A system identification problem motivated by robust control

Linear, dynamic model set estimation based on noisy, input-output data is addressed here from a confidence set standpoint. Following the usual robust control perspective a model set estimate for the ‘true’, but unknown, impulse response (truncated at the data length due to causality) is sought via a nominal model belonging to a pre-specified parametric class of approximating models (of ‘low order’) plus some quantitative information on the mismatch between the approximating model and the underlying (possibly, ‘high-order’) one. The solution proposed is based on the asymptotic, parameter estimation theory of Ljung (1978), and Ljung and Caines (1979). It hinges upon the characterization of a joint confidence set for an optimal approximation for the underlying system in a parametric class and the corresponding approximation error, which is shown to be consistent. This is done under relatively weak conditions on the system input (stationarity conditions are not imposed) and observation noise (a specific distribution form is not assumed), and without assuming that the approximating model class contains the underlying model.