﻿ 基于快速自适应超螺旋算法的制导律<sup>*</sup>
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1. 陆军工程大学石家庄校区 导弹工程系, 石家庄 050000;
2. 电子信息系统复杂电磁环境效应国家重点实验室, 洛阳 471003

Guidance law based on fast adaptive super-twisting algorithm
LIU Chang1, YANG Suochang1, WANG Liandong2, ZHANG Kuanqiao1
1. Department of Missile Engineering, Army Engineering University Shijiazhuang Campus, Shijiazhuang 050000, China;
2. State Key Laboratory of Complex Electromagnetics Environment Effects on Electronics and Information System, Luoyang 471003, China
Received: 2018-11-14; Accepted: 2019-01-23; Published online: 2019-02-26 17:18
Corresponding author. YANG Suochang, E-mail: ysuochang@163.com
Abstract: A new second-order sliding-mode guidance law with finite time stability is proposed for the design of the guidance law of surface-to-air missile attacking maneuvering target. Based on the relative motion model of the missile and the target, guidance problem is transformed into control problem of first-order system. A fast adaptive super-twisting (FAST) algorithm is proposed by introducing linear terms and a new parameter adaptive law in super-twisting (ST), which improves convergence speed without the prior knowledge of upper bound parameters of uncertainties. A quadratic Lyapunov function is adopted to verify the stability of the system in finite time and compute the convergence time. A comparison with adaptive sliding mode guidance, ST guidance and smooth second-order sliding-mode guidance shows that the proposed method can improve the convergence speed of sliding variable and avoid the difficulty of choosing parameters, and can guarantee the guidance accuracy at the same time.
Keywords: second-order sliding mode     super-twisting (ST) algorithm     guidance law     finite time stability     adaptive law

1 问题描述和相关引理 1.1 问题描述

 图 1 导弹和目标相对运动示意图 Fig. 1 Schematic diagram of relative motion of missile and target

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1.2 相关引理

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1) V(x)为正定函数。

2) 存在正实数ζ1>0和α∈(0, 1)，以及包含原点的开邻域U0U，使得下式成立：

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1) V(x)为正定函数。

2) 存在正实数ζ1>0，ζ2>0和α∈(0, 1)，以及包含原点的开邻域U0U，使得下式成立：

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2 FAST算法设计及稳定性证明 2.1 ST算法

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ρ1=E1(x1, t)，，式(20)的等价形式为

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2.2 FAST算法设计

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2.3 有限时间稳定性证明

ξT=[ϕ1(x1), x2]，由

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Q11 < 0，Q22 < 0，Q12=Q21=0，于是Q为半负定矩阵。令

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ϕ1(x1)=|x1|1/2sgn(x1)+x1可得

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1) 若k1k1*, k2k2*，且，由定理1可知，定理2显然成立。

2) 若k1 < k1*, k2 < k2*，选取类二次型Lyapunov函数：

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① 若|x1|>0，则

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② 若|x1|=0，则系统状态到达滑模面，且，因此k1=k1*k2=k2*，由引理1可知，系统有限时间收敛。证毕

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3 制导律设计

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FAST制导律对外界干扰具有鲁棒性，且能够在有限时间内收敛。由制导律的形式可以看出，参数k1k2随着s的变化实时改变，且不需要已知外部干扰的上界。

4 仿真分析

ASMG为[17]

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SSOSMG为[18]

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STG为[19]

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FASTG形式如式(64)所示，参数满足：a=0.5, b=1, c1=c2=1, ε2=0.1。

4.1 目标无机动(情形1)

MATLAB仿真仿真结果如表 1图 2~图 4所示。表 1为情形1的仿真结果，其中Δ为脱靶量，treach分别为滑模变量实际和理论收敛时间，tf为制导时间。图 2为导弹弹道曲线，黑色虚线为目标运动曲线，绿色、黑色、红色、蓝色实线分别为ASMG、SSOSMG、STG、FASTG四种制导律下的导弹运动曲线。图 3(a)图 3(b)分别为滑模变量s(t)及滑模变量的一阶导数变化曲线。图 4为导弹法向过载变化曲线。

 制导律 Δ/m treach/s tf/s ASMG 0.462 2 7.36 10.53 14.94 SSOSMG 0.518 6 12.2 21.83 14.91 STG 1.175 6 6.65 12.25 14.92 FASTG 0.616 0 5.30 14.64 14.90

 图 2 导弹弹道曲线(情形1) Fig. 2 Missile ballistic curves (Case 1)
 图 3 滑模变量及其一阶导数变化曲线(情形1) Fig. 3 Variation curves of sliding-mode variable and its first-order derivative (Case 1)
 图 4 导弹法向过载变化曲线(情形1) Fig. 4 Variation curves of missile normal overload (Case 1)

4.2 目标有机动(情形2)

MATLAB仿真结果如表 2图 5~图 7所示，其中曲线和变量含义与情形1相同。

 制导律 Δ/m treach/s tf/s ASMG 0.549 4 — 20.31 12.57 SSOSMG 0.566 2 10.91 21.30 11.92 STG 0.319 7 — 20.43 11.97 FASTG 0.875 3 1.79 18.98 11.74

 图 5 导弹弹道曲线(情形2) Fig. 5 Missile ballistic curve (Case 2)
 图 6 滑模变量及其一阶导数变化曲线(情形2) Fig. 6 Variation curves of sliding-mode variable and its first-order derivative (Case 2)
 图 7 导弹法向过载变化曲线(情形2) Fig. 7 Variation curves of missile normal overload (Case 2)

5 结论

1) 在标准ST算法的基础上，增加了自适应参数控制器和线性项，提出了FAST算法。在系统不确定性上界未知的前提下，一方面控制器参数能够自适应调节，避免参数过大造成系统不稳定；另一方面系统在远离平衡点时具有更快的收敛速度，提升了标准ST算法的收敛特性。

2) 利用二次型Lyapunov函数证明了FAST算法的有限时间稳定性，与其他证明方法相比，该方法计算较为简单，且能够得到收敛时间的估计公式。

3) 将FAST算法成功地应用于制导律设计。仿真结果表明，本文算法在保留标准ST算法有效抑制抖振、鲁棒性强等优点的同时，具有快速收敛特性且不需要已知不确定性的上界，使得制导系统拥有更高的命中精度和稳定性。

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文章信息

LIU Chang, YANG Suochang, WANG Liandong, ZHANG Kuanqiao

Guidance law based on fast adaptive super-twisting algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(7): 1388-1397
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0654