﻿ 线性伪谱模型预测能量最优姿态机动控制方法<sup>*</sup>
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Fuel-optimal attitude maneuver using linear pseudo-spectral model predictive control method
FENG Yijun, CHEN Wanchun, YANG Liang
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2017-12-12; Accepted: 2018-03-16; Published online: 2018-04-18 11:55
Corresponding author. CHEN Wanchun, E-mail:wanchun_chen@buaa.edu.cn
Abstract: In order to perform large angle attitude maneuvers of spacecraft outside the atmosphere, we propose a fuel-optimal large angle attitude maneuver strategy using modified linear pseudo-spectral model predictive control method. First, a fuel-optimal attitude maneuver trajectory satisfying initial and terminal constrains is planned offline. Then, the nonlinear equation of motion is linearized under the condition of little perturbation based on the planned trajectory, and thus the linear perturbation propagation equations are obtained. Finally, the analytical solution of fuel-optimal control correction to the planned trajectory has been derivated through the discretization of state variables and control variables using Gauss pseudo-spectral method. Numerical calculation and Monte Carlo simulations were performed to validate the feasibility and effectiveness of the proposed strategy, which can provide real-time control with terminal state satisfied in high accuracy and save almost 10% fuel cost under the same control precision compared with traditional linear quadratic regulator (LQR) tracking method.
Keywords: attitude control     linear pseudo-spectrum     model predictive control     fuel-optimal     trajectory tracking

1 飞行器模型及姿态机动问题

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G(σ)∈R3×3为飞行器姿态运动学矩阵，定义为

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2 线性伪谱能量最优姿态机动控制

2.1 线性伪谱姿态机动终端状态解析预测

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 图 1 高斯正交节点示意图 Fig. 1 Schematic diagram of Gauss quadrature points

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，则微分动力学约束不仅能转化为一组代数约束，并且能够表示为LGR节点上各个预测状态偏差和控制修正的函数：

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2.2 线性伪谱能量最优姿态机动控制指令修正

1) 求当前状态量与标称状态量的初始偏差δx0=(t0)-x(t0)，根据式(42)来预测终端状态偏差

2) 以当前状态作为初始状态，保持控制为标称控制，进行数值积分，从而预测终端状态偏差

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Φ展开，有

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tk(k=1, 2, …, N)为[t0, tf]上LGR节点所在的时间点，ũik为第i通道在tk时刻的标称控制，δûik为第i通道在tk时刻的控制修正，则可用LGR节点上的离散值δûik对全时域积分Φ*进行拟合：

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PR6×6，设其第ij列的元素为pij，则有

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2.3 能量最优姿态机动控制方法实施步骤

3 能量最优姿态机动控制仿真 3.1 飞行器模型参数与控制参数设定

 参数 J/(kg·m2) Umax/(N·m) 数值 diag(5, 70, 70) [5  10  10]

1) Q=100×I6×6, R=0.01×I3×3，采用LQR方法跟踪标称轨迹。

2) Q=100×I6×6, R=0.01×I3×3，采用LQR方法直接进行机动。

3) Q=100×I6×6, R=0.1×I3×3，采用LQR方法跟踪标称轨迹。

4) Q=100×I6×6, R=0.1×I3×3，采用LQR方法直接进行机动。

3.2 能量最优姿态机动单次仿真

 控制方法 [γf  θf  ψf]/(°) [ω1f  ω2f  ω3f]/((°)·s-1) Φ/(N2·m2·s) 线性伪谱模型预测控制 [0.15  -0.19  0.07] [0.008 1  0.94  -0.48] 247.98 LQR跟踪标称轨迹-参数1 [0.26  -0.16  0.35] [-0.097  1.04  -0.56] 260.54 LQR直接控制-参数1 [7.76  17.7  -9.78] [2.21  1.43  -3.28] 489.22 LQR跟踪标称轨迹-参数2 [1.14  0.7  0.62] [-0.13  0.75  -0.75] 240.58 LQR直接控制-参数2 [8.7  23.4  -12.1] [1.93  4.09  -2.76] 249.75

 图 2 能量最优姿态机动姿态角仿真曲线 Fig. 2 Simulation curves of attitude angles of fuel-optimal attitude maneuvers
 图 3 能量最优姿态机动角速度仿真曲线 Fig. 3 Simulation curves of angular velocities of fuel-optimal attitude maneuvers
 图 4 能量最优姿态机动控制力矩仿真曲线 Fig. 4 Simulation curves of control moment of fuel-optimal attitude maneuvers

3.3 能量最优姿态机动蒙特卡罗仿真

 终端项 线性伪谱模型预测控制 LQR跟踪标称轨迹-参数1 均值 标准差 均值 标准差 γf/(°) 0.004 7 0.004 7 -0.046 0.222 3 θf/(°) -0.056 5 0.056 5 -0.050 0.118 3 ψf/(°) 0.028 4 0.028 4 0.020 0.198 0 ω1f/((°)·s-1) 0.004 6 0.004 6 -0.008 3 0.076 8 ω2f/((°)·s-1) 0.994 6 0.994 6 1.014 8 0.031 6 ω3f/((°)·s-1) -0.496 2 0.496 2 -0.489 7 0.055 7

 仿真时间 LQR跟踪标称轨迹-参数1 线性伪谱模型预测控制 8个节点 10个节点 12个节点 指令生成时间/ms 14.6 59 70 78 全过程仿真时间/s 4.286 4.143 4.352 4.704

 图 5 姿态机动蒙特卡罗仿真姿态角曲线 Fig. 5 Curves of attitude angle maneuvers using Monte Carlo simulation
 图 6 姿态机动蒙特卡罗仿真角速度曲线 Fig. 6 Curves of angular velocities of attitude maneuvers using Monte Carlo simulation
 图 7 姿态机动蒙特卡罗仿真终端精度散布图 Fig. 7 Scatter diagram of terminal accuracy of attitude maneuvers using Monte Carlo simulation
 图 8 姿态机动蒙特卡罗仿真能量消耗对比图 Fig. 8 Comparison of energy consumption of attitude maneuvers using Monte Carlo simulation

4 结论

1) 能够实现限定时间内飞行器能量最优大角度姿态机动，相比于LQR跟踪规划轨迹的方法能够节省约10%的能量消耗。

2) 该方法能够得到的平滑且连续的控制量。

3) 该方法能够快速地进行控制修正，满足在线使用的要求。

4) 该方法的设计思路能推广到更为一般的具有终端约束的微分动力学系统跟踪问题上。

1) 实际使用中，需根据飞行器的计算能力进行节点数、积分步长和控制精度之间的权衡。

2) 修正得到的控制量无法保证一定满足过程约束，若直接在得到的修正控制量上增加限幅模块，则得到的修正控制无法保证能量最优。同时可能导致额外的控制更新。

3) 本文提出的方法只适用于连续控制，不适用于Bang-Bang控制(开关式控制)。

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#### 文章信息

FENG Yijun, CHEN Wanchun, YANG Liang

Fuel-optimal attitude maneuver using linear pseudo-spectral model predictive control method

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(10): 2165-2175
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0770