Balanced model reduction via the proper orthogonal decomposition

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 snapshots is used to obtain low-rank, reduced-range approximations to the system controllability and observability grammians in either the time or frequency domain. The approximations are then used to obtain a balanced reduced-order 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 reduced-order models are obtained that capture the required dynamics with just three states. The new models exhibit far superior performance than those derived using a conventional proper orthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.