在非线性系统中，受限初始条件下用多轨迹数据学习线性化模型。
Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data
- 基于多条轨迹的确定性数据采集与正则化最小二乘法
- 给出了有限样本下的线性化动态误差界，可一致收敛
- 适合需在非线性系统中建模的控制场景，尤其关注初始化约束
从数据中识别线性系统模型在控制理论中有广泛应用。现有工作通常假设系统真实为线性，并使用单条长轨迹、独立同分布随机输入的数据，提供有限样本保证。本文考虑在真实动力学为非线性的情况下，当实验初始化区域受约束时，如何识别线性化模型。我们提出一种基于多条轨迹的确定性数据采集算法，结合正则化最小二乘法，并给出所学线性化动态的有限样本误差界。该误差界表明：可在噪声与非线性误差之间实现一致学习，且二者存在权衡。数值实验验证了结果，同时展示了在存在非线性时，仅使用单条轨迹和i.i.d.输入进行线性识别可能不足。
原文摘要 · Abstract (English)
The identification of a linear system model from data has wide applications in control theory. The existing work that provides finite sample guarantees for linear system identification typically uses data from a single long system trajectory under i.i.d. random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear, given that there is a certain constraint on the region where one can initialize the experiments. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound shows that one can consistently learn the linearized dynamics, and demonstrates a trade-off between the error due to nonlinearity and the error due to noise. We validate our results through numerical experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d. random inputs, when nonlinearity does exist.
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