Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Flatness-Preserving Residuals for Pure-Feedback Systems

Fengjun Yang, Jake Welde, Nikolai Matni

TL;DR

This work proposes a framework for learning flatness-preserving residual dynamics in systems whose nominal model admits a pure-feedback form, and introduces a parameterization of flatness-preserving residuals using smooth function approximators, making them learnable from trajectory data with conventional algorithms.

Abstract

We study residual dynamics learning for differentially flat systems, where a nominal model is augmented with a learned correction term from data. A key challenge is that generic residual parameterizations may destroy flatness, limiting the applicability of flatness-based planning and control methods. To address this, we propose a framework for learning flatness-preserving residual dynamics in systems whose nominal model admits a pure-feedback form. We show that residuals with a lower-triangular structure preserve both the flatness of the system and the original flat outputs. Moreover, we provide a constructive procedure to recover the flatness diffeomorphism of the augmented system from that of the nominal model. We then introduce a learning algorithm that fits such residuals from trajectory data using smooth function approximators. Our approach is validated in simulation on a 2D quadrotor subject to unmodeled aerodynamic effects. We demonstrate that the resulting learned flat model enables tracking performance comparable to nonlinear model predictive control ($5\times$ lower tracking error than the nominal flat model) while also achieving over a $20\times$ speedup in computation.

Learning Flatness-Preserving Residuals for Pure-Feedback Systems

TL;DR

This work proposes a framework for learning flatness-preserving residual dynamics in systems whose nominal model admits a pure-feedback form, and introduces a parameterization of flatness-preserving residuals using smooth function approximators, making them learnable from trajectory data with conventional algorithms.

Abstract

We study residual dynamics learning for differentially flat systems, where a nominal model is augmented with a learned correction term from data. A key challenge is that generic residual parameterizations may destroy flatness, limiting the applicability of flatness-based planning and control methods. To address this, we propose a framework for learning flatness-preserving residual dynamics in systems whose nominal model admits a pure-feedback form. We show that residuals with a lower-triangular structure preserve both the flatness of the system and the original flat outputs. Moreover, we provide a constructive procedure to recover the flatness diffeomorphism of the augmented system from that of the nominal model. We then introduce a learning algorithm that fits such residuals from trajectory data using smooth function approximators. Our approach is validated in simulation on a 2D quadrotor subject to unmodeled aerodynamic effects. We demonstrate that the resulting learned flat model enables tracking performance comparable to nonlinear model predictive control ( lower tracking error than the nominal flat model) while also achieving over a speedup in computation.

Paper Structure

This paper contains 16 sections, 4 theorems, 50 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Learning Residual Dynamics
  4. Learning to Linearize Nonlinear Systems
  5. Contributions
  6. Preliminaries and Problem Formulation
  7. Differentially Flat Pure-Feedback Systems
  8. Flatness-Preserving Perturbations
  9. Learning Flatness-Preserving Residuals
  10. Numerical Experiments
  11. Setup
  12. Open-Loop Trajectory Evaluation
  13. Closed-Loop Tracking Control
  14. Conclusion
  15. Proof of Theorem \ref{['thm: regularity-unperturbed']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let the nominal model $\bar{f}$ satisfy Assumptions assm: nominal-pure-feedback and assm: nominal-dynamics-regular. Then $\bar{f}$ is locally differentially flat around $(\mathbf{x}^*, \mathbf{u}^*)$ with flat output $\mathbf{y} = \mathbf{x}_1$.

Figures (3)

  • Figure 1: Diagram of the 2D Quadrotor System.
  • Figure 2: Representative Open-Loop Trajectories: Adjusting the flatness diffeomorphism to account for the learned residual dynamics results in more accurate trajectory planning, leading to open-loop behavior that better matches the intended reference trajectory.
  • Figure 3: Closed-Loop Tracking Control Results: Figs. \ref{['fig: cl-circ']} and \ref{['fig: cl-lem']} show representative trajectories tracking the circular and lemniscate references, and Fig. \ref{['fig: compute-time']} shows mean and standard deviation of compute time per control update for both methods. The flatness-based controller with learned residual dynamics achieves similar performance to nonlinear MPC while requiring $20\times$ less time to compute control actions.

Theorems & Definitions (15)

  • Definition 1: See hagenmeyer2003exact
  • Remark 1
  • Theorem 1
  • proof : Proof Sketch
  • Example : Running Example: 2D Quadrotor
  • Definition 2: Lower-Triangular Residual Dynamics
  • Theorem 2
  • proof
  • Example : Running Example: 2D Quadrotor
  • Theorem 3
  • ...and 5 more