基于李群几何的神经网络控制器，实现未知系统在故障下的稳定跟踪控制。
Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups
- 利用李群的左不变性设计神经网络学习规则，避免参数化奇点。
- 保证所有误差信号（含权重、配置误差）最终有界，理论严格。
- 适用于多智能体编队控制等需全局一致性的复杂系统场景。
针对在矩阵李群上演化、具有未知动态、执行器故障和有界扰动的系统，提出一种基于几何神经网络的追踪控制器。通过将李群切丛视为向量空间 ℝ^{N×N} 的嵌入子流形，并利用其左不变性，设计了一组与李群结构内在兼容的神经网络权重学习规则，无需显式参数化。该方法规避了参数化奇点，支持对最优权重的全局搜索。采用李雅普诺夫直接法，证明了所有误差信号——包括神经网络权重、无坐标配置误差函数和追踪速度误差——的最终有界性。通过在特殊欧氏群上的多智能体分布式编队控制仿真实例，验证了所提方法的有效性。
原文摘要 · Abstract (English)
We present a geometric neural network-based tracking controller for systems evolving on matrix Lie groups under unknown dynamics, actuator faults, and bounded disturbances. Leveraging the left-invariance of the tangent bundle of matrix Lie groups, viewed as an embedded submanifold of the vector space $\R^{N\times N}$, we propose a set of learning rules for neural network weights that are intrinsically compatible with the Lie group structure and do not require explicit parameterization. Exploiting the geometric properties of Lie groups, this approach circumvents parameterization singularities and enables a global search for optimal weights. The ultimate boundedness of all error signals -- including the neural network weights, the coordinate-free configuration error function, and the tracking velocity error -- is established using Lyapunov's direct method. To validate the effectiveness of the proposed method, we provide illustrative simulation results for decentralized formation control of multi-agent systems on the Special Euclidean group.
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