In this work, we consider the problem of identifying an unknown linear dynamical system given a finite hypothesis class. In particular, we analyze the effect of the excitation input on the sample complexity of identifying the true system with high probability. To this end, we present sample complexity lower bounds that capture the choice of the selected excitation input. The sample complexity lower bound gives rise to a system theoretic condition to determine the potential benefit of experiment design. Informed by the analysis of the sample complexity lower bound, we propose a persistent excitation (PE) condition tailored to the considered setting, which we then use to establish sample complexity upper bounds. Notably, the PE condition is weaker than in the case of an infinite hypothesis class and allows analyzing different excitation inputs modularly. Crucially, the lower and upper bounds share the same dependency on key problem parameters. Finally, we leverage these insights to propose an active learning algorithm that sequentially excites the system optimally with respect to the current estimate, and provide sample complexity guarantees for the presented algorithm. Concluding simulations showcase the effectiveness of the proposed algorithm.

翻译：本文研究在给定有限假设类条件下识别未知线性动态系统的问题。具体而言，我们分析了激励输入对以高概率识别真实系统所需样本复杂度的影响。为此，我们提出了能够反映所选激励输入特性的样本复杂度下界。该下界引出了一个系统理论条件，用于评估实验设计的潜在效益。基于对样本复杂度下界的分析，我们提出了适用于当前场景的持续激励条件，并利用该条件建立了样本复杂度上界。值得注意的是，该PE条件弱于无限假设类情形，且能模块化地分析不同激励输入。关键的是，上下界在核心问题参数上具有相同的依赖关系。最后，我们基于这些洞见提出了一种主动学习算法，该算法能依据当前估计对系统进行序列最优激励，并给出了算法的样本复杂度保证。结论部分的仿真实验验证了所提算法的有效性。