Outline

1. Outline • Strix – Saab Bofors (BAE) • Precision Guided Mortar Munition (PGMM) – ATK • Projectile Equations of Motion • Controllability • Bang-Bang Control • Impulsive dynamical systems • “Naïve” control strategy • Simulations • Future directions

2. Terminally Guided Mortar Munition

3. STRIX Main Parts Package Programming unit Launch unit Projectile Sustainer

4. Projectile, Main Parts Fuze System Impact Sensor Electronics & Power Supply Control Rockets Assembly Warhead Fin Assembly Target Seeker

5. Data

6. Sequence of Events 3. Ballistic phase 4. Guidance phase Forward observer 2. Launch 1. Preparations

7. Guidance Phase Hit Find Sustainer separation Electric arming Guidance with control rockets Target seeker activation Target acquisition and selection Proportional navigation

8. STRIX Target Impact • KILL • Initiation of warhead from impact sensor • Penetration of ERA and main armour • Behind armour effect (pressure etc.)

9. Projectile Model Equations of Motion Forces:

10. Projectile Model cont’d Equations for Rotational rates Moments

11. Controllability Definition: Given the system Controllability: The pair (A,B) is said to be controllable iff at the initial time t0 there exist a control function u(t) which will transfer the system from its initial state x(t0) to the origin in some finite time. If this statement is true for all time, then the system is "Completely Controllable".

12. Controllability cont’d • Is the “full-information” nonlinear model of the projectile, with no wind, controllable such that it will land within a terminal set T, for a given number of discrete, fixed magnitude impulses? • Note that the control impulses have additional constraints which include: • each control impulse can only be fired once • presences of a dwell-time between firings • finite burn time

13. Controllability, cont’d • Three approaches to the nonlinear controllability problem with finite, discrete impulses are investigated: • Bang-Bang control • Impulsive dynamical systems • Naïve control design

14. Bang-Bang Control • The problem is to find a feasible bang-bang control action that takes the system from a given initial point to a given terminal point with time being a free parameter. • Minimum fuel optimal control problem • Unfortunately, the theory of minimum-fuel systems is not as well developed as the theory of minimum-time systems. Also the design of fuel-optimal controllers is more complex that time-optimal controllers. In fact there may not exist a fuel-optimal control, with a finite number of discrete thrusters, that drive the projectile from any initial state to the origin (controllable).

15. Impulsive Dynamical Systems • Many systems exhibit both continuous- and discrete-time behaviors which are often denoted as hybrid systems. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist of three elements: • a continuous-time differential equation, which governs the motion of the dynamical system between impulsive or resetting events; • a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs; • and a criterion for determining when the states of the system are to be reset.

16. Impulsive Dynamical Systems cont’d The projectile control problem can be viewed as an impulsive dynamical system, whose analysis can be quite involved. In the general situation, such systems can exhibit infinitely many switches, beating, etc. Controllability of hybrid systems is a hot topic currently, and despite the numerous papers on the topic efficient numerical algorithms that provide control algorithms is still lacking.

17. Naïve Control Strategy • Projectile control algorithms are often synthesized in an ad hoc manner. These solutions are logic based and involve testing a performance criteria at each time step. • Consider the following control strategy to drive a projectile from a given state to a target set: • If current state in the target set, STOP • Given current point (after apex). If an impulse is not active then compute the corresponding impact state and the miss distances. • Numerically integrate EOM to determine impact location • If miss distance is within tolerance, NO ACTION taken. • If miss distance is less than target set, FIRE in positive direction • If miss distance is more than target set, FIRE in negative direction • Under some conditions, NCS results in optimal solution.

18. Simulations • Ideal assumptions include: • Restricting the problem to two dimensions (x,y) • No wind, target location, projectile position/velocity known • Each impulse can be fired more than once • Infinitely many impulses • Point mass model • Physical parameters: • weight, 33 lbs • muzzle velocity and angle: 235 m/s, 50 degs • impulse duration and magnitude: 0.015 +/- .0002s, 5.0 +/-0.3 g • sample time,0.005s • Impact error tolerance: 0.1m • Unaided projectile path: 2772 m

19. Projectile Path Undisturbed + 300 Meters - 300 Meters

20. Point mass model Impact point computed exactly, 39 shots required • Due to numerical errors, two extra shots were needed • Chattering caused by numerical integration errors, which is typical of NCS algorithm

21. Rigid body model Impact point within 0.1m, 9 shots required • Due to numerical errors, series of extra shots were needed

22. Impact Distribution for +200m Target

23. Impulse Distribution Positive impulse Negative impulse 8 shots

24. Tradeoff Accuracy • Total impulse force constant, #shots x impulse = 24*5g • More shots: Increased accuracy, more complex • Less shots: Less accurate, cheaper

25. Initial Findings • It is easier to hit targets beyond the initial trajectory • Function of the limited flight time of the projectile and computation delay • If the target is overshot, the projectile may not be able to react fast enough to bring it down in time. • Current configurations allow for no more than a 225 meter overshoot and 310 meter undershoot

26. Summary • Interesting class of control systems for which there has been a limited amount of theoretical results • For the short term, focus on better understanding the naïve control strategy • Rigid body equations of motion • Atmospheric disturbances • Trajectory tracking versus end point control • Over the long term, develop a mathematical framework for control of nonlinear systems with a finite number of discrete, finite duration, fixed magnitude impulses.

27. References on Projectile Control • B. Burchett and M. Costello, “Model Predictive Lateral Pulse Jet Control of an Atmospheric Rocket,” Journal of Guidance, Control and Dynamics, V25, 5, 2002. • E. Cruck and P. Saint-Pierre, “Nonlinear Impulse Target Problems under State Constraint: A Numerical Analysis Based on Viability Theory,” Set-Valued Analysis, 12, pp. 383-416, 2004. • B. Friedrich, ATK, Private Communication. • S.K. Lucas and C.Y. Kaya, “Switching-Time Computation for Bang-Bang Control Laws,” Proceedings of the American Control Conference, Arlington, VA June 25-27, pp. 176-180, 2001 • C.Y. Kaya and J.L. Noakes, “Computations and time-optimal controls,” Optimal Control Applications and Methods, 17, pp. 171--185, 1996. • Y. Gao, J. Lygeros, M. Quincampoix and N. Seube, “On the control of uncertain impulsive systems: approximate stabilization and controlled invariance,” Int. J. Control, vol. 77, 16, pp. 1393-1407, 2004. • E.G. Gilbert and G.A. Harasty, “A Class of Fixed-Time Fuel-Optimal Impulsive Control Problems and an Efficient Algorithm for Their Solution,” IEEE Trans. Automatic Control, vol. 16, 1, pp.1-11, 1971 • Z.H. Guan, T.H. Qian and X. Yu, “On controllability and observability for a class of impulsive systems,” Systems and Control Letters, 47, p247-257, 2002.

28. References on Projectile Control • W.M. Haddad, V. Chellaboina and N.A. Kablar, “Non-linear impulsive dynamical systems. Part I: Stability and dissipativity,” Int. J. Control, vol. 74, 17, pp. 1631-1658, 2001. • W.M. Haddad, V. Chellaboina and N.A. Kablar, “Non-linear impulsive dynamical systems. Part II: Stability and dissipativity,” Int. J. Control, vol. 74, 17, pp. 1659-1677, 2001. • H. Ishii and B. A. Francis, “Stabilizing a Linear System by Switching Control with Dwell Time,” IEEE Trans. Automatic Control, pp.1962-1973, 2002. • T. Jitpraphai, B. Burchett and M. Costello, “A Comparison of different guidance schemes for a direct fire rocket with a pulse jet control mechanism,” AIAA-2001-4326, 2001. • R. Pytlak and R.B. Vinter, “An Algorithm for a general minimum fuel control problem,” Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, FL, December 1994. • G. N. Silva and R. B. Vinter, “Necessary conditions for optimal impulsive control problems,” SIAM J. Control Opt., vol. 35, 6, pp. 1829-1846, 1997. • G. Xie and L. Wang, “Necessary and sufficient conditions for controllability and observability of switched impulsive control systems,” IEEE Trans. Automatic Control, vol. 49, 6, pp.960-977, 2004.