Discrete-Time Approximations of Fliess Operators

TL;DR

This paper develops methods to approximate Fliess operators, a tool for representing nonlinear control systems, using iterated sums with error bounds, and connects these approximations to discrete-time state space models for rational operators.

Contribution

It introduces new approximation techniques for Fliess operators with error estimates and links rational Fliess operators to discrete-time state space models derived from bilinear realizations.

Findings

Error bounds are achievable for local and global convergence rates.

Iterated sum approximations can be realized as rational state space models.

Discrete-time models are derived from bilinear realizations of rational Fliess operators.

Abstract

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.