# ANALYSIS ON GENERALIZED IMPACT ANGLE CONTROL GUIDANCE LAW

• LEE, YONG-IN (DEPARTMENT OF GUIDANCE AND CONTROL, AGENCY FOR DEFENSE DEVELOPMENT)
• Received : 2015.08.10
• Accepted : 2015.08.24
• Published : 2015.09.25

#### Abstract

In this paper, a generalized guidance law with an arbitrary pair of guidance coefficients for impact angle control is proposed. Under the assumptions of a stationary target and a lag-free missile with constant speed, necessary conditions for the guidance coefficients to satisfy the required terminal constraints are obtained by deriving an explicit closed-form solution. Moreover, optimality of the generalized impact-angle control guidance law is discussed. By solving an inverse optimal control problem for the guidance law, it is found that the generalized guidance law can minimize a certain quadratic performance index. Finally, analytic solutions of the generalized guidance law for a first-order lag system are investigated. By solving a third-order linear time-varying ordinary differential equation, the blowing-up phenomenon of the guidance loop as the missile approaches the target is mathematically proved. Moreover, it is found that terminal misses due to the system lag are expressed in terms of the guidance coefficients, homing geometry, and the ratio of time-to-go to system time constant.

#### References

1. C. K. Ryoo, H. Cho, and M. J. Tahk, Optimal guidance laws with terminal impact angle constraints, J. Guid. Control. Dynam., 28(4) (2005), 724-732. https://doi.org/10.2514/1.8392
2. M. Kim and K. V. Grider, Terminal guidance for impact attitude angle constrained flight trajectories, IEEE T. Aero. Elec. Sys., 9(6) (1973), 852-859.
3. A. E. Bryson, Jr. and Y-C Ho, Applied Optimal Control, John Wiley & Sons, (1975), 154-155.
4. J. Z. Ben-Asher, Optimal trajectories for an unmanned air-vehicle in the horizontal plane, J. Aircraft, 32(3) (1995), 677-680. https://doi.org/10.2514/3.46773
5. Y. I. Lee, C. K. Ryoo, and E. Kim, Optimal guidance with constraints on impact angle and terminal acceleration, P. AIAA Guid. Nav. Cont. Conf., Austin, TX, Aug. (2003).
6. C. K. Ryoo, H Cho, and M. J. Tahk, Time-to-go weighted optimal guidance with impact angle constraints, IEEE T. Cont. Sys. Tech., 14(3) (2006), 483-492. https://doi.org/10.1109/TCST.2006.872525
7. J. I. Lee, I. S. Jeon, and M. J. Tahk, Guidance law to control impact time and angle, IEEE T. Aero. Elec. Sys., 43(1) (2007), 301-310. https://doi.org/10.1109/TAES.2007.357135
8. B. S. Kim, J. G. Lee, and H. S. Han, Biased PNG law for impact with angular constraint, IEEE T. Aero. Elec. Sys., 34(1) (1998), 277-288. https://doi.org/10.1109/7.640285
9. R. E. Kalman, When is a linear control system optimal?, Trans. ASME, J. Basic Eng., Ser. D, 86 (1964), 51-60. https://doi.org/10.1115/1.3653115
10. E. Kreindler and A. Jameson, Optimality of linear control systems, IEEE T. Automat. Contr., 17 (1972), 349-351. https://doi.org/10.1109/TAC.1972.1099985
11. E. Kreindler, Optimality of proportional navigation, AIAA J., 11(6) (1973), 878-880. https://doi.org/10.2514/3.50527
12. A. Jameson and E. Kreindler, Inverse problem of linear optimal control, SIAM J. Control, 11(1) (1973), 1-19. https://doi.org/10.1137/0311001
13. M. Guelman, The closed-form solution of the true proportional navigation, IEEE T. Aero. Elec. Sys., 12(4) (1976), 472-482.
14. C. K. Ryoo, H. Cho, and M. J. Tahk, Closed-form solutions of optimal guidance with terminal impact angle constraints, P. IEEE Int'l Conf. Contr. Appl., Istanbul, Turkey, (2003), 504-509.
15. A. C. Baker and H. L. Porteous, Linear Algebra and Differential Equations, Ellis Horwood, (1990), 102-140.
16. F. B. Hilderbrand, Advanced Calculus for Applications, Second Edition, Prentice-Hall, (1976), 118-185.
17. B. Noble and J. W. Daniel, Applied Linear Algebra, Third Edition, Prentice-Hall, Englewood Cliffs, New Jersey, (1988).
18. M. A. Sanchis-Lozano, On the connection between generalized hypergeometric functions and dilogarithms, hep-ph/9511322, IFIC-95-51 (1995), 1-11.
19. Y. I. Lee, S. H. Kim, and M. J. Tahk, Analytic solutions of optimal angularly constrained guidance for firstorder lag system, Proc. IMechE Part G: J. Aero. Eng., DOI: 10.1177/0954410012442617, (2012). https://doi.org/10.1177/0954410012442617
20. Y. I. Lee, S. H. Kim, and M. J. Tahk, Optimality of Linear Time-Varying Guidance for Impact Angle Control, IEEE T. Aero. Elec. Sys., 48(3) (2012), 2802-2817. https://doi.org/10.1109/TAES.2012.6324662
21. Y. I. Lee, J. I. Lee, S. H. Kim, and M. J. Tahk, Analytic solutions of Generalized Impact-Angle-Control Guidance Law for First-Order Lag System, J. Guid. Control Dynam, 36(1) (2013), 96-112. https://doi.org/10.2514/1.57454