Asymptotic Variance Expression in Output Over-sampling Based Closed-loop Identification

In this work, asymptotic variance expression is derived for the transfer function estimates in output over-sampling based closed-loop identification. The result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance, respectively, is proportional to the "generalized" noise-to-signal ratio, where the spectrum is in the form of cyclo-stationary spectrum due to the cyclo-stationary properties of the observed signals. The factor of proportionality is the ratio of model order to number of data. Based on the result, it can be shown that only when the output disturbance contains some energy beyond the bandwidth of the plant, the closed-loop identifiability of the over-sampling scheme without external excitation is guaranteed. Numerical examples are used to illustrate the asymptotic variance expression and closed-loop identifiability. Finally, when test signal is used, both in open and closed-loop, the result points out that using the over-sampling scheme can increase model accuracy in real world applications.