论文标题
Willems的基本引理及其含义的定量和建设性证明
A quantitative and constructive proof of Willems' Fundamental Lemma and its implications
论文作者
论文摘要
Willems的基本引理提供了基于一个持续令人兴奋（PE）输入的一个轨迹的可控线性时间变变系统的所有轨迹的强大数据驱动参数化。在本文中，我们提供了一个新的证明，该证明是受经典自适应控制文献的启发，与多个方面的现有证明不同。证明涉及定量和定向的PE概念，可以通过奇异值界限表征强大的PE性能，而不是基于二进制等级的PE条件。此外，证明是建设性的，即，我们为生成的数据得出了一个明确的PE下限。作为独立兴趣的贡献，我们从自适应控制文献中概括了现有的PE结果，并揭示了系统零的关键作用。
Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel proof of this result which is inspired by the classical adaptive control literature and differs from existing proofs in multiple aspects. The proof involves a quantitative and directional PE notion, allowing to characterize robust PE properties via singular value bounds, as opposed to binary rank-based PE conditions. Further, the proof is constructive, i.e., we derive an explicit PE lower bound for the generated data. As a contribution of independent interest, we generalize existing PE results from the adaptive control literature and reveal a crucial role of the system's zeros.
