Data-driven nonlinear model reduction to spectral submanifolds via oblique projection

The dynamics in a primary spectral submanifold (SSM) constructed over the slowest modes of a dynamical system provide an ideal reduced-order model for nearby trajectories. Modeling the dynamics of trajectories further away from the primary SSM, however, is difficult if the linear part of the system exhibits strong non-normal behavior. Such non-normality implies that simply projecting trajectories onto SSMs along directions normal to the slow linear modes will not pair those trajectories correctly with their reduced counterparts on the SSMs. In principle, a well-defined nonlinear projection along a stable invariant foliation exists and would exactly match the full dynamics to the SSM-reduced dynamics. This foliation, however, cannot realistically be constructed from practically feasible amounts and distributions of experimental data. Here, we develop an oblique projection technique that is able to approximate this foliation efficiently, even from a single experimental trajectory of a significantly non-normal and nonlinear beam.