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 12, 2004 john.koo@vanderbilt.edu http://www.vuse.vanderbilt.edu/~kootj All rights reserved.

2. 2 Analysis: Reachability - Continuous

3. 3 Topics Definitions Ordinary Differential Equations Reachable Sets Computation of Reachable Sets Computational tools 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.

4. 4 Hybrid Automaton Hybrid Automaton (Lygeros, 2003)

5. 5 Hybrid Automaton

6. 6 Q X Execution

7. 7 Hybrid Automaton

8. 8 Ordinary Differential Equations

9. 9

10. 10 Ordinary Differential Equations

11. 11 Reachable Sets

12. 12 Reachable Sets

13. 13 Reachable Sets

14. 14 Reachable Sets

15. 15 Reachable Sets

16. 16 Reachable Sets

17. 17 Computation of Reachable Sets

18. 18 Computation of Reachable Sets

19. 19 Computation of Reachable Sets

20. 20 Computation of Reachable Sets

21. 21 Computation of Reachable Sets

22. 22 Computational Tools Reachable Sets Computation Finite Automata Timed Automata Linear Automata Linear Hybrid Systems Nonlinear Hybrid Systems d/dt CheckMate Timed COSPAN KRONOS Timed HSIS VERITI UPPAAL HYTECHCOSPAN SMV VIS … Requiem

23. 23 End