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.

本研究解决了在未知动态、执行器故障和有界干扰下的矩阵李群上系统的跟踪控制问题。我们提出了一种基于几何神经网络的控制器，通过利用李群的几何特性，开发了一套与李群结构兼容的学习规则，从而规避了参数化奇异性，并能够进行全局权重优化。研究表明，该方法能保证所有误差信号的最终有界性，对多智能体系统的去中心化形成控制进行了验证。

未知系统在矩阵李群上的几何容错神经网络跟踪控制