A linear regression approach to state-space subspace system identification

Recently, state-space subspace system identification (4SID) has been suggested as an alternative to the more traditional prediction error system identification. The aim of this paper is to analyze the connections between these two different approaches to system identification. The conclusion is that 4SID can be viewed as a linear regression multistep-ahead prediction error method with certain rank constraints. This allows us to describe 4SID methods within the standard framework of system identification and linear regression estimation. For example, this observation is used to compare different cost-functions which occur rather implicitly in the ordinary framework of 4SID. From the cost-functions, estimates of the extended observability matrix are derived and related to previous work. Based on the estimates of the observability matrix, the asymptotic properties of two pole estimators, namely the shift invariance method and a weighted subspace fitting method, are analyzed. Expressions for the asymptotic variances of the pole estimation error are given. From these expressions, difficulties in choosing user-specified parameters are pointed out. Furthermore, it is found that a row-weighting in the subspace estimation step does not affect the pole estimation error asymptotically.