论文标题
在单个轨迹上稳定LTI系统的样品复杂性
On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory
论文作者
论文摘要
稳定未知的动力系统是控制理论中的核心问题之一。在本文中，我们研究了单个轨迹上线性时间流动（LTI）系统中学习到稳定问题的样本复杂性。当前的最新方法需要以$ n $（状态维度为单位的样本复杂性线性，这会导致状态规范在$ n $中呈指数式增长。我们提出了一种基于光谱分解的新型算法，该算法只需要学习作用于其不稳定子空间的动力矩阵的“一小部分”。我们表明，在适当的假设下，我们的算法可以使用$ \ tilde {o}（k）$样本在单个轨迹上稳定LTI系统，其中$ k $是系统的不稳定性索引。这代表了当$ k = o（n）$时，LTI系统稳定的第一个子线性样品复杂性结果。
Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (LTI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in $n$, the state dimension, which incurs a state norm that blows up exponentially in $n$. We propose a novel algorithm based on spectral decomposition that only needs to learn "a small part" of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our algorithm stabilizes an LTI system on a single trajectory with $\tilde{O}(k)$ samples, where $k$ is the instability index of the system. This represents the first sub-linear sample complexity result for the stabilization of LTI systems under the regime when $k = o(n)$.
