Initial Conditions in Time- and Frequency-Domain System Identification: Implications of the Shift Operator Versus the Z and Discrete Fourier Transforms

This article has two main objectives. The first objective is to demonstrate that, unlike frequency-domain models, time-domain models exactly account for the effect of initial conditions without an explicit expression for the free response. Although this property is obvious in state-space models, it has been the subject of confusion and misconceptions within the context of input-output models. In contrast to time-domain models, frequency-domain models (not to be confused with Laplace-domain and Z-transfer-domain models, which separately include the free response) do not account for initial conditions.