1. 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 February 5, 2004 john.koo@vanderbilt.edu http://www.vuse.vanderbilt.edu/~kootj

2. 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)

3. 3 Modeling: Hybrid Automata

4. 4 Topics Hybrid Automata Definitions Examples Bouncing Ball Thermostat Properties Executions Non-Determinism Blocking Zeno Executions Ref: [1] J.Lygeros, Lecture Notes on Hybrid Systems, Cambridge, 2003. [2] J. Lygeros, C. Tomlin, and S. Sastry, The Art of Hybrid Systems, July 2001. [3] Thomas A. Henzinger, The theory of hybrid automata, Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, pp. 278-292, 1996.

5. 5 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

6. 6 Proposed Framework Control Theory Control of individual agents Continuous models Differential equations Computer Science Models of computation Communication models Discrete event systems Hybrid Systems

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

8. 8 Hybrid Automaton

9. 9 Q X Execution

10. 10 Examples: Bouncing Ball

11. 11 HyVisual http://ptolemy.eecs.berkeley.edu http://ptolemy.eecs.berkeley.edu/ptolemyII/hyvisual2.2/

12. 12 Examples: Bouncing Ball

13. 13 Examples: Bouncing Ball

14. 14 Examples: Bouncing Ball

15. 15 Hybrid Automaton t i 0 1 2 3 4

16. 16 Hybrid Automaton i 0 1 2 3 4 t

17. 17 Hybrid Automaton i 0 1 2 3 4 t

18. 18 Hybrid Automaton

19. 19 Hybrid Automaton i 0 1 2 t finite i 0 1 2 t infinite

20. 20 Hybrid Automaton i 0 1 2 t finite i 0 1 2 t Zeno

21. 21 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.

22. 22 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

23. 23 Examples: Thermostat

24. 24 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 ?

25. 25 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

26. 26 Examples: Bouncing Ball

27. 27 End