Fading memory and the problem of approximating nonlinear operators with volterra series

Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.

[1]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[2]  M. Fliess,et al.  Fonctionnelles causales non linaires et indtermines non commutatives , 1981 .

[3]  J. Barrett.  The Use of Functionals in the Analysis of Non-linear Physical Systems† , 1963 .

[4]  William L. Root.  On the Modeling of Systems for Identification. I: $\varepsilon $-Representations of Classes of Systems , 1975 .

[5]  H. Dang Van Mien,et al.  Nonlinear State-Affine Identification Methods; Applications to Electrical Power Plants , 1980 .

[6]  D. A. George.  Continuous nonlinear systems , 1959 .

[7]  Dorothée Normand-Cyrot,et al.  Nonlinear state affine identification methods: Applications to electrical power plants , 1984, Autom..

[8]  Stephen P. Boyd,et al.  Analytical Foundations of Volterra Series , 1984 .

[9]  Philip G. Gallman,et al.  Representations of nonlinear systems via the stone-weierstrass theorem , 1976, Autom..

[10]  V. Costanza,et al.  Universal approximations of discrete-time control systems over finite time , 1983 .

[11]  W. L. Root,et al.  On system measurement and identification. , 1965 .

[12]  L. Chua,et al.  A qualitative analysis of the behavior of dynamic nonlinear networks: Steady-state solutions of nonautonomous networks , 1976 .

[13]  P. M. Prenter.  A Weierstrass theorem for real, separable Hilbert spaces , 1970 .

[14]  Roger W. Brockett,et al.  CONVERGENCE OF VOLTERRA SERIES ON INFINITE INTERVALS AND BILINEAR APPROXIMATIONS , 1977 .

[15]  H. Sussmann.  Semigroup Representations, Bilinear Approximation of Input-Output Maps, and Generalized Inputs , 1976 .

[16]  I. Sandberg.  Series expansions for nonlinear systems , 1983 .

[17]  A. Friedman.  Foundations of modern analysis , 1970 .

[18]  Martin B. Brilliant,et al.  Theory of the analysis of nonlinear systems , 1958 .

[19]  William A. Porter,et al.  Causality structure and the Weierstrass theorem , 1975 .

[20]  Rui J. P. de Figueiredo,et al.  A best approximation framework and implementation for simulation of large-scale nonlinear systems , 1980 .