Analytical Sensitivities for Principal Components Analysis of Dynamical Systems

Principal Components Analysis (PCA), which is more recently referred to as Proper Orthogonal Decomposition (POD) in the literature, is a popular technique in many fields of engineering, science, and mathematics. The benefit of PCA 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. Most notable is the use of PCA to generate efficient basis sets for developing reduced order models for fluid flow dynamics and structural dynamics problems. Similarly, PCA has been investigated for system identification in experimental modal analysis. Additional application in the structural dynamics area is calibration of linear and nonlinear finite element models. These are but a few applications of PCA in the science and engineering literature. The objective of this work is to address the issue of sensitivity analysis for PCA. Sensitivity analysis is standard tool used by analysts and has the potential to impact many applications of PCA. In this paper, analytical approaches for sensitivity analysis of PCA are developed. These analytical approaches are developed in contrast to numerical approaches such as finite differencing. Methods are developed for both time domain and frequency domain implementations of PCA. For the time domain case, both a general complete method and a computationally economical method are developed. Each approach is verified by comparing the analytical sensitivities with numerical finite difference calculations.