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 asymptotically achievable for certain worst case inputs. The paper then focuses on the special case where the operators are rational, i.e., they have rational generating series, and thus are realizable in terms of bilinear ordinary differential state equations. In particular, it is shown that a discretization of the state equation via a kind of Euler approximation coincides exactly with the discrete-time Fliess operator approximator of the continuous-time rational operator.
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Notes
The polynomial \(1\emptyset \) is abbreviated throughout as 1.
Note here that \(\hat{u}^i(N)\) is the i-th power of \(\hat{u}(N)\) and not the i-th component of \(\hat{u}(N)\).
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W. Steven Gray was supported by Grant SEV-2011-0087 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. This research was also supported by a grant from the BBVA Foundation.
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Gray, W.S., Duffaut Espinosa, L.A. & Ebrahimi-Fard, K. Discrete-time approximations of Fliess operators. Numer. Math. 137, 35–62 (2017). https://doi.org/10.1007/s00211-017-0874-x
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DOI: https://doi.org/10.1007/s00211-017-0874-x