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Learning-based Observer for Coupled Disturbance

Jindou Jia, Meng Wang, Zihan Yang, Bin Yang, Yuhang Liu, Kexin Guo, Xiang Yu

TL;DR

This work tackles high-precision control for robotic systems under a tightly coupled disturbance that depends on both internal uncertainties and external factors. It introduces a learning-based observer that first decomposes the coupled disturbance $\mathbf{\Delta}(\mathbf{x},\mathbf{d})$ into an unknown parameter matrix $\mathbf{\Theta}$ and structure-defined components via Chebyshev polynomials, then learns $\mathbf{\Theta}$ offline with a regularized least-squares formulation and estimates the remaining time-varying part online with a polynomial disturbance observer. The approach yields convergent disturbance estimation by combining an explicit offline parameter learning with an online DO that leverages learned structure, avoiding strict bounded-disturbance assumptions and offering a lightweight, interpretable alternative to deep learning methods. Empirical results from nonlinear function learning, a second-order dynamics example, and drone flight tests demonstrate improved disturbance reconstruction and tracking performance, including generalization to unseen disturbances and low computational overhead. This framework provides a pathway to integrate learning into control loops for robust, high-precision robotic applications.

Abstract

Achieving high-precision control for robotic systems is hindered by the low-fidelity dynamical model and external disturbances. Especially, the intricate coupling between internal uncertainties and external disturbances further exacerbates this challenge. This study introduces an effective and convergent algorithm enabling accurate estimation of the coupled disturbance via combining control and learning philosophies. Concretely, by resorting to Chebyshev series expansion, the coupled disturbance is firstly decomposed into an unknown parameter matrix and two known structures dependent on system state and external disturbance respectively. A regularized least squares algorithm is subsequently formalized to learn the parameter matrix using historical time-series data. Finally, a polynomial disturbance observer is specifically devised to achieve a high-precision estimation of the coupled disturbance by utilizing the learned portion. The proposed algorithm is evaluated through extensive simulations and real flight tests. We believe this work can offer a new pathway to integrate learning approaches into control frameworks for addressing longstanding challenges in robotic applications.

Learning-based Observer for Coupled Disturbance

TL;DR

This work tackles high-precision control for robotic systems under a tightly coupled disturbance that depends on both internal uncertainties and external factors. It introduces a learning-based observer that first decomposes the coupled disturbance into an unknown parameter matrix and structure-defined components via Chebyshev polynomials, then learns offline with a regularized least-squares formulation and estimates the remaining time-varying part online with a polynomial disturbance observer. The approach yields convergent disturbance estimation by combining an explicit offline parameter learning with an online DO that leverages learned structure, avoiding strict bounded-disturbance assumptions and offering a lightweight, interpretable alternative to deep learning methods. Empirical results from nonlinear function learning, a second-order dynamics example, and drone flight tests demonstrate improved disturbance reconstruction and tracking performance, including generalization to unseen disturbances and low computational overhead. This framework provides a pathway to integrate learning into control loops for robust, high-precision robotic applications.

Abstract

Achieving high-precision control for robotic systems is hindered by the low-fidelity dynamical model and external disturbances. Especially, the intricate coupling between internal uncertainties and external disturbances further exacerbates this challenge. This study introduces an effective and convergent algorithm enabling accurate estimation of the coupled disturbance via combining control and learning philosophies. Concretely, by resorting to Chebyshev series expansion, the coupled disturbance is firstly decomposed into an unknown parameter matrix and two known structures dependent on system state and external disturbance respectively. A regularized least squares algorithm is subsequently formalized to learn the parameter matrix using historical time-series data. Finally, a polynomial disturbance observer is specifically devised to achieve a high-precision estimation of the coupled disturbance by utilizing the learned portion. The proposed algorithm is evaluated through extensive simulations and real flight tests. We believe this work can offer a new pathway to integrate learning approaches into control frameworks for addressing longstanding challenges in robotic applications.
Paper Structure (29 sections, 4 theorems, 33 equations, 9 figures, 1 algorithm)

This paper contains 29 sections, 4 theorems, 33 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Method
  4. Decomposition of the coupled disturbance
  5. Learning the parameter matrix
  6. Estimation via a polynomial disturbance observer
  7. Empirical Study
  8. Learning performance on nonlinear functions
  9. Setup.
  10. Results.
  11. Estimation performance on a second-order dynamics
  12. Setup.
  13. Results.
  14. Control performance on a flying drone
  15. Setup.
  16. ...and 14 more sections

Key Result

Lemma 1

o2022neural Assume an analytic function $\bm{\Delta}_i\left({\bm{x}, \bm{d}}\right): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ for all $\left[\bm{x},\bm{d}\right] \in \mathcal{X} \times \mathcal{D}$. For any small value $\epsilon > 0$, there always exist $p = O\left(\frac{{\log \left( and $s = \left(p+1\right)^m = O\left(log\left(1/\epsilon\right)^m\right)$.

Figures (9)

  • Figure 1: Overall framework of our proposed learning-based observer for coupled disturbances.
  • Figure 2: Learning errors (MAE) of three nonlinear functions \ref{['nonlinear_functions']} under different noise variances $\bm{\sigma}_x^2$ and parameters $p$ on the test dataset.
  • Figure 3: The tracking and estimation performance of the second-order dynamics example. A, The trajectory tracking performance. B, The disturbance estimation performance. C, MAEs of all compared methods: NDO NDO_bounded_variation, ESO GU2022105158ESO, $\mathcal{L}_1$ adaptive huang2023datthanover2021performance, EVOLVER 10288520, the baseline controller (without compensation) and the proposed one \ref{['DO_higher']}.
  • Figure 4: Eexperimental scenarios in the drone example. A, Hovering under wind disturbance. B, Circle flight under model uncertainty. C, Circle flight under model uncertainty and wind disturbance.
  • Figure 5: The tracking and estimation performance in the drone example.A-B, Test #1. C-D, Test #2. E-F, Test #3. G, Tracking error and estimated disturbance along the $X$-axis in Test #1.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • ...and 2 more