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Adaptive terminal sliding mode guidance law with impact angle constraint
YANG Suochang , ZHANG Kuanqiao , CHEN Peng
Department of Missile Engineering, Ordnance Engineering College, Shijiazhuang 050003, China
Received: 2015-07-28; Accepted: 2015-09-18; Published online: 2015-10-10 10:13
Corresponding author. YANG Suochang, Tel.:0311-87994404, E-mail:yangsuochang_jx@sina.com
Abstract: Aimed at the requirement of zero miss-distance and terminal impact angle constraint for some missiles attacking the targets, an adaptive nonsingular terminal sliding mode control algorithm based on the theories of terminal sliding mode control and finite-time control is proposed first. The algorithm avoids the singularity of terminal sliding mode control, and makes the state variables achieve the equilibrium point by improving a fast nonsingular terminal sliding mode function to construct the sliding mode surface, and employing an adaptive exponential reaching law. Then the algorithm is utilized to design the guidance law, and an adaptive nonsingular and finite-time convergent guidance law with impact angle constraint is proposed. Realizing the requirement of miss distance and attack angle of the missiles. Finite-time control theory is used to analyze the convergence of the guidance law, and proves the fast and finite-time convergence of guidance system states during the whole process. Compared with conventional nonsingular terminal sliding mode guidance law, the designed guidance law can attack the targets with less miss-distance and higher precision of expected impact angle in a shorter time. A large number of simulation experiments verify the validity of the proposed law.
Key words: guidance law     impact angle constraint     nonsingular terminal sliding mode control     exponential reaching law     finite-time convergence

1 弹目相对运动方程

 图 1 弹目相对运动关系 Fig. 1 Relationship of missile-to-target relative motion

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 图 2 场景1的仿真实验结果 Fig. 2 Simulation experimental results of Situation 1

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2 自适应非奇异终端滑模控制

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，将式(19)代入式(20)得

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x2 ≠ 0时，ρ(x)>0，则

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x2=0时，将式(19)代入式(7)得

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3 带攻击角度约束的导引律设计

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1) 滑模到达阶段。系统状态在趋近律作用下，趋近于滑模面的过程。

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2) 沿滑模面运动阶段。系统状态到达滑模面后，沿滑模面趋近于平衡点的过程。

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4 仿真分析

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1) 场景1。打击固定目标，θm0=0°，λd=90°，vt=0 m/s，at=0 m/s2，仿真实验结果如图 2所示。

 导引律 攻击时间/s 脱靶量/m 攻击角度偏差/(°) ANTSMG 12.01 0.51 0.01 12.17 0.25 0.01 12.29 0.43 0.01 12.30 0.37 0.04 12.45 0.50 0.10 NTSMG 12.31 0.58 0.20 12.50 0.29 0.10 12.44 0.57 1.50 12.51 0.89 0.30 12.71 0.87 0.50 SMG 12.85 0.34 3.40 13.59 0.61 1.60 13.05 0.58 2.80 13.71 0.84 0.80 14.69 0.51 2.30

2) 场景2。打击匀速运动目标，θm0=0°，λd=-60°，vt=15 m/s2at=0 m/s，θt0=120°，仿真实验结果如图 3所示。在ANTSMG的作用下， sλ均能严格收敛到0和-60°；在NTSMG作用下，sλ只能分别收敛到|s| < 0.05，-59.3° < λ < -59.0°；在SMG的作用下，sλ只能分别收敛到|s| < 0.01，-56.7° < λ < -56.0°，且前者较后两者收敛速度更快，时间更短。

 图 3 场景2的仿真实验结果 Fig. 3 Simulation experimental results of situation 2

 导引律 攻击时间/s 脱靶量/m 攻击角度偏差/(°) ANTSMG 11.26 0.12 0.08 11.40 0.31 0.01 NTSMG 11.60 0.34 0.30 11.76 0.28 0.20 SMG 12.07 0.97 3.90 12.77 0.62 2.50

3) 场景3。打击机动目标，θm0=0°，λd=-60°，vt=-15 m/s，at=10cos t m/s2θt0=120°。仿真实验结果如图 4所示，为节省篇幅，滑模面曲线未给出。可以看出，在ANTSMG的作用下，sλ均能严格收敛到|s| < 0.01和-60°；在NTSMG和SMG的作用下，sλ分别收敛到|s| < 0.1、-58.1° < λ < -57.4°和|s| < 0.03、-54.3° < λ < -53.0°。说明相比非有限时间收敛的SMG，具有有限时间收敛特性的ANTSMG和NTSMG对未知的目标机动这一干扰有着很好的抑制作用。

 图 4 场景3的仿真实验结果 Fig. 4 Simulation experimental results of situation 3

5 结论

1) 采用快速收敛的非奇异终端滑模面，在不存在奇异问题的情况下，能够实现制导系统状态在有限时间内严格收敛的平衡点，且比传统非奇异终端滑模控制的收敛速度更快，选取自适应指数趋近律，能够根据系统状态距平衡点的远近自适应增大趋近速率，同时减小变结构项系数，实现系统状态全局有限时间快速收敛的同时有效削弱了抖振。

2) 所设计导引律相比非奇异终端滑模导引律和线性滑模导引律，具有更高的命中精度和更小的攻击角度跟踪误差，以及更短的攻击时间。

3) 本文所设计导引律不需要任何近似处理，形式较为简单，易于工程实现。

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#### 文章信息

YANG Suochang, ZHANG Kuanqiao, CHEN Peng

Adaptive terminal sliding mode guidance law with impact angle constraint

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(8): 1566-1574
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0502