Consistency analysis of orthogonal projection based closed-loop subspace identification methods

Based on the orthogonal projection approach, a few closed-loop subspace identification methods have been developed recently in the literature. To guarantee consistent estimation of these methods against measurement noise, this paper analyzes the rank condition for performing the corresponding singular value decomposition (SVD) involved in deriving the extended observability matrix and the lower triangular block-Toeliptz matrix of the plant state-space model, together with the requirement on the input excitation signal, such that no mistake will be made in picking up the corresponding eigenvalues for numerical computation. Also, no knowledge of the closed-loop controller is required for the computation of the model state matrices. An example from the literature is used to demonstrate the proposed conditions for consistent estimation.