  A convenient way to represent a nonlinear input-output system in control
theory is via a Chen-Fliess functional expansion or Fliess operator. The
general goal of this paper is to describe how to approximate Fliess operators
with iterated sums and to provide accurate error estimates for two different
scenarios, one where the series coefficients are growing at a local convergence
rate, and the other where they are growing at a global convergence rate. In
each case, it is shown that the error estimates are achievable in the sense
that worst case inputs can be identified which hit the error bound. The paper
then focuses on the special case where the operators are rational, i.e., they
have rational generating series. It is shown in this situation that the
iterated sum approximation can be realized by a discrete-time state space model
which is a rational function of the input and state affine. In addition, this
model comes from a specific discretization of the bilinear realization of the
rational Fliess operator.
