In this paper, we consider the task of designing a Kalman Filter (KF) for an unknown and partially observed autonomous linear time invariant system driven by process and sensor noise. To do so, we propose studying the following two step process: first, using system identification tools rooted in subspace methods, we obtain coarse finite-data estimates of the state-space parameters and Kalman gain describing the autonomous system; and second, we use these approximate parameters to design a filter which produces estimates of the system state. We show that when the system identification step produces sufficiently accurate estimates, or when the underlying true KF is sufficiently robust, that a Certainty Equivalent (CE) KF, i.e., one designed using the estimated parameters directly, enjoys provable sub-optimality guarantees. We further show that when these conditions fail, and in particular, when the CE KF is marginally stable (i.e., has eigenvalues very close to the unit circle), that imposing additional robustness constraints on the filter leads to similar sub-optimality guarantees. We further show that with high probability, both the CE and robust filters have mean prediction error bounded by $\tilde O(1/\sqrt{N})$, where $N$ is the number of data points collected in the system identification step. To the best of our knowledge, these are the first end-to-end sample complexity bounds for the Kalman Filtering of an unknown system.

本文研究了利用系统辨识方法设计 Kalman 滤波器的问题，给出了一种两步法：第一步是获取有限数据的状态空间参数和 Kalman 增益的粗略估计，第二步是利用这些估计参数设计产生系统状态估计的滤波器，研究发现当获取的参数精确度较高，或核心的 Kalman 滤波器具有足够的鲁棒性时，计算得到的等价 Kalman 滤波器具有可证明的次优保证，也证明对于较脆弱的滤波器，采取附加的鲁棒约束可提高次优保证性能，同时提出用样本复杂度度量此问题的最小观测数据数。

未知系统的卡尔曼滤波样本复杂度