Variance estimation of modal parameters from the poly-reference least-squares complex frequency-domain algorithm

Modal parameter estimation from input/output data is a fundamental task in engineering. The poly-reference least-squares complex frequency-domain (pLSCF) algorithm is a fast and robust method for this task, and is extensively used in research and industry. As with any method using noisy measurement data, the modal parameter estimates are afflicted with uncertainty. However, their uncertainty quantification has been incomplete, in particular for the case of real-valued polynomial coefficients in the modeling of the frequency response functions (FRFs) in the pLSCF algorithm, and no expressions have been available for the covariance of participation vectors and mode shapes that are subsequently estimated with the least-squares frequency domain (LSFD) approach. This paper closes these gaps. Uncertainty expressions for the modal parameters, including participation vectors and mode shapes, are derived and presented. It is shown how to estimate the covariance between different modal parameters, and a complete method is provided for modal parameter covariance estimation from pLSCF. The method is propagating the uncertainty of FRFs through the algorithm using first-order perturbation theory and the delta method. The method is validated via extensive Monte-Carlo simulations and the applicability is illustrated using a laboratory experiment.