通过特征分解参数化惩罚矩阵，提升航天控制性能。
Eigendecomposition Parameterization of Penalty Matrices for Enhanced Control Design: Aerospace Applications
- 用特征分解隐式保证惩罚矩阵正定性，支持非对角线项。
- 在三个航天控制问题中实现最高65%的性能提升。
- 适合需要精细控制设计的航空航天领域研究者。
现代控制算法需调整出现在二次型/代价函数中的方阵权重/惩罚矩阵，以提升性能或稳定性。由于调参简便且能保证正定性，对角惩罚矩阵广泛用于线性二次调节器（LQR）、模型预测控制及基于李雅普诺夫的控制方法。本文提出一种特征分解参数化方法，使惩罚矩阵在保持正定性的同时允许非零非对角元素，不仅带来显著的计算与实现优势，还拓展了可实现控制的范围。我们求解了三个控制问题：1）泽梅洛导航问题变体；2）基于LQR与李雅普诺夫方法的最小能量航天器姿态控制；3）基于李雅普诺夫方法的最小燃料与最短时间低推力轨迹设计。采用粒子群优化来优化决策变量，进而参数化惩罚矩阵。结果表明，在示例问题中，该方法可实现高达65%的性能目标改善。
原文摘要 · Abstract (English)
Modern control algorithms require tuning of square weight/penalty matrices appearing in quadratic functions/costs to improve performance and/or stability output. Due to simplicity in gain-tuning and enforcing positive-definiteness, diagonal penalty matrices are used extensively in control methods such as linear quadratic regulator (LQR), model predictive control, and Lyapunov-based control. In this paper, we propose an eigendecomposition approach to parameterize penalty matrices, allowing positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied, which not only offers notable computational and implementation advantages, but broadens the class of achievable controls. We solve three control problems: 1) a variation of Zermelo's navigation problem, 2) minimum-energy spacecraft attitude control using both LQR and Lyapunov-based methods, and 3) minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design. Particle swarm optimization is used to optimize the decision variables, which will parameterize the penalty matrices. The results demonstrate improvements of up to 65% in the performance objective in the example problems utilizing the proposed method.
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