  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.
