Data-Dependent analysis of model validation errors for linear system identification

This study addresses the fundamental question of how accurately dynamical systems can be learned using a limited amount of data. First, we show that a state-space model constructed via a prediction error method perfectly fits a dataset, with no assumptions regarding the data themselves. Motivated by this result, we then analyze how accurately the identified model can mimic the true system for any arbitrary input. We derive necessary and sufficient conditions for the validation error to be exactly zero. Moreover, we introduce a notion of approximation to this characterization from the perspectives of both reachability and observability. Subsequently, we show two upper bounds of the l(2)-norm of the validation error, for any unit impulse input and any input signals with an l(2)-norm less than one. The legitimacy of our analyses is investigated through numerical simulations. (C) 2022 The Author(s). Published by Elsevier Ltd on behalf of European Control Association.