Multi-Order Covariance Computation for Estimates in Stochastic Subspace Identification Using QR Decompositions

For applications as Operational Modal Analysis (OMA) of vibrating structures, an output-only LTI system with state and measurement noise can be identified using subspace methods. While these identification techniques have been very suitable for the identification of such mechanical, aeronautical or civil structures, covariance expressions of the estimates of the system matrices are difficult to obtain and theoretical results from literature are hard to implement for output-only systems with unknown noise properties in practice. Moreover, the model order of the underlying system is generally unknown and due to noise and model errors, usual statistical criteria cannot be used. Instead, the system is estimated at multiple model orders and some GUI driven stabilization diagram containing the resulting modal parameters is used by the structural engineer. Then, the covariance of the estimates at these different model orders is an important information for the engineer, which, however, would be computationally expensive to obtain with the existing tools. Recently a fast multi-order version of the stochastic subspace identification approach has been proposed, which is based on the use of the QR decomposition of the observability matrix at the largest model order. In this paper, the corresponding covariance expressions for the system matrix estimates at multiple model orders are derived and successfully applied on real vibration data.