  This paper considers the problem of system identification (ID) of linear and
nonlinear non-autonomous systems from noisy and sparse data. We propose and
analyze an objective function derived from a Bayesian formulation for learning
a hidden Markov model with stochastic dynamics. We then analyze this objective
function in the context of several state-of-the-art approaches for both linear
and nonlinear system ID. In the former, we analyze least squares approaches for
Markov parameter estimation, and in the latter, we analyze the multiple
shooting approach. We demonstrate the limitations of the optimization problems
posed by these existing methods by showing that they can be seen as special
cases of the proposed optimization objective under certain simplifying
assumptions: conditional independence of data and zero model error.
Furthermore, we observe that our proposed approach has improved smoothness and
inherent regularization that make it well-suited for system ID and provide
mathematical explanations for these characteristics' origins. Finally,
numerical simulations demonstrate a mean squared error over 8.7 times lower
compared to multiple shooting when data are noisy and/or sparse. Moreover, the
proposed approach can identify accurate and generalizable models even when
there are more parameters than data or when the underlying system exhibits
chaotic behavior.
