# Space-Frequency Localized Basis Function Networks for Nonlinear
System Estimation and Control

## Reference

*Space-Frequency Localized Basis Function Networks for Nonlinear
System Estimation and Control*, M. Cannon and J.J.E. Slotine,
Neurocomputing, 9(3), 1995.

## Abstract

Stable neural network control and estimation may be viewed formally as
a merging of concepts from nonlinear dynamic systems theory with tools
from multivariate approximation theory. This paper extends earlier
results on adaptive control and estimation of nonlinear systems using
gaussian radial basis functions to the on-line generation of
irregularly sampled networks, using tools from multiresolution
analysis and wavelet theory. This yields much more compact and
efficient system representations while preserving global closed-loop
stability.
Approximation models employing basis functions that are localized in
both space and spatial frequency admit a measure of the approximated
function's spatial frequency content that is not directly dependent on
reconstruction error. As a result, these models afford a means of
adaptively selecting basis functions according to the local spatial
frequency content of the approximated function. An algorithm for
stable, on-line adaptation of output weights simultaneously with
node configuration in a class of non-parametric models with wavelet
basis functions is presented. An asymptotic bound on the error in the
network's reconstruction is derived and shown to be dependent solely
on the minimum approximation error associated with the steady state
node configuration. In addition, prior bounds on the temporal
bandwidth of the system to be identified or controlled are used to
develop a criterion for on-line selection of radial and ridge wavelet
basis functions, thus reducing the rate of increase in network's size
with the dimension of the state vector.

Experimental results obtained by using the network to predict the path
of an unknown light bluff object thrown through air, in an
active-vision based robotic catching system, are given to illustrate
the network's performance in a simple real-time application.

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