Box–Jenkins identification revisited—Part I: Theory

In classical time domain Box–Jenkins identification discrete-time plant and noise models are estimated using sampled input/output signals. The frequency content of the input/output samples covers uniformly the whole unit circle in a natural way, even in case of prefiltering. Recently, the classical time domain Box–Jenkins framework has been extended to frequency domain data captured in open loop. The proposed frequency domain maximum likelihood (ML) solution can handle (i) discrete-time models using data that only covers a part of the unit circle, and (ii) continuous-time models. Part I of this series of two papers (i) generalizes the frequency domain ML solution to the closed loop case, and (ii) proves the properties of the ML estimator under non-standard conditions. Contrary to the classical time domain case it is shown that the controller should be either known or estimated. The proposed ML estimators are applicable to frequency domain data as well as time domain data.