Asymptotic analysis of vector ARMA identification

This paper investigates the asymptotic analysis of the subspace approach for vector ARMA process estimation. To overcome the statistical insufficiency of data, a new and straightforward LMI-based approach is proposed to obtain a positive real covariance model. Numerical results show this approach performs well even if the system poles are very close to the unit circle. 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, several central limit theorems for the controllability matrix, observability matrix, and the state-space matrices of the associated covariance model are derived, as well as the norm bounds of Kalman gain and the innovations covariance matrix in the innovations model. By combining these asymptotic results, the ℋ
₂
norm bound of the error system, i.e. the difference between the identified transfer function and the true one, is derived with a given confidence level.