On the frequency domain accuracy of closed-loop estimates

It has been argued that the frequency domain accuracy of high model-order estimates obtained on the basis of closed-loop data is largely invariant to whether direct or indirect approaches are used. The analysis underlying this conclusion has employed variance expressions that are asymptotic both in the data length and the model order, and hence are approximations when either of these are finite. However, recent work has provided variance expressions that are exact for finite (possibly low) model order, and hence can potentially deliver more accurate quantification of estimation accuracy. This paper, and a companion one, revisits the study of identification from closed-loop data in light of these new quantifications and establishes that, under certain assumptions, there can be significant differences in the accuracy of frequency response estimates. These discrepencies are established here and in the companion paper to be dependent on what type of direct, indirect or joint input-output identification strategy is pursued.