Small Noise Analysis of Non-Parametric Closed-Loop Identification

We revisit the problem of non-parametric closedloop identification in frequency domain; we give a brief survey of the literature and provide a small noise analysis of the direct, indirect, and joint input-output methods when two independent experiments with identical excitation are used. The analysis is asymptotic in the noise variance (i.e., as the standard deviation of the innovations $\sigma \rightarrow 0$), for a finite data record of length N. We highlight the relationship between the estimators accuracy and the loop shape via asymptotic variance expressions given in terms of the sensitivity function. The results are illustrated using a numerical simulation example.