论文标题
从多个轨迹中学习自主线性系统的动力学
Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories
论文作者
论文摘要
我们考虑了从对这些系统的多个轨迹的观察以及具有有限样本保证的多个轨迹的观察结果来学习自主线性系统（即不受外部控制输入影响的系统）动态的问题。自主线性系统识别的学习率和一致性的现有结果取决于单个长轨迹的稳态行为的观察，并且不适用于不稳定的系统。相比之下，我们考虑了基于多个短轨迹的学习系统动力学的方案，那里没有容易观察到的稳态行为。我们提供了有限的样本分析，该分析表明，对于稳定和不稳定的系统，可以以$ \ Mathcal {o}（\ frac {1} {\ frac {1} {\ sqrt {n}} $ $的稳定系统，$ n $是轨迹的数量，当系统的初始状态具有零平均值（这是常见的文献中），$ n $是轨迹的数量。我们将结果进一步概括为初始状态具有非零均值的情况。我们表明，人们可以调整轨迹的长度，以达到$ \ Mathcal {o}的学习率（\ sqrt {\ frac {\ frac {\ log {n}} {n} {n}）} $ $ \ mathcal {o}（\ frac {（\ log {n}）^d} {\ sqrt {n}}）$用于边缘稳定的系统，其中$ d $有些常数。
We consider the problem of learning the dynamics of autonomous linear systems (i.e., systems that are not affected by external control inputs) from observations of multiple trajectories of those systems, with finite sample guarantees. Existing results on learning rate and consistency of autonomous linear system identification rely on observations of steady state behaviors from a single long trajectory, and are not applicable to unstable systems. In contrast, we consider the scenario of learning system dynamics based on multiple short trajectories, where there are no easily observed steady state behaviors. We provide a finite sample analysis, which shows that the dynamics can be learned at a rate $\mathcal{O}(\frac{1}{\sqrt{N}})$ for both stable and unstable systems, where $N$ is the number of trajectories, when the initial state of the system has zero mean (which is a common assumption in the existing literature). We further generalize our result to the case where the initial state has non-zero mean. We show that one can adjust the length of the trajectories to achieve a learning rate of $\mathcal{O}(\sqrt{\frac{\log{N}}{N})}$ for strictly stable systems and a learning rate of $\mathcal{O}(\frac{(\log{N})^d}{\sqrt{N}})$ for marginally stable systems, where $d$ is some constant.
