Identification of linear systems via spectral analysis given time-domain data: consistency, reduced-order approximation, and performance analysis

Estimation of a parametric input-output (I/O) infinite impulse response transfer function given time-domain I/O data is considered. Some of the desirable properties of any approach to this problem are: unimodality of the performance surface, consistent identification in the sufficient-order case, and stability of the fitted model under undermodeling. We first consider a frequency-domain solution to the least squares equation error identification problem using the power spectrum and the cross-spectrum of the I/O data to estimate the I/O parametric transfer function. The proposed approach is shown to yield a unimodal performance surface, consistent identification in colored noise and sufficient-order case, and stable fitted models under undermodeling for arbitrary stationary inputs so long as they are persistently exciting of sufficiently high order. Asymptotic performance analysis is carried out for both sufficient-order and reduced-order cases. These asymptotic results are then used to derive statistics on the corresponding estimated transfer function. We also investigate an iterative pseudomaximum likelihood approach and analyze its performance under sufficient-order modeling. Finally, computer simulation examples are provided to illustrate the two approaches.