论文标题
Willems对国家空间系统的基本引理及其扩展到多个数据集
Willems' Fundamental Lemma for State-space Systems and its Extension to Multiple Datasets
论文作者
论文摘要
Willems等人的基本引理断言，假设激发条件的持续性存在，则可以从单个给定的系统中获得所有线性系统的轨迹。该结果对系统识别和数据驱动的控制具有深远的影响，并且在过去几年中已经复兴了。本文的目的是将Willems的引理扩展到给出多个（可能是短）系统轨迹而不是单个长长的情况的情况。为此，我们介绍了集体持续的激发概念。然后，我们将证明线性系统的所有轨迹都可以从给定的有限数量的轨迹中获得，只要这些轨迹集体令人兴奋。我们将证明该结果可以从数据集中识别有丢失数据示例的数据集的线性系统。此外，我们表明结果在数据驱动的不稳定系统的控制中具有实际意义。
Willems et al.'s fundamental lemma asserts that all trajectories of a linear system can be obtained from a single given one, assuming that a persistency of excitation condition holds. This result has profound implications for system identification and data-driven control, and has seen a revival over the last few years. The purpose of this paper is to extend Willems' lemma to the situation where multiple (possibly short) system trajectories are given instead of a single long one. To this end, we introduce a notion of collective persistency of excitation. We will then show that all trajectories of a linear system can be obtained from a given finite number of trajectories, as long as these are collectively persistently exciting. We will demonstrate that this result enables the identification of linear systems from data sets with missing data samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.
