Stochastic Subspace Identification: Valid Model, Asymptotics and Model Error Bounds

This paper investigates the ability of the stochastic subspace identification technique to return a valid model from finite measurement data, its asymptotic properties as the data set becomes large, and asymptotic error bounds of the identified model (in terms of H2 and H∞ norms). First, a new and straightforward LMI-based approach is proposed, which returns a valid identified model even in cases where the system poles are very close to unit circle and there is insufficient data to accurately estimate the covariance matrices. The approach, which is demonstrated by numerical examples, provides an altenative to other techniques which often fail under these circumstances. Then, an explicit expression for the variance of the asymptotically normally distributed sample output covariance matrices and blockHankel matrix are derived. From this result, together with perturbation techniques, error bounds for the state-space matrices in the innovations model are derived, for a given confidence level. This result is in turn used to derive several error bounds for the identified transfer functions, for a given confidence level. One is an explicit H2 bound. Additionally, two H∞ error bounds are derived; one via perturbation analysis, and the other via an LMI-based technique. Index Terms Asymptotic variance, stochastic subspace identification, positive realness, H2 norm error bound, H∞ norm error bound, linear matrix inequalities. Q. Li is currently with the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: [email protected]). J.T. Scruggs is with the Department of Civil & Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA (e-mail: [email protected]).