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Identification of ARX and ARARX Models in the Presence of Input and Output Noises

ARX (AutoRegressive models with eXogenous variables) are the simplest models within the equation error family but are endowed with many practical advantages concerning both their estimation and their predictive use since their optimal predictors are always stable. Similar considerations can be repeated for ARARX models where the equation error is described by an AR process instead of a white noise. The ARX and ARARX schemes can be enhanced by introducing the assumption of the presence of additive white noise on the input and output observations. These schemes, that will be denoted as “ARX + noise” and “ARARX + noise”, can be seen as errors-in-variables models where both measurement errors and process disturbances are taken into account. This paper analyzes the problem of identifying ARX + noise and ARARX + noise models. The proposed identification algorithms are derived on the basis of the procedures developed for the solution of the dynamic Frisch scheme. The paper reports also Monte Carlo simulations that confirm the effectiveness of the proposed procedures.