Balanced Model Reduction via the Proper Orthogonal Decomposition

TLDR

The exact computation of system grammians is impractical for large models, yet the proposed approach can be extended to nonlinear systems. The authors present a new balanced reduction technique for high‑order linear systems. The method combines proper orthogonal decomposition with balanced realization theory, using snapshot‑based low‑rank approximations of controllability and observability grammians to construct a balanced reduced‑order model. Applied to a linearized airfoil model, the approach produces highly accurate three‑state reduced models that excel when few outputs are of interest and outperform conventional POD.

Abstract

A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.