A new interpretation of subspace methods by using Schur complement

In this paper, we present a new interpretation of the subspace-based state space system identification (4SID) method by using a subspace extraction via Schur complement (SES). Several efficient implementations of subspace-based identification methods (e.g., N4SID, etc) have been thus far developed and proposed. We consider the MIMO output error state space model identification, the so-called MOESP. Making a comparison between the matrix obtained by the QR factorization in the MOESP and the Schur complement matrix of data product moments shows that the estimate of the extended observability matrix can be obtained by using the SES procedure. Viewing that fact that the Schur complement yielded by the subspace extraction coincides with the error covariance in the least squares (LS), our interpretation will pave the way for the derivation of the recursive algorithm of the 4SID method.