1. EE291E - UC BERKELEY EE291E: Hybrid Systems T. John Koo and S. Shankar Sastry Department of EECS University of California at Berkeley Spring 2002 http://robotics.eecs.berkeley.edu/~koo/EE291E/

2. EE291E - UC BERKELEY Controller Synthesis  A Game Theoretic Approach – In Church[1], solutions to digital circuits are studied by posing the controller synthesis problem as a discrete game between the system and its environment. – A version of the von Neumann-Morgenstern discrete game[2] is used for deriving the solution by Buchi and Landweber[3] and Rabin[4]. – Games on automata are discussed in [5]. – In [6] and [7], a survey of infinite discrete games on automata is presented. – Controller synthesis on times automata was first developed in [8] and [9]. – An algorithm for controller synthesis on linear automata is presented in [10]. – The notion of control invariance for continuous systems is described in [11]. – The notion of control invariance for hybrid systems is discussed in [12].

3. EE291E - UC BERKELEY Notation  Discrete and Continuous Systems

4. EE291E - UC BERKELEY Infinite Game on Finite Automata  System Definition

5. EE291E - UC BERKELEY Infinite Game on Finite Automata  Wining Condition

6. EE291E - UC BERKELEY Infinite Game on Finite Automata  State Space Partition

7. EE291E - UC BERKELEY Infinite Game on Finite Automata  State Space Partition

8. EE291E - UC BERKELEY Infinite Game on Finite Automata  State Space Partition Check union or intersection

9. EE291E - UC BERKELEY Infinite Game on Finite Automata  State Space Partition

10. EE291E - UC BERKELEY Infinite Game on Finite Automata  The Value Function

11. EE291E - UC BERKELEY Infinite Game on Finite Automata  The Value Function

12. EE291E - UC BERKELEY Infinite Game on Finite Automata

13. EE291E - UC BERKELEY Infinite Game on Finite Automata

14. EE291E - UC BERKELEY Infinite Game on Finite Automata

15. EE291E - UC BERKELEY Infinite Game on Finite Automata

16. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

17. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

18. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

19. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

20. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems  The Value Function

21. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems  The Value Function

22. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems  The Value Function

23. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

24. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems  Computation – Optimal Control Theory 1. 2. 3.

25. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

26. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

27. EE291E - UC BERKELEY Dynamics Games on Nonlinear Systems

28. EE291E - UC BERKELEY Reference  Synthesizing Controllers for Nonlinear Hybrid Systems Claire J. Tomlin, John Lygeros, and Shankar Sastry Volume 1386, LNCS series, Springer-Verlag, 1998. Synthesizing Controllers for Nonlinear Hybrid Systems Volume 1386LNCS series