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Vector-Valued Native Space Embedding for Adaptive State Observation

Shengyuan Niu, Haoran Wang, Heejip Moon, Andrea L'Afflitto, Andrew Kurdila, Daniel Stilwell

TL;DR

This work develops a non-parametric adaptive observer by embedding uncertain nonlinear dynamics into a vector-valued native space ${\bm{\mathcal{H}}}$ within a vector-valued reproducing kernel Hilbert space. A distributed parameter system approach with an infinite-dimensional adaptive law is shown to yield bounded state estimation error, with an explicit dead-zone bound $E_0$ and Lyapunov-based guarantees; practical implementations use finite-dimensional approximations with a smoothed dead-zone and a coordinate form that updates kernel coefficients. The method is demonstrated on rigid-body translational and rotational estimation using a Sobolev-Materln kernel and a lattice of kernel centers, highlighting the trade-off between dead-zone width, approximation accuracy, and computational cost. The results indicate the framework can handle infinite-dimensional matched uncertainties and bounded disturbances, offering a principled, non-parametric alternative to parametric MRAC-like observers for complex MIMO systems. Future work aims to address non-deterministic disturbances and to integrate adaptive estimation with MRAC in native spaces for broader applicability.

Abstract

This paper combines vector-valued reproducing kernel Hilbert space (vRKHS) embedding with robust adaptive observation, yielding an algorithm that is both non-parametric and robust. The main contribution of this paper lies in the ability of the proposed system to estimate the state of a plan model whose matched uncertainties are elements of an infinite-dimensional native space. The plant model considered in this paper also suffers from unmatched uncertainties. Finally, the measured output is affected by disturbances as well. Upper bounds on the state observation error are provided in an analytical form. The proposed theoretical results are applied to the problem of estimating the state of a rigid body.

Vector-Valued Native Space Embedding for Adaptive State Observation

TL;DR

This work develops a non-parametric adaptive observer by embedding uncertain nonlinear dynamics into a vector-valued native space within a vector-valued reproducing kernel Hilbert space. A distributed parameter system approach with an infinite-dimensional adaptive law is shown to yield bounded state estimation error, with an explicit dead-zone bound and Lyapunov-based guarantees; practical implementations use finite-dimensional approximations with a smoothed dead-zone and a coordinate form that updates kernel coefficients. The method is demonstrated on rigid-body translational and rotational estimation using a Sobolev-Materln kernel and a lattice of kernel centers, highlighting the trade-off between dead-zone width, approximation accuracy, and computational cost. The results indicate the framework can handle infinite-dimensional matched uncertainties and bounded disturbances, offering a principled, non-parametric alternative to parametric MRAC-like observers for complex MIMO systems. Future work aims to address non-deterministic disturbances and to integrate adaptive estimation with MRAC in native spaces for broader applicability.

Abstract

This paper combines vector-valued reproducing kernel Hilbert space (vRKHS) embedding with robust adaptive observation, yielding an algorithm that is both non-parametric and robust. The main contribution of this paper lies in the ability of the proposed system to estimate the state of a plan model whose matched uncertainties are elements of an infinite-dimensional native space. The plant model considered in this paper also suffers from unmatched uncertainties. Finally, the measured output is affected by disturbances as well. Upper bounds on the state observation error are provided in an analytical form. The proposed theoretical results are applied to the problem of estimating the state of a rigid body.
Paper Structure (12 sections, 2 theorems, 29 equations, 3 figures)

This paper contains 12 sections, 2 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation on Vector-valued Reproducing Kernel Hilbert Space
  3. Problem Statement
  4. A Distributed Parameter System Solution
  5. Implementable Adaptive Laws
  6. Smoothed Dead-Zone Modification
  7. Coordinate Implementation
  8. Numerical Example
  9. Estimating the Translational Position
  10. Estimating the Angular Position
  11. Numerical Results
  12. Conclusion

Key Result

Theorem 1

Consider the nonlinear dynamics eqn_plant_dynamics and the observer eqn_state_estimator. If the dynamical system given by eqn_plant_dynamics and eqn_state_estimator has a unique solution $(\hat{x},\tilde{f}) \in {\mathbb{R}}^n \times C^1([t_0,\infty),{\bm{\mathcal{H}}})$ for every initial condition,

Figures (3)

  • Figure 1: Actual translational position and the estimated translational position as functions of time
  • Figure 2: Actual angular position and estimated angular position as functions of time
  • Figure 3: Actual angular rate and estimated angular rate as functions of time

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Definition 1: boffi2022nonparametric
  • Theorem 2