Projection properties of the L 2 optimal reduced order model

Abstract

The paper investigates the optimal L 2 reduced order model in the context of reduced order models obtained by projection. It deals first with the general case of such systems and shows that the existence and uniqueness of a projection that relates two models of different orders can be inferred from the Kronecker canonical form of a certain linear pencil. Combining the L 2 optimality conditions into that framework gives a different point of view on the optimal reduced order model. It is shown that the zeros of the optimal approximation error transfer function are the mirror image of the reduced model poles with the same multiplicity in non-square systems, and doubled multiplicity in square systems. The latter case enhances an existing result, for SISO systems, that the multiplicity of the error zeros exceeds that of the optimal reduced model poles by one. It is also shown that the optimal projection, which is obtained in some solutions of the problem, is discontinuous at the optimal solution if the reduced order model has at least one real pole, and that in some cases the optimal L 2 reduced order model resides on the boundary of the set of models that can be obtained by projection.