A continuous-time global estimator of the parameters of a SISO radial basis function network

A class of single-input single-output radial basis function networks (RBFN) is considered with both linear and nonlinear tuning parameters, the latter representing the radial basis functions centers. All parameters are assumed to be unknown except a known common input scaling factor. It is shown that the static RBFN hides a dynamic model in the input variable, in particular the RBFN output coincides with the output of a suitable observable autonomous system of linear first order differential equations with respect to a scalar variable if this quantity coincides with the RBFN output. By virtue of this property when the RBFN input is a time function, under regularity assumptions on the RBFN input trajectory, it is constructed a novel continuous time observer yielding global exponential estimates of the radial basis function centers as well as its linear parameters. If the RBFN is the approximation of an arbitrary nonlinear mapping, the parameters estimates convergence is shown to be robust with respect to the neural network output approximation error.