Error analysis of state approaches in subspace identification methods

Explicit formulation of dominant parts of the estimation errors of A and C matrices in shift invariance approach and state approach are derived based on a proposing lemma on perturbations to the singular subspaces. In the state approach, two methods are considered: one estimates the state from the left singular subspaces of a certain matrix, while the other estimates the state from the right singular subspaces of the same matrix. It is shown that the estimate in the state approach has an asymptotic bias when the past horizon p is finite, while the estimate in the shift invariance approach is unbiased.