Electronics and Electrical Engineering and Control

Power series solution for miss distance of higher-order linear proportional navigation guidance systems

  • HE Tailong ,
  • CHEN Wanchun ,
  • ZHOU Hao
Expand
  • School of Astronautics, Beihang University, Beijing 100083, China

Received date: 2018-04-24

  Revised date: 2018-05-28

  Online published: 2018-08-16

Supported by

Aeronautical Science Fundation of China (20150151002)

Abstract

Miss distance is a most important performance index for the design and evaluation of a missile guidance system. For a first-order linear propprtional guidance system, closed-form solutions miss distance can be obtained. However, such solutions do not exist for a generic higher-order guidance system matching reality better, in which case miss distance is usually achieved by direct simulation or adjoint technique. In this paper, power series solutions for miss distance of high-order linear proportional navigation are explored to provide a new method for calculation of miss distance. First, the adjoint system of a generic higher-order linear proportional navigation guidance system is constructed and normalized, and the solutions in form of the product of a power series and an exponential decay are assumed. Then a recursion relation for coefficients is derived by using the power series method. Moreover, it is proved that these power series solutions converge everywhere. For single-lag and higher-order binomial guidance systems, the recursion relation is simplified significantly by selecting the appropriate exponential decay constant. In practice, partial sums of power series are used to numerically calculate miss distance, and how many terms of partial sums are adequate is related to the exponential decay constant. Therefore, the influence of the exponential decay constant on the convergence rate of power series solutions is analyzed, and selection methods for the constant are proposed, laying the foundation for practical applications of power series solutions.

Cite this article

HE Tailong , CHEN Wanchun , ZHOU Hao . Power series solution for miss distance of higher-order linear proportional navigation guidance systems[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2018 , 39(11) : 322241 -322250 . DOI: 10.7527/S1000-6893.2018.22241

References

[1] ZARCHAN P. Tactical and strategic missile guidance[M]. 6th ed. Reston, VA: AIAA, 2012: 35-105.
[2] NESLINE F W, ZARCHAN P. A new look at classical vs modern homing missile guidance[J]. Journal of Guidance, Control and Dynamics, 1981, 4(1): 78-85.
[3] SU W S, YAO D N, LI K B, et al. A novel biased proportional navigation guidance law for close approach phase[J]. Chinese Journal of Aeronautics, 2016, 29(1): 228-237.
[4] YU W B, CHEN W C, YANG L, et al. Optimal terminal guidance for exoatmospheric interception[J]. Chinese Journal of Aeronautics, 2016, 29(4): 1052-1064.
[5] 陈峰, 肖业伦, 陈万春. 基于零控脱靶量的大气层外超远程拦截制导[J]. 航空学报, 2009, 30(9): 1583-1589. CHEN F, XIAO Y L, CHEN W C. Guidance based on zero effort miss for super-range exoatmospheric intercept[J]. Acta Aeronautica et Astronautica Sinica, 2009, 30(9): 1583-1589 (in Chinese).
[6] 陈峰, 王育林, 肖业伦, 等. 基于预测脱靶量的远程拦截速度增益导引[J]. 航空学报, 2008, 29(6): 1665-1672. CHEN F, WANG Y L, XIAO Y L, et al. Velocity-to-be-gained guidance based on predicted miss distance for long-range intercept[J]. Acta Aeronautica et Astronautica Sinica, 2008, 29(6): 1665-1672 (in Chinese).
[7] WEISS M. Adjoint method for missile performance analysis on state-space models[J]. Journal of Guidance, Control, and Dynamics, 2005, 28(2): 236-248.
[8] HE T L, CHEN W C. A new interpretation of adjoint method in linear time-varying system analysis[C]//IEEE International Conference on CIS & RAM. Piscataway, NJ: IEEE Press, 2017: 58-63.
[9] 赫泰龙, 陈万春, 刘芳. 高超声速飞行器平稳滑翔扰动运动伴随分析[J/OL]. 北京航空航天大学学报, (2018-04-18)[2018-04-20]. http://kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html HE T L, CHEN W C, LIU F. Adjoint analysis of steady glide with disturbance motion for hypersonic vehicle[J/OL]. Journal of Beijing University of Aeronautics and Astronautics, (2018-04-18) [2018-04-20]. http://kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html (in Chinese).
[10] 王辉, 林德福, 祁载康, 等. 时变最优的增强型比例导引及其脱靶量解析解[J]. 红外与激光工程, 2013, 42(3): 692-698. WANG H, LIN D F, QI Z K, et al. Time-varying optimal augmented proportional navigation and miss distance closed-form solutions[J]. Infrared and Laser Engineering, 2013, 42(3): 692-698 (in Chinese).
[11] KABAMBA P T, GIRARD A R. Fundamentals of aerospace navigation and guidance[M]. Cambridge: Cambridge University Press, 2014: 141-144.
[12] INCE E L. Ordinary differential equations[M]. New York: Dover Publication, Inc., 1978: 158-185.
[13] WASOW W. Asymptotic expansions for ordinary differential equations[M]. New York: Dover Publication, Inc., 1987: 9-48.
[14] CODDINGTON E A, CARLSON R. Linear ordinary differential equations[M]. Philadelphia: Society for Industrial and Applied Mathematics, 1997: 163-220.
[15] BENDER C M, ORSZAG S A. Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory[M]. Berlin: Springer, 1999: 61-145.
[16] DUNKEL O. Regular singular points of a system of homogeneous linear differential equations of the first order[J]. Proceedings of the American Academy of Arts and Sciences, 1902, 38(9): 341-370.
[17] VAZQUEZ-LEAL H, SARMIENTO-REYES A. Power series extender method for the solution of nonlinear differential equations[J/OL]. Mathematical Problems in Engineering, 2015, 15(7): 1-7.
[18] HOLT G C. Linear proportional navigation: An exact solution for a 3rd-order missile system[J]. Proceedings of the Institution of Electrical Engineers, 1977, 124(12): 1230-1236.
[19] BOAS R P. Partial sums of infinite series, and how they grow[J]. The American Mathematical Monthly, 1977, 84(4): 237-258.
[20] ASCHER, U, GREIF C. A first course in numerical methods[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2011: 35-55.
[21] HIGHAM N. Accuracy and stability of numerical algorithms[M]. 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics, 2002: 93-104.
Outlines

/