1. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg1 Final Presentation Online-implementable robust optimal guidance law - Raghunathan T., Ph.D. student (On behalf of Late Dr. S Pradeep, Associate Professor, Aerospace Engineering Department)

2. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg2 Two dimensional missile-target engagement model

3. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg3 Background and motivation: Miss distances for the linear model

4. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg4 Background and motivation Optimal guidance law (OGL) Assumptions a) linear model of missile-target engagement : b) unbounded control : infinite lateral acceleration c) t go known accurately d) constant target maneuver Yields an analytical/closed form solution that is implementable online

5. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg5 Reality : how valid are the assumptions? a) Missile-target engagement kinematics is highly nonlinear b) Lateral acceleration is limited by saturation c) t go cannot be known accurately d) Constant target maneuver?

6. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg6 Result of applying OGL to the nonlinear kinematic model Miss distances for the plant

7. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg7 Objective An improved, robust guidance law i ) that nullifies or at least mitigates the effect of assumptions made ii) implementable in real-time

8. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg8 Solution Methodology (i) Make use of the solution (i.e. OGL) that we know, as a starting point (ii) Explore the solution space around this starting point for the best solution

9. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg9 The starting point: optimal guidance law (OGL) Minimise subject to

10. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg10 Linear model

11. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg11 The starting point: OGL (cont’d) The solution/control input/lateral acceleration/OGL: Cancellation of system dynamics

12. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg12 Own problem formulation Minimize subject to free and free Control input/guidance law :

13. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg13 Nonlinear kinematic model

14. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg14 Challenges 1) lack of optimal control methods to deal with inequality constraints 2) real-time implementation

15. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg15 Our approach: The Differential Evolution Tuned Optimal Guidance Law (DE-OGL): Control input/guidance law : (Differential Evolution is one of the evolutionary computation (EC) methods)

16. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg16 Differential Evolution (DE) parameters used: Crossover constant, CR = 0.9 Weighting factor, F = 0.8 Population size, NP = 12 Stopping criterion: max. no. of generations = 4 or solution < tolerance limit

17. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg17 Real-time implementation: The Optimal Control Problem and evolutionary computation(EC) In general, EC is computationally intensive! Which leads to the second set of challenges : System dynamics slow enough A ‘good enough’ (suboptimal) solution Massively parallel implementation

18. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg18 EC for the Missile Guidance Problem Fast dynamics Acceptable: almost the best solution Limited onboard computation power Must be available in real-time !

19. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg19 Online Implementation actuator plant/ guidance system OGL OGL model plant model OGL DE + + DE- OGL

20. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg20 Acceleration signatures

21. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg21 Comparison of total acceleration PNAPNOGLDE-OGL 100 %81.3 %85.9 %55.5 %

22. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg22 Miss distances for all guidance laws

23. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg23 N’ for OGL and DE-OGL

24. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg24 Convergence of the solution

25. 29 Oct 200725 Future work For more practical maneuvers of target More complex model? Applicability to a larger range of initial conditions?

26. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg26 Publications Papers: (a) Raghunathan T. and S. Pradeep, “A Differential Evolution Tuned Optimal Guidance Law,” in The 15th Mediterranean Conference on Control and Automation - MED’07 held at Athens, Greece during June 27-29, 2007. (b) Raghunathan T. and S. Pradeep, “An online Implementable Differential Evolution Tuned Optimal Guidance Law,” in Genetic and Evolutionary Computation Conference - GECCO 2007, held at London, United Kingdom, during July 7-11, 2007. Technical Report: Raghunathan T. and S. Pradeep, “Online-implementable Robust Optimal Guidance Law,” Technical Report No. TR-PME-2007-12 dated 20 December 2007, under DRDO-IISc Programme on Advanced Research in Mathematical Engineering. The financial support provided for the above by DRDO-IISc Program on Advanced Research in Mathematical Engineering is gratefully acknowledged

27. 15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg27 The End