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

Estimation of 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.