1. Aditya Zutshi Sriram Sankaranarayanan Ashish Tiwari TIMED RELATIONAL ABSTRACTIONS FOR SAMPLED DATA CONTROL SYSTEMS

2. SAMPLED DATA CONTROL SYSTEMS

3. CONTROL SYSTEMS

5. SAMPLED CONTROL SYSTEMS Discrete Controller compute wait Ts actuatesense Plant: hybrid system Hybrid Plant M2 ODE2 M1 ODE1 M3 ODE3 Controller: software

6. System speed time SAMPLED CONTROL SYSTEMS Discrete Controller Physical Plant SA Desired Speed

7. System speed time SAMPLED CONTROL SYSTEMS Discrete Controller Physical Plant SA uphill!

8. PLANT – HYBRID AUTOMATON Down shift Up shift

9. RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Hybrid Automaton] Actuate (Ts) Sense (Ts) Abstract the plant dynamics using relations

10. RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Hybrid Automaton] Actuate (Ts) Sense (Ts)

11. RELATIONALIZATION Discrete System [Discrete Transition System] Actuate (Ts) Sense (Ts) ODE 1 ODE 2

12. RELATIONALIZATION Discrete System [Discrete Transition System] Actuate (Ts) Sense (Ts) R1R1 R2R2

13. RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Discrete Transition System] Actuate (Ts) Sense (Ts) Use existing tools to verify safety properties

14. TIMED RELATIONAL ABSTRACTIONS Plant state time Plant Dynamics

15. TIMED RELATIONAL ABSTRACTIONS Plant state time System Dynamics

16. TIMED RELATIONAL ABSTRACTIONS Plant state time Relational Abstraction R R R

17. TIMED RELATIONAL ABSTRACTIONS  Relation R  Captures states reachable in one sampling period  Resulting abstraction  is equivalent: when only controlled transitions are present  sound: when autonomous transitions are present Plant state time R R R

18. CONTROLLED TRANSITIONS Relationalize

19. AUTONOMOUS TRANSITIONS

20. time m1 ODE1 Controlled Transitions m1 ODE1 m2 ODE2 Autonomous Transitions M1 ODE1 M2 ODE2 M5 ODE5 M3 ODE3 M4 ODE4 Dwell Time Restriction

21. AUTONOMOUS TRANSITIONS time m1 ODE1 Controlled Transitions m1 ODE1 m2 ODE2 Autonomous Transitions

22. The resulting abstraction is a quantified formula over exponentials. AUTONOMOUS TRANSITIONS Relationalize

23.  Solution  Using interval arithmetic rewrite the formula as a Interval linear inequalities  Reformulate as a Linear Complementarity Problem  Linearize the dynamics around the midpoint and iteratively find the bounds AUTONOMOUS TRANSITIONS

24. IMPLEMENTATION

25.  Experiments:  NAV and Heat benchmark set [Ivancic + Fehnker]  Benchmarks formulated in the paper  Results:  Promising for systems with many controlled transitions + few autonomous transitions  Precision loss as number of autonomous transition increases  Our Approach:  Is sound  Provides proofs when the property is inductive  Is exact for controlled transitions EXPERIMENTAL RESULTS

27. QUESTIONS?