Identification of stochastic linear systems via spectral analysis: reduced-order approximation, performance analysis and transfer function bias

Estimation of the parametric input-output (IO) infinite impulse response (IIR) transfer function given time-domain IO 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. Some of the well-known approaches fail to satisfy one or more of these properties. The time-domain equation error method (EEM) yields a unimodal performance surface, biased estimates in colored noise and sufficient-order case, and stable fitted models under undermodeling if the input is autoregressive. In this paper we propose a frequency-domain solution to the least-squares equation error identification problem using the power spectrum and the cross-spectrum of the IO data to estimate the IO 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 can then used to derive statistics on the corresponding estimated transfer function. We also investigate an iterative pseudo-maximum likelihood approach and analyse its performance under sufficient-order modeling. Finally a computer simulation example is presented