Interpolated mapping system identification as a numerical algorithm

Abstract

A new method for nonlinear system identification, based on the technique of Interpolated Mapping, is formulated. The input to the procedure is a map, taking initial conditions on a regular grid to their images after a fixed time step. It is assumed that the underlying dynamics evolves in continuous time. The given map is then replaced by a cascade of maps with progressively smaller time steps. When the time step becomes sufficiently small, the vector field underlying the map is estimated by way of difference quotients. This first stage of the method is really a numerical differentiation procedure that is distinguished by producing estimates consistent with the overall dynamical flow, as opposed to being only local. A function fit is then performed on the approximate derivatives to produce the equations of motion, constituting a second, independent stage of the method. By reconstructing the continuous-time dynamics from observed discrete-time behavior. shortcomings of parametric discrete-time models in the presence of nonlinearity (such as loss of insight and restricted domain of applicability) are avoided.