Adaptive Estimation in Reproducing Kernel Hilbert Spaces

This paper introduces a novel framework for the study of adaptive or online estimation problems for a common class of nonlinear systems governed by ordinary differential equations (ODEs) on R-d. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), H. The nonlinear adaptive estimation problem is then cast as a time-varying estimation problem in the product space R-d x H of finite dimensional state estimates and infinite dimensional estimates of the unknown function. The adaptive estimation problem thereby constitutes a type of distributed parameter system, even though the original system is a collection of ODEs. The unknown function that lies in the RKHS is the distributed parameter. This paper derives (1) the sufficient conditions for the existence and uniqueness of solutions, (2) the stability and convergence of the state estimation error, and (3) the convergence of finite dimensional approximate solutions to the solution on the infinite dimensional state space. The new formulation provides a succinct and direct way of rigorously posing adaptive estimation problems using bases that are amenable to scattered approximation. A numerical example on adaptive estimation of road or terrain maps is presented to illustrate the convergence of the function estimates derived in this paper.