Dynamical Realizations of Finite Volterra Series

In this paper, realizations of finite Volterra series are viewed as nonlinear analytic input–output systems, with state space described by an analytic manifold. For a minimal realization guaranteed by H. J. Sussmann, the state space, which is unique up to diffeomorphism, is shown to have the homogeneous space structure of a nilmanifold, the quotient of two nilpotent Lie groups. The structure of nilmanifolds as described by A. Malcev is used to show that for these systems, the state space has a vector space structure. As a consequence of this result, it is shown that a minimal realization of a finite Volterra series can be described as a cascade of linear subsystems with polynomial link maps, in which the dimension o f each linear subsystem is independent of the realization considered.