Balanced Model Reduction via the Proper Orthogonal Decomposition

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 snapshots is used to obtain low-rank, reduced-range approximationsto the system controllability and observability grammiansin either the time or frequency domain.The approximationsare then used to obtain a balanced reducedorder 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 superior performance than those derived using a conventionalproper orthogonaldecomposition. Although further development is necessary, the concept also extends to nonlinear systems. W Nomenclature h = airfoil plunge displacement K = proper orthogonal decomposition (POD) kernel m = number of POD snapshots n = number of states in computational � uid dynamics (CFD) model nr = number of states in reduced-ordermodel R = correlation matrix T = matrix whose columns contain the balancing transformation vectors u; U = vector containing inputs for models, time and frequency domain Wc = controllabilitygrammian Wco = grammian product Wo = observability grammian x; X = aerodynamic state vector for CFD model, time and frequency domain xr = aerodynamic state vector for reduced-ordermodel y; Y = vector containing outputs of CFD model, time and frequency domain yr = vector containing outputs of reduced-ordermodel z = dual state vector for CFD model i = ith Hankel singular value = basis vector! = forcing frequency