The Observation Space and Realizations of Finite Volterra Series

In this paper we extend the results on the theory of realizations of finite Volterra series by exploiting the structural properties of the observation space. In general, two distinct minimal realizations of a finite Volterra series may be constructed, one based on the properties of the observation space and the other based on the properties of the Lie algebra. Both these realizations display the canonical structure found earlier for such systems. The results given here also yield information on the observation algebra generated by functions in the observation space, just as previous work gave information on the Lie algebra. As an application, the structure found here is applied to the finite-dimensional filtering problem for these systems.