Title: Adaptive Estimation for Nonlinear Systems using Reproducing Kernel Hilbert Spaces,
Abstract: This paper extends a conventional, general framework for online adaptive
estimation problems for systems governed by unknown nonlinear ordinary
differential equations. The central feature of the theory introduced in this
paper represents the unknown function as a member of a reproducing kernel
Hilbert space (RKHS) and defines a distributed parameter system (DPS) that
governs state estimates and estimates of the unknown function. This paper 1)
derives sufficient conditions for the existence and stability of the infinite
dimensional online estimation problem, 2) derives existence and stability of
finite dimensional approximations of the infinite dimensional approximations,
and 3) determines sufficient conditions for the convergence of finite
dimensional approximations to the infinite dimensional online estimates. A new
condition for persistency of excitation in a RKHS in terms of its evaluation
functionals is introduced in the paper that enables proof of convergence of the
finite dimensional approximations of the unknown function in the RKHS. This
paper studies two particular choices of the RKHS, those that are generated by
exponential functions and those that are generated by multiscale kernels
defined from a multiresolution analysis.