Learning Flatness-Preserving Residuals for Pure-Feedback Systems (2504.04324v2)

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.

• The paper introduces a novel learning framework that integrates lower-triangular residuals to preserve differential flatness in pure-feedback systems.
• It demonstrates significant improvements, achieving approximately five times better tracking accuracy and twenty times faster computation relative to NMPC.
• The method employs neural network-based learning on trajectory data to construct augmented diffeomorphisms while maintaining flat outputs for efficient control.

Learning Flatness-Preserving Residuals for Pure-Feedback Systems

Abstract

The paper investigates the augmentation of nominal models for differentially flat systems with learned residuals to account for unmodeled dynamics. The focus is on preserving flatness, a property crucial for efficient trajectory planning and control. The authors propose a framework specific to systems that can be expressed in pure-feedback form, demonstrating that using lower-triangular residual structures preserves flatness. This ensures that flat outputs remain unchanged and allows constructing flatness diffeomorphisms for the augmented system. The research is validated through simulations on a 2D quadrotor affected by aerodynamic disturbances, showcasing tracking enhancements and computational efficiency improvements compared to existing nonlinear model predictive control approaches.

Introduction

Differential flatness simplifies control of nonlinear systems by transforming these systems into chains of integrators. This is particularly beneficial for systems like quadrotors, where flat outputs facilitate easier planning and feedback design. However, typical flat representations are based on ideal models and can fail to address practical disturbances.

The proposed research addresses this by learning residual dynamics that preserve the flatness of the nominal model. The paper restricts its paper to pure-feedback form systems, wherein a systematic condition on residuals ensures the maintenance of differential flatness. The residuals, primarily learned from trajectory data using smooth function approximators, are structured into lower-triangular forms.

Key Contributions

1. Flatness-preserving residuals: Identification of residual dynamics that maintain the differential flatness in pure-feedback systems.
2. Learning algorithm: Development of a framework for parameterizing residual dynamics that allow for the preservation of flat outputs and their diffeomorphisms.
3. Empirical validation: Implementation of the proposed method using a 2D quadrotor model. Results show significant improvement in tracking performance and computational efficiency.

Preliminaries and Problem Formulation

The core of the research lies in the exploration of dynamical systems that can be expressed in a pure-feedback form, defined as sequences of integrators. The residual dynamics are structured to preserve this form, ensuring that essential properties like flatness remain intact after augmentation.

Definitions

• Residual Dynamics: The difference between actual system dynamics and the nominal model.
• Pure-feedback Form: A representation of system dynamics integrating sub-states of equal relative degrees.
• Differential Flatness: The existence of a flat output through which the entire state and control inputs can be reconstructed via a finite sequence of their derivatives.

Main Results

Flatness-Preserving Perturbations

The paper introduces lower-triangular residual dynamics that retain the structural format necessary for differential flatness. This approach allows the augmented model to maintain computational advantages while reflecting real-world dynamics more accurately.

Figure 1: Diagram of the 2D Quadrotor System.

Constructing Flatness Diffeomorphisms

Given a nominal model's diffeomorphisms, the paper provides a constructive procedure for obtaining augmented diffeomorphisms post residual incorporation. This procedure ensures that the transformed system can utilize flatness-based algorithms for efficient computation.

Learning Algorithm

The implementation involves parameterizing residual dynamics using neural networks that operate on trajectory data to fine-tune the model while enforcing the prescribed structural conditions.

Experiments

Open-Loop Evaluation

Experiments on trajectory generation, specifically focusing on circular and lemniscate paths, show significantly reduced position errors when comparing the augmented model against the nominal approach.

Figure 2: Circular Trajectory.

Closed-Loop Tracking Control

A comparison between flatness-based and nonlinear model predictive control reveals that the former achieves near-equivalent tracking precision with orders of magnitude less computational demand.

Figure 3: Lemniscate Trajectory.

Figure 4: Average Compute Time Comparison.

Performance Metrics

Notably, the learned flat model enhances tracking accuracy by approximately five times over the nominal approach and achieves a speedup factor of twenty in computation time when contrasted with NMPC.

Conclusion

This research provides crucial insights into leveraging learned residuals for ensuring flatness in feedback systems. It demonstrates both theoretical soundness and practical applicability by aligning computational demands with real-world scenarios. Future directions could explore relaxing structural assumptions and deploying adaptive controllers for dynamic environments.