Projection of state space realizations

1.0.1 Description of the problem We consider two m × p strictly proper transfer functions T (s) = C(sI n − A) −1 B, ˆ T (s) = ˆ C(sI k − ˆ A) −1ˆB, (1.1) of respective Mc Millan degrees n and k < n. We are interested in finding the necessary and sufficient conditions for the existence of projecting matrices Z, V ∈ C n×k such that and in characterizing the set of all transfer functionsˆT (s) that can be obtained via the projection equations (1.1,1.2). Only the image of the projecting matrices Z and V are important since choosing other bases satisfying the bi-orthogonality condition (1.2) amounts to a state-space transformation of the realization ofˆT (s). 1.0.2 Motivation and history of the problem Equation (1.2) arises naturally in the general framework of model reduction of large scale linear systems [1]. In this context we are given a transfer function T (s) of Mc Millan degree n which we want to approximate by a transfer function 1