1. Optimal Missile Guidance system By Yaron Eshet & Alon Shtakan Supervised by Dr. Mark Mulin

2. Equations of motion Target Missile

3. Interception - Overview Parameters: Interception: Necessary condition for interception for all initial conditions: The problem: non-linear and complex relation between the parameters The solution: a)Guidance law in (RTPN) to achieve b) Guidance law in to complete the interception process

4. Test case Simple maneuver simulates realistic missile

5. Test case – Guidance law RTPN: Realistic true proportional navigation compensation of target missile acceleration Guidance law perpendicular to line of sight (LOS(:

6. Test case – Guidance law Target Missile LOS distance decrease: has a projection in direction

7. Test Case: equations of motion Initial conditions

8. Interception in 37.57 sec

9. Interception time vs. The influence of depends on the initial conditions difference in interception time of order 0.1 sec difference in interception time of order 10 sec 

10. Interception time vs. For values under a certain bound, there is no guarantee for interception Interception time diverges for small values of

11. Interception time vs. Saturation zone: minor influence of. Critical influence for initial conditions and maneuver interception time ~ 37 sec interception time ~ 200 sec

12. Analytical analysis Necessary condition for interception for all initial conditions resulting condition (1) (2)

13. =1.49 (interception) =0.9 (miss) =3.85 (interception) =1.51 (interception) Behavior of with respect to (comparison with theory) extreme divergence Edge of divergence divergence occurs around r  0, as  starts varying rapidly

14. ? The solution: guidance law also in direction ensures interception for all initial conditions depends on maneuver

15. ?

16. Summery: sufficient conditions for interception K>0 1.5< These conditions ensure interception for all initial conditions and for any target missile maneuver.

17. Gain Scheduling - K The case: delay in data acquisition about the target missile maneuver limited sensitivity sec 203.73  212.08

18. Gain Scheduling - The case: adjusting for distance increase/decrease sec 47.34  49.68 not negligible

19. Constraints on interception time: Optimal control

20. Example: Limited angular acceleration

21. No limit No interception

22. Transition from failure to successful interception (green plot – previous page) Conclusion: Under realistic constraints, one gets an upper bound for K, which means a lower bound for interception time

23. Ideal interception vs. interception under constraint (blue vs. red plots) No constraint Conclusion: Interception with no constraint is faster indeed. However, it requires homing missile with higher performance and greater control effort.

24. Project summery Analysis of the equations of motion of the system Introduction of guidance laws and study of their function in ensuring interception Applying “Gain Scheduling” methods for improved performance Analysis of the system behavior under realistic constraints and restrictions