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Hybrid and Embedded Systems: Computation and Analysis

This paper discusses hybrid systems, which are built from atomic discrete and continuous components using parallel and serial composition. The behaviors and interactions of components are governed by models of computation. The paper also explores modeling, analysis, and verification techniques for hybrid systems.

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Hybrid and Embedded Systems: Computation and Analysis

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  1. EECE 396-1Hybrid 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 john.koo@vanderbilt.edu http://www.vuse.vanderbilt.edu/~kootj

  2. Summary

  3. 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)

  4. 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

  5. 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

  6. Hybrid Automaton

  7. Hybrid Automaton • Hybrid Automaton (Lygeros, 2003)

  8. Hybrid Automaton Execution Q X

  9. Examples: Thermostat

  10. Examples: Bouncing Ball

  11. Motivating Examples:Two Tanks

  12. Hybrid Automaton i 4 3 2 1 0 t

  13. Hybrid Automaton i 4 3 2 1 0 t

  14. Hybrid Automaton i 4 3 2 1 0 t

  15. Hybrid Automaton

  16. Examples: Bouncing Ball

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

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

  19. 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.

  20. 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

  21. 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 ?

  22. 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

  23. Exercise

  24. Examples: Bouncing Ball • Is this model: • Deterministics? • Non-Blocking? • Zeno?

  25. 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?

  26. 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?

  27. Examples: Bouncing Ball

  28. Examples: Bouncing Ball

  29. Examples: Bouncing Ball

  30. Examples: Bouncing Ball • Is this model: • Deterministics? • Non-Blocking? • Zeno? • Yes, it is Zeno since the time sequence converges.

  31. Thermostat • Is this model: • Deterministics? • Non-Blocking? • Zeno?

  32. Thermostat • Is this model: • Deterministics? No. • Non-Blocking? Yes. • Zeno? No.

  33. Two Tanks • Is this model: • Deterministics? Yes. • Non-Blocking? Yes. • Zeno? Yes.

  34. If Water Tank Automaton Zeno—infinitely many jumps in finite time

  35. Timed Automata • Is this model: • Deterministics? • Non-Blocking? • Zeno?

  36. Timed Automata • Is this model: • Deterministics? No. • Non-Blocking? Yes. • Zeno? No.

  37. In Summary

  38. In Summary Verification Special Attention in Simulation Mapping Verification

  39. Computational Tools

  40. 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

  41. Computational Tools • Simulation Masaccio CHARON Ptolemy II Dymola Modelica StateFlow/Simulink System Complexity ABACUSS SHIFT OmSim Models of Computation

  42. 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

  43. End

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