Order Reduction by Proper Orthogonal Decomposition (POD) Analysis

Abstract

The method of proper orthogonal mode decomposition (POD) or KarhunenLoeve decomposition (KLD) is a means of extracting spatial information from a set of time-series data available from a set of sensing locations over a domain. The POD can be used to obtain low-dimensional models or discrete or distributed dynamical systems by computing an orthogonal set of eigen-functions through a finite-dimensional eigenvalue problem that is obtained by post processing of time-series measurements at different spatial locations. Interestingly enough, these eigenfunctions form an orthogonal basis (irrespective of the linear or nonlinear nature of the measured signals) which is optimal in the sense that fewer POD modes are needed to capture a given amount of energy of the measured signal than any other linear set of modes, including vibration modes [219]. Moreover, the eigenvalue corresponding to a given eigenfunction quantifies the amount of energy of the measured signal that is captured by the specific POD mode. Hence, the POD method not only provides a linear orthogonal basis of modes, but also a quantitative measure of the relative importance of these modes with regard to the energy of the signal captured by the POD analysis. This feature of the method makes it a valuable tool in the analysis, system identification and order reduction of the dynamics of engineering systems. As pointed by Kerschen [102] the POD analysis resembles the Singular Value Decomposition Method, with the later method providing additional information related to amplitude modulations of the identified waveforms.