A Schur Method for Balanced Model Reduction

The calculation of the balancing transformation required in Moore\´s "balanced-transformation" model reduction procedure tends to be badly conditioned, especially for non-minimal models that stand to benifit the most from model reduction. In this paper it is shown that the Moore reduced model can be computed directly without balancing via projections defined in terms of arbitary bases for the left and right eigenspaces associated with the "large" eigenvalues of the product PQ of the reachability and controllablility grammians. Two methods for computing these bases are proposed, one based on the ordered Schur decomposition of PQ and the other based on the Cholesky factors of P and Q. The algorithms perform reliably even for non-minimal models.