We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. It is well known in the literature that the learn-to-stabilize problem suffers from exponential blow-up in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the state space dimension. This blow-up is due to the open-loop instability when exploring the $n$-dimensional state space. To address this issue, we develop a novel algorithm that decouples the unstable subspace of the LTI system from the stable subspace, based on which the algorithm only explores and stabilizes the unstable subspace, the dimension of which can be much smaller than $n$. With a new singular-value-decomposition(SVD)-based analytical framework, we prove that the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where $k$ is the dimension of the unstable subspace. Critically, this bound avoids exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the previous works, and to the best of our knowledge, this is the first paper to avoid exponential blow-up in dimension for stabilizing LTI systems with noise.

我们研究了在单一轨迹上学习稳定未知噪声线性时不变系统的问题，并提出了一种新算法，通过解耦系统的不稳定子空间和稳定子空间来避免状态空间维度的指数级增长。使用基于奇异值分解的分析框架，我们证明了在状态范数达到2^(O(k log n))之前，系统能够被稳定，避免了传统方法中状态维度指数级增长的问题。本文是第一篇避免状态空间维度指数级增长的稳定带噪声线性时不变系统的文章。

学习在单轨迹下稳定未知线性时不变系统的方法在随机噪声下