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Neural Robust Control on Lie Groups Using Contraction Methods (Extended Version)

Yi Lok Lo, Longhao Qian, Hugh H. T. Liu

Abstract

In this paper, we propose a learning framework for synthesizing a robust controller for dynamical systems evolving on a Lie group. A robust control contraction metric (RCCM) and a neural feedback controller are jointly trained to enforce contraction conditions on the Lie group manifold. Sufficient conditions are derived for the existence of such an RCCM and neural controller, ensuring that the geometric constraints imposed by the manifold structure are respected while establishing a disturbance-dependent tube that bounds the output trajectories. As a case study, a feedback controller for a quadrotor is designed using the proposed framework. Its performance is evaluated using numerical simulations and compared with a geometric controller.

Neural Robust Control on Lie Groups Using Contraction Methods (Extended Version)

Abstract

In this paper, we propose a learning framework for synthesizing a robust controller for dynamical systems evolving on a Lie group. A robust control contraction metric (RCCM) and a neural feedback controller are jointly trained to enforce contraction conditions on the Lie group manifold. Sufficient conditions are derived for the existence of such an RCCM and neural controller, ensuring that the geometric constraints imposed by the manifold structure are respected while establishing a disturbance-dependent tube that bounds the output trajectories. As a case study, a feedback controller for a quadrotor is designed using the proposed framework. Its performance is evaluated using numerical simulations and compared with a geometric controller.

Paper Structure

This paper contains 11 sections, 3 theorems, 59 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem statement and Preliminaries
  3. Preliminaries
  4. Neural Robust Control on Lie Groups Using Contraction Methods
  5. RCCMs on Lie groups
  6. Conditions for the existence of RCCM controllers
  7. Offline learning of RCCMs and feedback controller
  8. Online implementation of neural feedback controller
  9. Case Study: A Quadrotor Example
  10. Training and simulation results
  11. Conclusion

Key Result

Proposition 1

A closed-loop system satisfying Assumption assum: sys_lie admits a universal $\mathcal{L}_\infty$ gain bound of $\alpha>0$, if there exist a uniformly bounded symmetric metric $\boldsymbol{\mathcal{M}}(\boldsymbol{x})$, a feedback controller $\boldsymbol{u}$ of the form of eqn: fb_ctrl_format and co where $\dot{\boldsymbol{\mathcal{M}}} = \partial_{\dot{\boldsymbol{x}}}\boldsymbol{\mathcal{M}}$ wi

Figures (4)

  • Figure D1: Closed-loop system with a CCM feedback controller and a UDE.
  • Figure D2: Quadrotor tracking a spiral trajectory under the four test cases.
  • Figure D3: Output deviation norm under the four test cases.
  • Figure D4: Disturbance estimation error of UDE.

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1
  • proof
  • proof
  • Corollary 2
  • proof
  • Proposition 3
  • proof