AI 中文总结
针对非线性系统观测器校正项设计难题，本文提出基于学习的方法，将收缩条件嵌入训练损失函数确定校正项和收缩度量，扩展到非自治系统，建立学习误差边界与鲁棒性，还提出鲁棒观测器并经数值模拟评估。
AI 中文摘要
设计观测器的常用方法是给系统副本添加校正项，但为非线性系统设计校正项仍是长期重大挑战。收缩理论通过求解矩阵偏微分不等式（MPDI）并确定收缩度量提供统一方法，但求解MPDI极具挑战性。本文旨在提出基于学习的方法，将收缩要求纳入学习过程来确定观测器校正项和收缩度量。该方法依赖科学机器学习公式，将收缩条件嵌入训练损失函数，扩展到非自治系统时保持校正项和收缩度量静态。建立了观测器学习误差的可计算边界，在指数输入到状态稳定意义上确定了对测量噪声和学习误差的鲁棒性，并进一步提出鲁棒基于学习的收缩非线性观测器，通过数值模拟评估了不同收缩率和测量噪声水平下的观测器。
英文摘要
A common way to design observers is to add a correction term to a copy of the system; however, designing the correction term for nonlinear systems remains a significant long-standing challenge. Contraction theory offers a unified approach to designing this correction term by solving a matrix partial differential inequality (MPDI) and identifying a contraction metric. However, solving the MPDI for both the correction term and the contraction metric is highly challenging, both analytically and numerically. Therefore, the aim of this paper is to propose a learning-based approach to determine both the observer's correction term and the contraction metric by incorporating the contraction requirements into the learning process. The proposed approach relies on a scientific machine learning formulation that embeds the contraction conditions into the training loss function. The proposed approach is then extended to non-autonomous systems while keeping the correction term and contraction metric static to avoid generalization issues arising from time-dependent training. Computable bounds on the learning errors of the proposed observer are established as a function of the training residual and the sampling resolution. Furthermore, the robustness of the proposed observer to measurement noise and learning errors are established in an exponential input-to-state stability sense. Based on the robustness analysis, the present paper takes a further step by proposing a robust learning-based contraction nonlinear observer. The proposed observers are evaluated in numerical simulations for different contraction rates and measurement noise levels.
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