﻿ 考虑机动效率的多约束导引律设计<sup>*</sup>
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Design of guidance law with multiple constraints considering maneuvering efficiency
WANG Chaolun, XUE Lin, YAN Xiaoyong
The Second Academy of China Aerospace Science & Industry Corp., Beijing 100854, China
Received: 2016-07-05; Accepted: 2016-08-10; Published online: 2016-09-02 14:39
Foundation item: National Natural Science Foundation of China (11202023)
Corresponding author. XUE Lin, E-mail:majie283@sina.com
Abstract: Due to the guidance law with terminal intercept angle which will cause big angle error when available payload is insufficient, a guidance law considering time-varying overload constraint has been elicited, which would bring on more energy loss when much maneuver is achieved at the same time. Given this, this paper elicits a guidance law with multiple constraints considering maneuvering efficiency. First, a closed-loop guidance law with time-varying control weight coefficient is elicited according to optimal quadratic theory. Second, drag coefficient when maneuvering is introduced into time-varying control weight coefficient, and the constraint boundaries of available payload and maneuvering efficiency are obtained through iterations. Finally, the time-varying weight coefficient is changed into function of time-to-go, and the trajectories are simulated with guidance law considering available payload and maneuvering efficiency. The simulation results indicate that both the two guidance laws can meet the requirement of trajectory shaping, and the acceleration command of guidance law with constraint considering maneuvering efficiency is more reasonable, which reduces the velocity loss effectively and enhances the guidance accuracy and damage effect. Moreover, balance solution of time-varying weight coefficient is not necessary with this method, so iteration speed will be highly improved when accuracy is guaranteed.
Key words: air-defense missile     guidance law     terminal intercept angle     multiple constraints     optimal control     maneuvering efficiency

2005年，Cloutier[5]对考虑常值过载边界饱和的导引律进行了研究。2007年，Hexner和Shima[6]对考虑常值过载约束的随机最优导引律进行了研究。2014年，董晨等[7]提出了一种带落角约束与控制约束的纵向制导律，并给出了制导律参数的在线整定方法。这些方法都是基于控制饱和的前提，即过载约束在导弹拦截阶段为时不变的常值，这与稠密大气中的拦截情况有较大出入。

2003年，Cho及其团队研究了二维平面内的一种最优制导律，该制导律的关键在于考虑到导弹速度时变对制导带来的影响，推导出适用于反舰导弹的带落角约束的导引律[8-9]。2014年，王辉和林德福等[10]以导弹剩余飞行时间的幂函数为基础构建扩展的目标罚函数，并探讨了不同指数取值对弹道成型的影响。以上方法均可认为是对时变可用过载边界的近似估计，与实际情况差别较大，工程上难以实现。

1 弹目相对运动数学模型

 图 1 弹目相对运动关系示意图[3] Fig. 1 Schematic diagram of missile-target relative movement geometry[3]

R为弹目相对距离，有

 (1)

γ为弹目交会角，则弹目相对速度和相对交会角速度分别为

 (2)

q为小量时，对式(2) 线性化：

 (3)

 (4)

x求导，得

 (5)

 (6)
 (7)

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2 多约束导引律

 (9)

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F(t)求微分，得

 (11)

 (12)

 (13)

 (14)

 (15)

 (16)

 (17)

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 (24)

 (25)

 (26)

 (27)

 (28)

 (29)

3 机动性能权系数的确定

3.1 时变可用过载权系数

 (30)

 (31)

 图 2 可用过载拟合曲线与实际可用过载曲线对比 Fig. 2 Contrast between available overload fitted curve and practical available overload curve

 (32)

3.2 时变机动效率权系数

 (33)

 (34)

 图 3 常规防空导弹升阻比曲线 Fig. 3 Common air-defense missile's lift-drag ratio curves
 (35)

 图 4 常规防空导弹Cy/Cx(max)随Ma变化 Fig. 4 Variation of common air-defense missile Cy/Cx(max) with Ma

 (36)

4 弹道仿真分析

 图 5 导弹机动效率随迭代次数变化 Fig. 5 Variation of missile maneuvering efficiency with iteration times
 图 6 性能约束边界与剩余时间关系 Fig. 6 Relationship between performance constrained boundary and time-to-go
 图 7 3种导引律末制导弹道对比(匀速目标) Fig. 7 Three guidance laws' terminal trajectory contrast(constant target)
 图 8 3种导引律弹目交会角对比(匀速目标) Fig. 8 Three guidance laws' missile-target intercept angle contrast(constant target)
 图 9 3种导引律指令加速度对比(匀速目标) Fig. 9 Three guidance laws' command acceleration contrast (constant target)

 导引律 脱靶量/m 末端速度/(m·s-1) 末端交会角/(°) 拦截时间/s 交会角误差/(°) 交会角约束 5.75 455.2 84.29 13.63 3.29 交会角/可用过载约束 0.43 514.5 80.31 13.38 -0.69 交会角/机动效率约束 0.51 519.4 81.05 13.29 0.05

 图 10 3种导引律末制导弹道对比(机动目标) Fig. 10 Three guidance laws' terminal trajectory contrast (manevering target)
 图 11 3种导引律指令加速度对比(机动目标) Fig. 11 Three guidance laws' command acceleration contrast (manevering target)

 导引律 脱靶量/m 末端速度/(m·s-1) 末端交会角/(°) 拦截时间/s 交会角误差/(°) 交会角约束 1.58 570.2 -1.00 12.30 -1.00 交会角/可用过载约束 0.14 569.5 -0.66 12.29 -0.66 交会角/机动效率约束 0.22 571.1 -0.60 12.29 -0.60

5 结论

1) 推导了一种通过迭代确定时变控制项权系数的闭环制导规律。

2) 考虑以可用过载约束的导引律在机动同时会带来较大的附加阻力，在以可用过载为时变权系数的基础上，引入当地阻力系数，提出一种不仅满足时变过载约束又能保证导弹机动损耗能量较小的导引形式。

3) 将3种导引律进行仿真验证，证明所设计导引律的有效可行，特别对过载要求较高的情况下效果明显。同时极大提高了迭代速度，为防空导弹拦截高速机动目标提供了方法参考。

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#### 文章信息

WANG Chaolun, XUE Lin, YAN Xiaoyong

Design of guidance law with multiple constraints considering maneuvering efficiency

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(8): 1594-1601
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0567