  This work investigates the problem of estimating the weight matrices of a
stable time-invariant linear dynamical system from a single sequence of noisy
measurements. We show that if the unknown weight matrices describing the system
are in Brunovsky canonical form, we can efficiently estimate the ground truth
unknown matrices of the system from a linear system of equations formulated
based on the transfer function of the system, using both online and offline
stochastic gradient descent (SGD) methods. Specifically, by deriving concrete
complexity bounds, we show that SGD converges linearly in expectation to any
arbitrary small Frobenius norm distance from the ground truth weights. To the
best of our knowledge, ours is the first work to establish linear convergence
characteristics for online and offline gradient-based iterative methods for
weight matrix estimation in linear dynamical systems from a single trajectory.
Extensive numerical tests verify that the performance of the proposed methods
is consistent with our theory, and show their superior performance relative to
existing state of the art methods.
