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Enhanced Robust Tracking Control: An Online Learning Approach

Ao Jin, Weijian Zhao, Yifeng Ma, Panfeng Huang, Fan Zhang

TL;DR

This work addresses tracking control for nonlinear systems subject to unknown disturbances by integrating a neural-network–driven Control Contraction Metric (CCM) with an online disturbance learning module. A learnable metric $M(\bm{x})$ and tracking law $\bm{u}=\bm{k}(\bm{x},\bm{x}^*)+\bm{u}^*$ enforce contraction at rate $\lambda$, while an online estimator $\bm{H}(t,\bm{x})$ uses historical data to compensate disturbances via $\bm{u}_{\text{ol}}=\bm{u}_{\text{ccm}}-\bm{H}(t,\bm{x})$, tightening the robust invariant tube (RCI). The approach combines a neural network–based CCM synthesis with a memory-buffer–driven moving-horizon disturbance learning to improve tracking accuracy and reduce conservatism in motion planning, demonstrated on a TSR and a PVTOL with substantial RMSE reductions; open-source code is provided. Overall, the method offers real-time disturbance compensation without prior disturbance models, enhancing robustness and planning efficiency for nonlinear robotic systems.

Abstract

This work focuses the tracking control problem for nonlinear systems subjected to unknown external disturbances. Inspired by contraction theory, a neural network-dirven CCM synthesis is adopted to obtain a feedback controller that could track any feasible trajectory. Based on the observation that the system states under continuous control input inherently contain embedded information about unknown external disturbances, we propose an online learning scheme that captures the disturbances dyanmics from online historical data and embeds the compensation within the CCM controller. The proposed scheme operates as a plug-and-play module that intrinsically enhances the tracking performance of CCM synthesis. The numerical simulations on tethered space robot and PVTOL demonstrate the effectiveness of proposed scheme. The source code of the proposed online learning scheme can be found at https://github.com/NPU-RCIR/Online_CCM.git.

Enhanced Robust Tracking Control: An Online Learning Approach

TL;DR

This work addresses tracking control for nonlinear systems subject to unknown disturbances by integrating a neural-network–driven Control Contraction Metric (CCM) with an online disturbance learning module. A learnable metric and tracking law enforce contraction at rate , while an online estimator uses historical data to compensate disturbances via , tightening the robust invariant tube (RCI). The approach combines a neural network–based CCM synthesis with a memory-buffer–driven moving-horizon disturbance learning to improve tracking accuracy and reduce conservatism in motion planning, demonstrated on a TSR and a PVTOL with substantial RMSE reductions; open-source code is provided. Overall, the method offers real-time disturbance compensation without prior disturbance models, enhancing robustness and planning efficiency for nonlinear robotic systems.

Abstract

This work focuses the tracking control problem for nonlinear systems subjected to unknown external disturbances. Inspired by contraction theory, a neural network-dirven CCM synthesis is adopted to obtain a feedback controller that could track any feasible trajectory. Based on the observation that the system states under continuous control input inherently contain embedded information about unknown external disturbances, we propose an online learning scheme that captures the disturbances dyanmics from online historical data and embeds the compensation within the CCM controller. The proposed scheme operates as a plug-and-play module that intrinsically enhances the tracking performance of CCM synthesis. The numerical simulations on tethered space robot and PVTOL demonstrate the effectiveness of proposed scheme. The source code of the proposed online learning scheme can be found at https://github.com/NPU-RCIR/Online_CCM.git.
Paper Structure (10 sections, 4 theorems, 29 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 4 theorems, 29 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminaries on Control Contraction Metrics
  4. Main Results
  5. Neural Network-Driven CCM Synthesis
  6. Enhanced Robust Tracking Control via Online Learning
  7. Numerical Simulations
  8. Tethered Space Robot
  9. PVTOL
  10. Conclusion

Key Result

Proposition 1

If the condition (condition_ccm) is satisfied with a uniformly bounded metric $\bm M(\bm{x})$, i.e., $\underline{\alpha}\bm{I}\preceq \bm M(\bm{x})\preceq \overline{\alpha}\bm{I}$ and a feedback tracking controller $\bm{u}=\bm k(\bm{x},\bm{x}^*)+\bm{u}^*$, then the displacement between actual trajec where $\lambda$ is the contracting rate and the constant $R=\sqrt{{\overline{\alpha}}/{\underline{\

Figures (5)

  • Figure 1: Illustration of proposed online learning scheme for enhancing the robust tracking control.
  • Figure 2: The schematic of construction of memory buffer.
  • Figure 3: The tracking performance of proposed framework for TSR. The deployment of TSR is generally done when the inplane angle and tether length reach the $1\%$ precision zone.
  • Figure 4: The tracking performance of proposed framework for PVTOL. The black dotted line denotes the trajectory without online learning ($\bm{u}=\bm{u}_{\text{ccm}}$), the solid purple line represents the trajectory with online learning ($\bm{u}=\bm{u}_{\text{ol}}$). The light blue dashed lines represent the boundary of tube computed by (\ref{['RCI_set']}).
  • Figure : Enhanced Robust Tracking Control vis Online Learning

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof