Error Bounds for Native Space Embedding Observers with Operator-Valued Kernels

This paper derives new rates of convergence for observers for a class of uncertain systems governed by non-linear ODEs. We assume that the generally nonlinear function appearing in the ODEs that represents the unknown dynamics is an element of a vector-valued reproducing kernel Hilbert space ℍ (RKHS) that is induced by an operator-valued kernel. The vector-valued RKHS embedding method described in the paper yields a nonparametric adaptive estimator that takes the form of distributed parameter system (DPS). The original ODEs in ℝ
^d
are thus embedded in a product space ℝ
^d
× ℍ in which state and functional uncertainty estimates evolve. We first discuss the well-posedness of the DPS formulation, and subsequently describe an approximation scheme in finitedimensional subspaces based on certain types of history dependent bases. We derive sufficient conditions that ensure the consistency of the finite-dimensional approximation scheme, and further derive rates of convergence in some particular cases when the samples that define the scattered bases are dense in some sufficiently regular and invariant subset of the observation space. A numerical example is given to illustrate the qualitative behavior of implementations of the theoretical results derived in the paper.