Willems’ Fundamental Lemma for State-Space Systems and Its Extension to Multiple Datasets
Willems’ Fundamental Lemma for State-Space Systems and Its Extension to Multiple Datasets
复制标题
威廉斯状态空间系统基本引理及其对多个数据集的扩展
作者：
H. V. Waarde;Kanat M. Camlibel;P. Tesi
Willems等人的S基本引理断言，假设激励的持久性和可控性条件成立，线性系统的所有轨迹都可以从一个给定的轨迹得到。这一结果对系统识别和数据驱动控制具有深远的影响，并在过去几年中出现了复苏。这封信的目的是将Willems引理推广到给出多个(可能是短的)系统轨迹而不是一个长的系统轨迹的情况。为此，我们引入了集体持续激发的概念。我们将证明，一个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，只要这些轨迹是共同的持久令人兴奋的。我们将证明这一结果能够从缺失样本的数据集中识别线性系统。此外，研究结果对不稳定系统的数据驱动控制具有实际意义。
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 and a controllability condition hold. 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 letter 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 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 samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.