﻿ 基于虚拟域预测控制的轨迹跟踪方法<sup>*</sup>
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Trajectory tracking method based on predictive control in virtual domain
FAN Yu, YAN Liang, ZHU Wuxuan, BAI Guangzhou
Institute of Aerospace Engineering, Beijing 100094, China
Received: 2016-09-20; Accepted: 2016-10-14; Published online: 2016-11-07 16:38
Foundation item: National High-tech Research and Development Program of China
Corresponding author. YAN Liang, E-mail:yanliangbj@163.com
Abstract: A trajectory tracking method based on predictive control in the virtual domain is proposed for the algorithm of online real-time trajectory planning. The method uses polynomial approximate system model and introduces virtual path and inverse dynamics to convert time domain into virtual domain. Its advantage is that the decoupling effect is good and it takes less compute time than the control method of approximate model in the time domain. Through inverse dynamics, the import configuration of nonlinear programming ensures the continuous controls, which cannot be guaranteed by the nonlinear predictive control of traditional soft constraint method. In the background of intercepting ballistic missiles, simulation verification is carried out under the condition of initial minor disturbances and terminal condition change. The simulation results demonstrate that, compared with nonlinear feedback tracking method, the curve is smooth, the miss distance, pitch angle error and yaw angle error are small, and instantaneity can meet the control requirements.
Key words: missile     control theory     nominal guidance law     trajectory tracking     predictive control     interception

NPC设计的主要难点为选择设计代价函数和相应的优化方法。这将直接影响轨迹跟踪精度以及跟踪的实时性。可将优化方法划分为“精确优化”和“近似优化”。“精确优化”的优势是精度高，不足是计算量大，耗时多。“近似优化”相对精度低，但实时性强。目前，“近似优化”主流采用NPC设计。

Joos等[23]将代价函数用雅可比矩阵线性化，然后采用NPC跟踪轨迹，但其计算量大，实时性不能保证。Chen和Allgöwer[24-25]采用泰勒展开式近似代价函数，该方法仅适用于单输入单输出的控制中。Slegers等[26]将Chen和Allgöwer[24-25]的控制方法扩展到多输入多输出系统的控制中。Hess和Jung[6, 27]取系统的5个动态平衡点，采用插值的方法得到系统的近似模型。以上非线性控制方法的共性是将系统模型近似为简单易求出优化近似解的近似模型，且为了保证时效性，需根据预定的轨迹及设定的环境模型，离线计算参数，但对于实时规划的轨迹误差较大。因此，本文提出了一种基于虚拟域预测控制的轨迹跟踪方法，该方法使用多项式近似系统模型，引入虚拟域及反动力方法，时效性强，能够在线计算模型参数，能有效跟踪实时规划的轨迹。

1 轨迹跟踪方法 1.1 轨迹跟踪控制器实现方案

 图 1 控制器实现流程图 Fig. 1 Flowchart of controller realization
1.2 被控模型近似

1.2.1 虚拟域内的多项式近似方法

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1.2.2 τ域内控制输入及控制输出计算

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1.3 滚动优化

 图 2 τ域内滚动优化策略 Fig. 2 Rolling optimization strategy in domain τ

τ域内，在每一个节点k，确定从该节点起的M个控制增量Δu(k)，Δu(k+1)，…，Δu(k+M－1)，使被控对象在其作用下未来P个节点的实际输出z尽可能接近给定的参考输出zr(k+l)，l=1, 2, …, PMP分别称为控制τ域与预测τ域。通常规定MP

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1.4 反馈校正

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1.5 轨迹跟踪控制结构

 图 3 轨迹跟踪控制结构 Fig. 3 Control architecture of trajectory tracking

2 仿真验证 2.1 仿真条件

 参数 数值 位置(x, y, z)/km (100, 100, 8.460) V/(m·s-1) 566 γ/(°) 0 ψ/(°) -133

 参数 数值 (x, y, z)/km (20.533, 0, 29.216) γ/(°) 0 ψ/(°) -107.647

2.2 初始条件下的小扰动仿真

 方案 (x, y, z)/km V/(m·s-1) γ/(°) ψ/(°) 脱靶量/m Eγ/(°) Eψ/(°) CPU计算时间/s 1 (105, 105, 8.883)(+5%) 566 0 -133 0.231 0.119 0.163 0.011 5 2 (100, 100, 8.460) 594.3(+5%) 0 -133 0.437 0.147 0.203 0.014 5 3 (100, 100, 8.460) 566 2 -133 0.195 0.298 0.097 0.013 7 4 (100, 100, 8.460) 566 0 -130.3(+2%) 0.275 0.147 0.359 0.013 9 5 (105, 105, 8.883)(+5%) 594.3(+5%) 2 -130.3(+2%) 0.493 0.510 0.529 0.015 2 FL[20] (105, 105, 8.883)(+5%) 594.3(+5%) 2 -130.3(+2%) 1.543 1.589 1.641 0.009 2

 图 4 拦截弹轨迹 Fig. 4 Trajectories of interceptor
 图 5 拦截弹路径倾角、路径偏角、速度、侧向过载及法向过载变化 Fig. 5 Trajectory angle, yaw angle, velocity, side load and normal load variation of interceptor

2.3 末端条件改变的轨迹重生仿真

 方案 时间/s 拦截弹位置/km 目标位置/km 脱靶量/m Eγ/(°) Eψ/(°) 1 3 (104.383, 104.337, 8.481) (20, 0, 29) 0.193 0.119 0.163 2 20.1 (76.923, 77.943, 11.649) (14.6, 0, 20) 0.165 0.127 0.161 3 30.8 (45.648, 50.222, 18.144) (11, 0, 15) 0.183 0.115 0.174

 图 6 遭遇点位置改变时的拦截弹轨迹 Fig. 6 Trajectories of interceptor when point of encounter is changed
 图 7 遭遇点位置改变时的拦截弹路径倾角、路径偏角、速度、侧向过载及法向过载变化 Fig. 7 Trajectory angle, yaw angle, velocity, side load and normal load variation of interceptor when point of encounter position is changed

3 结论

1) 在虚拟域内，使用多项式近似系统模型，以虚拟路径的节点为滚动优化的基准，将有限时域内的控制输入作用于系统，利用实时信息进行反馈校正，以准确跟踪预定轨迹。

2) 考虑了参考轨迹与实际轨迹偏差过大，预测优化失败或者目标状态发生改变3种情况下的轨迹跟踪问题，将轨迹控制结构分为内外环路，完成了轨迹跟踪的控制结构的设计。

3) 在初始状态量添加小扰动及末端条件改变的条件下，以拦截弹道导弹为背景，进行仿真验证。结果表明：与非线性反馈跟踪方法相比，曲线平滑，在遭遇点脱靶量、末端路径倾角及偏角误差较小，实时性也可满足控制需求。

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#### 文章信息

FAN Yu, YAN Liang, ZHU Wuxuan, BAI Guangzhou

Trajectory tracking method based on predictive control in virtual domain

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(9): 1813-1823
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0751