EECE 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 17,

2 Analysis: Reachability - Hybrid

3 Topics Definitions Transition System Reachable Sets Reachability Problem Reachability Algorithm Decidability Reachable Sets - Hybrid Ref: [1] J.Lygeros, Lecture Notes on Hybrid Systems, Cambridge, [2] J. Lygeros, C. Tomlin, and S. Sastry, The Art of Hybrid Systems, July [3] G.Pappas, Hybrid Systems: Computation and Abstraction, PhD Dissertation, EECS Dept., UC Berkeley, [4] Thomas A. Henzinger, The theory of hybrid automata, Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, pp , 1996.

4 Hybrid Automaton Hybrid Automaton (Lygeros, 2003)

5 Hybrid Automaton

6 Q X Execution

7 Transition System

8 Unsafe Set Transition System Initial set

9 Unsafe Set Transition System Initial set

10 Transition System

11 Reachability Problem

12 Reachability Problem

13 Reachability Problem

14 Reachability Problem

15 Reachability Algorithm

16 Reachability Algorithm

17 Reachability Algorithm Unsafe!!

18 Reachability Algorithm Safe!!

19 Reachability Algorithm Keep iterating until when!?

20 Reachability Algorithm

21 Reachability Algorithm

22 Reachability Algorithm Keep iterating until when!?

23 Deciability Keep iterating until when!?

24 Deciability Keep iterating until when!?

25 Hybrid Automaton

26 Hybrid Automaton

27 Reachable Sets

28 Reachable Sets

29 Reachable Sets

30 Reachable Sets

31 Reachable Sets

32 Reachable Sets

33 Reachable Sets

34 Reachable Sets

35 Reachable Sets

36 Reachable Sets

37 Reachable Sets

38 Reachable Sets

39 Reachable Sets

40 Reachable Sets

41 Reachable Sets

42 Reachable Sets Q X

43 End