1. APPLICATION OF GEOMETRICAL EXTRAPOLATION METHOD BASED HYBRID SYSTEM CONTROLLER ON PURSUIT-AVOIDANCE DIFFERENTIAL GAME Ginzburg Pavel and Slavnaya Lyudmila Supervisory by Dr. Mark Moulin

2. List of contents: Optimal control and Hamiltonian formalism Hybrid control Intuitive Geometrical Controller Circle Extrapolation Controller Second Order Approximation in Geometrical Extrapolation Nonlinear Heading Estimator The Supper Hybrid Controller Conclusion

3. Hamiltonian Formalism

4. System representation

5. Solutions of Backward Reachable Set Capture radius 5 Linear velocities 5 Angular velocities 1 Capture radius 5 Linear velocities 5 Angular velocity Evader 2 Angular velocity Pursuer 1 Capture radius 5 Linear velocities 5 Angular velocity Evader 1 Angular velocity Pursuer 2

6. Open loop via closed loop control Open loop: + Safe for initial condition check - Sensitive to measurement noise - Non flexible to task changing - Slow and massive calculation (offline ) - And more… Closed loop: + Safe control + Online and fast calculations - Non optimal

7. Intuitive Geometrical Controller Plane division to provide control signal Problematic situation for Simple Controller

8. Results of Intuitive Geometrical Controller Initial conditions (-2, 2, π/2) Initial conditions (-2, 2, π/3) Control signal output

9. Circle Extrapolation Controller Red Cross – estimated position Blue Star – measurement data, stored in controller memory Assumptions: - The opponent control signal unchanged during the sampling Principe: - 3 points define the only one circle - Forth point is the opponent estimated position -Controller chooses the optimal output depends on the estimated position - Each step correction provided (like Kalman filter measurement correction)

10. Results of Circle Extrapolation Controller Initial conditions (-2, 2, π/2) Control signal output Initial conditions (-2, 2, π/3) Control signal output

11. Second Order Approximation in Geometrical Extrapolation

12. Results Geometrical Extrapolation Controller Initial conditions (-2, 2, π/3) Initial conditions (-2, 2, π/2) Control signal output

13. Geometrical Extrapolation Controller used by both players Border point (0, 3.44, -π) Applied control signals Border point (2, 1.42, -π/2) (0, 3, -π) (1.6, 1, -π/2)

14. Results Geometrical Extrapolation Controller with “arctan” “Sign” function replaced by “Arctangent” Initial conditions (-2, 2, π/2) Control signal output Initial conditions (-2, 2, π/3)

15. Noise in coordinate measurement 1 variance noise Border point (2, 1.42, -π/2) 0.1 variance noise Border point (2, 1.42, -π/2)

16. Nonlinear Heading Estimator Assumptions: - The opponent heading signal unchanged during the sampling Principe: - 5 points is a trade off between noise filtering and device flexibility Controller provides the most fitted line (MSE) Red points – measurement data Blue lines – calculated slopes, the estimator output

17. Performances of Nonlinear Heading Estimator Constant control input (0), noisy case (0.01 variance) no noise exist Constant control input (0.1), noisy case (0.01 variance).

18. The Supper Hybrid Controller Idea: - fitting noisy data to estimated orbit - verify the noise level by R-condition checking - choose more successful controller implementation Travels between hybrids controllers (double hybridism exists)

19. Conclusions Advantages of Hybrid Control Implementation The geometrical extrapolation method in Hybrid system provides wise plane division Noise filtering using The Method More levels in Hybrid Implementation

20. Directions Non linear filtering (fit the surface to measurement data) Analytical approximated solution of HJB equations More complex differential games General case