Formulating Hankel singular values of closely-spaced modes in time domain

It is well-known that the identifiability of closely-spaced modes is deteriorated due to the near rank-deficiency of the data matrix. To assess the interactions among close modes, it would be helpful if the evolution of matrix condition could be predicted analytically. To this end, Singular Value Decomposition is considered to be the most suitable tool for this purpose. The singular values not only represent the modal powers, but also reflect the condition of the data matrix, usually of Hankel form. The objective of this paper is thus aimed at developing analytical expressions of time-domain Hankel Singular Values (HSVs) for two closely-spaced modes. The approach takes advantage of the formulation of Hankel SVD for widely spaced modes using Signal Subspace Correlation (SSC) techniques being developed earlier. With the aid of this work, the predictions of HSVs for a two-mode system may be transformed from the SVD of a full Hankel matrix into that of an SSC matrix of size 4-by-4. Computational time is therefore dramatically reduced. Meanwhile, the mechanisms of matrix condition as a function of frequency separation, damping ratio, matrix size (or data points used), sampling rate, etc, can be clearly observed from the derived formulae. Computer simulations are also provided to compare with analytical predictions in the end. Their excellent agreement confirms the effectiveness of the proposed method.