Properties of the system matrix of a generalized state-space system†

Abstract

For an irreducible polynomial system matrix P(s)= ((1970, p. 111) has shown that the polar structure of the associated transfer function R(s)= V(s)T−² (s) U (s) at any finite frequency is isomorphic to the zero structure of T(s) at that frequency, while the zero structure of R(s) at any finite frequency is isomorphic to that of P(s) at the same frequency. In this paper we obtain the appropriate extensions for the structure at infinite frequencies in the particular case of systems for which T(a) =. sE − A (with E possibly singular), U(s) = B, V(s) = C, and W(a) =D, under a strengthened irreducibility condition. We term such systems generalized state-space systems, and note that any rational R(s) may bo realized in this form. We also demonstrate in this case that a minimal basis (in the sense of Forney (1975) for the left or right null space of P(s) directly generates one with the same minimal indices for the corresponding null space of R(s), and vice versa. These results also enable us to identify the pole-zero excess of R(s) as being equal to the sum of the minimal indices of its null spaces. Connections with Kronecker's theory of matrix pencils are made.