Analytical Sensitivities of Principal Components in Time-Series Analysis of Dynamical Systems

摘要

Principal components analysis, which is also referred to as proper orthogonal decomposition in the literature, is a useful technique in many fields of engineering, science, and mathematics for analysis of time-series data. The benefit of principal components analysis for dynamical systems comes from its ability to detect and rank the dominant coherent spatial structures of dynamic response, such as operating deflection shapes or mode shapes. In this work, an original method for calculating the analytical sensitivities of the principal components of dynamical systems is developed. Methods for analytical sensitivity calculations are developed for both the singular-value decomposition and eigenanalysis-based approaches for principal component calculation. Sensitivities with respect to state initial conditions and system parameters are enabled by state transition matrix calculations for augmented state and parameter differential equations. A novel approach to compute principal component sensitivities with respect to transient forcing-function parameters is introduced by transforming nonhomogenous differential equations (forced system) into homogenous differential equations (unforced system). These new developments are applied to several example problems in dynamics analysis, with an emphasis on structural dynamics analysis. Analytical sensitivities provide the necessary derivatives for gradient-based optimization algorithms and provide an analytical framework for evaluating structural modifications based on principal components analysis.