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 $\mathcal{H}2$ and $\mathcal{H}{\infty}$ 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 block-Hankel 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 $\mathcal{H}2$ bound. Additionally, two $\mathcal{H}{\infty}$ error bounds are derived, one via perturbation analysis, and the other via an LMI-based technique.