Learning Flatness-Preserving Residuals for Pure-Feedback Systems

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. Building on these insights, we introduce a parameterization of flatness-preserving residuals using smooth function approximators, making them learnable from trajectory data with conventional algorithms. Our approach is validated in simulation on a 2D quadrotor subject to unmodeled aerodynamic effects. We demonstrate that the resulting learned flat model achieves a tracking error $5\times$ lower than the nominal flat model, while being $20\times$ faster over a structure-agnostic alternative.