A Shift Procedure for Identifying Low Rank Behavior from Non-Stationary Dynamical System Data

Many model order reduction techniques for dynamical systems leverage pe-

riodicity or statistical stationarity properties in collected data to identify the

underlying physics and to build reduced order models. However, many sys-

tems of interest, such as scramjet unstart, do not exhibit these properties and

traditional approaches to applying data-driven decomposition techniques, such

as Proper Orthogonal Decomposition (POD), yield results that are difficult to

interpret, or use for reduced order models. However, shifting the data into a

suitably defined reference frame that travels with an identified feature of interest

can recover many of the attractive properties of typical decomposition methods,

at least to first-order. We propose a robust, data-driven way to execute this shift

by computing an instantaneously varying translational velocity vector from the

Empirical Mode Decomposition (EMD) of time series data which tracks the lo-

cation of targeted space-time features. Application of decomposition techniques

in the shifting reference frame recovers many of the low rank dynamics that

are not easily apparent in the original data. The approach is demonstrated by

application to four example problems ranging in complexity from a convecting

Gaussian pulse to unstarting shocks in a scramjet isolator.