Partial Persistence of Excitation in RKHS Embedded Adaptive Estimation

In this article, an adaptive nonparametric method is proposed to estimate the unknown scalar-valued function that appears in systems governed by ordinary differential equations (ODEs). We recast the nonlinear estimation problem in a finite-dimensional Euclidean space into a linear one in an infinite-dimensional reproducing kernel Hilbert space (RKHS) by viewing the unknown function as a functional parameter in the RKHS, for which an RKHS embedded adaptive estimator is developed. The convergence analysis is facilitated by introducing a novel condition of partial persistent excitation (partial PE), which is defined for a subspace $\mathcal {H}_{\Omega }\subseteq \mathcal {H}_{X}$ of the RKHS $\mathcal {H}_{X}$. Using this condition, we prove that the projection of the functional estimation error onto the PE subspace $\mathcal {H}_{\Omega }$ converges in norm asymptotically to zero. While this is an abstract notion of convergence that depends implicitly on the kernel used to define the RKHS, we derive conditions that ensure the pointwise convergence of the function estimates over the subset $\Omega$ that generates the subspace $\mathcal {H}_{\Omega }$. In this article, we also introduce a weaker but geometrically intuitive notion of a partial PE condition, one that resembles PE conditions as they have been formulated historically in Euclidean spaces. Sufficient conditions are derived that describe when the two conditions are equivalent. Finally, qualitative properties of the convergence proofs derived in this article are illustrated with numerical simulations.