Non-intrusive balancing transformation of highly stiff systems with lightly damped impulse response

Abstract

Balanced truncation (BT) is a model reduction method that uses a coordinate transformation to retain eigen-directions that are highly observable and reachable. To address realizability and scalability of BT applied to highly stiff and lightly damped systems, a non-intrusive data-driven method is developed for balancing discrete-time linear systems via the eigensystem realization algorithm (ERA). The advantage of ERA for balancing transformation makes full-state outputs tractable. Further, ERA enables balancing despite stiffness, by eliminating computation of balancing modes and adjoint simulations. As a demonstrative example, we create balanced reduced-order models (ROMs) for a one-dimensional reactive flow with pressure forcing, where the stiffness introduced by the chemical source term is extreme (condition number 10 13 ), preventing analytical implementation of BT. We investigate the performance of ROMs in prediction of dynamics with unseen forcing inputs and demonstrate stability and accuracy of balanced ROMs in truly predictive scenarios, whereas without ERA, proper orthogonal decomposition-Galerkin and least-squares Petrov–Galerkin projections fail to represent the true dynamics. We show that after the initial transients under unit impulse forcing, the system undergoes lightly damped oscillations, which magnifies the influence of sampling properties on predictive performance of the balanced ROMs. We propose an output domain decomposition approach and couple it with tangential interpolation to resolve sharp gradients at reduced computational costs. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.