Realization theory for rational systems: Minimal rational realizations

The study of realizations of response maps is a topic of control and system theory. Realization theory is used in system identification and control synthesis. A minimal rational realization of a given response map p is a rational realization of p such that the dimension of its state space equals the transcendence degree of the observation field of p. We relate minimality of rational realizations with their rational observability, algebraic controllability and canonicity. We show that the existence of a minimal rational realization is implied by the existence of a rational realization. We also specify the relation between birational equivalence of rational realizations and the properties of being canonical and minimal. Furthermore, we briefly discuss the procedures for checking various properties of rational realizations.