In industrial process control engineering, using routine operating data obtained from closed-loop operation of a process for model identification would be extremely valuable in applications such as control performance monitoring, root cause diagnosis, and controller retuning. However, the conditions for closed-loop identifiability using routine operating data are still largely unknown or untried. In this paper, criteria for closed-loop identification of an autoregressive, moving average process with exogenous input (ARMAX) regulated with an arbitrary, rational, polynomial controller are derived. The theoretical criteria that are developed for the closed-loop identification of an ARMAX process are compared with Monte Carlo simulations and previous theoretical results. It is shown that the newly-proposed theoretical results are in agreement with the simulation results. (C) 2011 Elsevier Ltd. All rights reserved.