飞机多约束轨迹优化奇异性分析与自动解算方法

doi:10.7527/S1000-6893.2025.32147

Abstract

Abstract:

The aircraft trajectory optimization problem usually involves complex situations such as singular arc type control， bang-bang control， and active constraint control， where the control variable is switched. The existing solution methods are difficult to achieve a balance between accuracy and numerical stability when dealing with the problem containing switch control. In this paper， the control singularity and the influence of active constraints on the switch of control law in the aircraft trajectory optimization problem are analyzed， and an automatic solution method is constructed. The proposed method requires no prior knowledge of the control structure； instead， based on the solution by the Radau pseudospectral method， an automatic estimation method for the control switch moment is added. According to these estimations， the entire control process is divided into multiple stages. Then， the control types of each stage are identified and the solution strategies are constructed in a targeted manner to gradually approximate the optimal solution of the problem in the optimization iteration. Taking Airbus A320 aircraft as an example， both the proposed method and the optimal control software GPOPS-Ⅱ are employed to solve trajectory optimization problem with singularity and active constraints. Numerical experiments show that the proposed method has better advantages in numerical stability and convergence accuracy for the aircraft trajectory optimization problem with control switching.

Key words: trajectory optimization, singular arc, Bang-bang control, active constraint, Radau pseudospectral method

CLC Number:

V271

Yonghui WU, Xiang LI, Hao HUANG, Xu LIU. Singularity analysis and automatic solution method for aircraft multi-constrained trajectory optimization[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(2): 332147.

Figures/Tables 17

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对比项目	本文方法	GPOPS-Ⅱ
燃油消耗量/kg	4 459.03	4 451.81
总飞行时间/s	4 204.19	4 214.73
优化迭代次数	4	15
最终LGR配点数	64	1 203
CPU平均运行时间/s	1.804 2	10.594 8

约束项	本文方法	GPOPS-Ⅱ
动力学约束残差最大值/10^-11	3.91	995
最大马赫数^*	0.82	0.82
最大飞行高度/m	7 483.87	7 778.27
最小飞行高度/m	0.00	0.00

CPU运行时间/s	初始LGR配点数目					占总CPU运行时间的比值/%
CPU运行时间/s	8	16	32	48	64	占总CPU运行时间的比值/%
总时间	8.103 5	5.945 4	2.638 0	1.804 2	1.935 9
NLP优化求解	6.095 8	4.449 7	1.931 0	1.291 2	1.330 3	68.03~76.37
正则化迭代参数更新	0.674 3	0.493 7	0.156 8	0.094 3	0.123 4	5.23~8.32
残差计算	0.287 6	0.215 4	0.102 3	0.087 6	0.098 7	3.55~5.10
网格调整与伪谱矩阵计算	0.221 5	0.112 8	0.083 4	0.058 7	0.062 3	1.90~3.25
控制突变时刻的估计	0.356 7	0.287 6	0.154 3	0.093 4	0.105 6	4.40~5.85
控制类型的识别	0.278 4	0.215 4	0.102 3	0.087 6	0.098 7	3.44~5.10
协态变量的估计	0.115 4	0.087 6	0.016 5	0.007 6	0.008 3	0.42~1.47
其他	0.073 8	0.083 2	0.091 4	0.083 8	0.108 6	0.91~5.61

对比项目	案例2求解结果		案例3求解结果		案例4求解结果
对比项目	本文方法	GPOPS-Ⅱ	本文方法	GPOPS-Ⅱ	本文方法	GPOPS-Ⅱ
燃油消耗量/kg	4 088.96	4 085.24	2 763.68	2 762.57	4 310.55	4 306.35
总飞行时间/s	4 283.04	4 983.11	2 632.22	2 632.60	4 185.29	4 187.51
优化迭代次数	4	21	3	16	4	18
最终LGR配点数	93	1 786	59	1076	76	1 315
CPU平均运行时间/s	2.09	43.45	1.67	11.75	1.81	16.86
最大动力学约束残差/10^-10	4.56	98.4	41.1	31.4	0.35	90.5
最大马赫数	0.82	0.82	0.82	0.82	0.82	0.82
最大飞行高度/m	10 648.44	11 000	7 483.87	7 511.78	7 483.87	7 563.98
最小飞行高度/m	0	0	0	0	0	0

控制突变时刻	迭代历程
控制突变时刻	E=1	E=2	E=3	E=4
t₁/s	22.42	29.64	29.64	29.64
t₂/s	1 080.93	1 039.85	1 039.85	1 039.85
t₃/s	3 409.15	3 441.01	3 441.22	3 441.22
t₄/s	4 103.81	4 087.63	4 080.94	4 080.94
t₅/s	4 170.64	4 149.61	4 149.98	4 149.98

Singularity analysis and automatic solution method for aircraft multi-constrained trajectory optimization

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阶段	时间区间/s	控制类型
阶段	时间区间/s	推力比	爬升角
1	［0， 29.64］	Bang-bang型	主动约束型
2	［29.64，1 039.85］	Bang-bang型	奇异弧型
3	［1 039.85，3 441.22］	主动约束型	主动约束型
4	［3 441.22，4 080.94］	奇异弧型	Bang-bang型
5	［4 080.94，4 149.98］	Bang-bang型	Bang-bang型
6	［4 149.98，4 204.19］	Bang-bang型	主动约束型

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