In this contribution, variance properties of L2 model reduction are studied. That is, given an estimated model of high order we study the resulting variance of an L2 reduced approximation. The main result of the paper is showing that estimating a low-order output error (OE) model via L2 model reduction of a high-order model gives a smaller variance compared to estimating a low-order model directly from data in case of undermodeling. This has previously been shown to hold for Finite Impulse Response models, but is in this paper extended to general linear OE models.

In this contribution, the variance properties of a two-step ARX estimation scheme are discussed. An expression for the covariance of the final low-order model is calculated and it is shown how this covariance can be minimised (at least for high-model orders). The implication of the results is that identification of the dynamics of a system can very easily be performed with standard linear least squares (two times), even if the measurement noise is heavily colored. A numerical example is included, where this two-step method gives a variance which is close (but not equal) to the Cramèr-Rao lower bound. Moreover, the point estimate of the covariance is close to the one obtained through Monte Carlo simulations.

In this contribution we examine certain variance properties of model reduction. The focus is on L₂ model reduction, but some general results are also presented. These general results can be used to analyze various other model reduction schemes. The models we study are finite impulse response (FIR) and output error (OE) models. We compare the variance of two estimated models. The first one is estimated directly from data and the other one is computed by reducing a high order model, by L₂ model reduction. In the FIR case we show that it is never better to estimate the model directly from data, compared to estimating it via L₂ model reduction of a high order FIR model. For OE models we show that the reduced model has the same variance as the directly estimated one if the reduced model class used contains the true system.

This note deals with the problem of estimating the variance of an undermodeled model. Undermodeling means that the model class used is not flexible enough to describe the underlying system. The proposed solution to the problem is an algorithm that is based on the bootstrap. A simulation example shows that the variance estimates based on the proposed algorithm are in very good agreement with Monte Carlo simulations.