Open-loop versus closed-loop identification of Box-Jenkins models: a new variance analysis

We present formulae for the analysis of variance of estimated transfer functions, which are valid for Box-Jenkins (BJ) and Output Error (OE) model structures of finite order, identified in either open-loop or closed-loop, using Prediction Error (PE) Identification. The formulae are based on the asymptotic (in number of data) expression of the parameter covariance. They do not require special assumptions on the generation of the external signals. One of the results of our analysis is to show that, under reasonable assumptions on the signal powers, the variance of the estimated input-output model is smaller with closed-loop than with open-loop identification.