The realizations of response maps by rational systems

This paper treats the problem of realization of response maps by rational systems. We provide sufficient and necessary conditions for an analytic response map to be realizable by a rational system. The existence of rational realizations which are algebraically ob- servable, algebraically controllable, or minimal is also studied. We relate the properties of rational realizations such as algebraic observability, algebraic controllability, minimal- ity, and minimal-dimensionality. We also explain the relation between minimal/canonical rational realizations and a birational equivalence of rational realizations.