO Zeno 37
In Summary Deterministic Non-Blocking Thermostat Verification NO Bouncing Ball YES Mapping Two Tanks YES Time NO YES YES YES YES Zeno NO YES Special Attention in Simulation YES NO 38
Computational Tools 39
Computational Tools
Simulation
Ptolemy II: ptolemy.eecs.berkeley.edu
Modelica: www.modelica.org
SHIFT: www.path.berkeley.edu/shift Dymola: www.dynasim.se
OmSim: www.control.lth.se/~cace/omsim.html
ABACUSS: yoric.mit.edu/abacuss/abacuss.html
Stateflow: www.mathworks.com/products/stateflow CHARON: http://www.cis.upenn.edu/mobies/charon/
Masaccio: http://www-cad.eecs.berkeley.edu/~tah/Publications/masaccio.html
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Computational Tools
Simulation Masaccio CHARON StateFlow/Simulink Dymola Modelica System Complexity ABACUSS SHIFT OmSim Models of Computation Ptolemy II 41
Computational Tools
Verification Finite Automata Timed Automata Linear Automata Linear Hybrid Systems Nonlinear Hybrid Systems COSPAN SMV VIS … Timed COSPAN KRONOS Timed HSIS VERITI UPPAAL HYTECH Requiem d/dt CheckMate 42
End 43