A counterexample to the uniqueness of the asymptotic estimate in ARMAX model identification via the correlation approach

This paper deals with the identifiability of an ARMAX system when the correlation approach is adopted. In general, identifiability depends on both the parametrization of the model class and on the informativeness of the data. Here, we focus on the latter aspect and, therefore, a full-order model class is considered. The main goal is to provide a counterexample to the uniqueness of the asymptotic estimate when a persistently exciting input is adopted. This shows the somehow counterintuitive fact that the identifiability of ARMAX systems within the correlation approach is related to the “color” of the input.