Geometric Closure of Linear Controlled Models via Tangent-Normal Decomposition

Linear models remain central in engineering for their simplicity and interpretability, but their local validity often requires data-driven closure to capture nonlinear effects. This paper proposes a geometric closure framework for nominal linear controlled models, where the mismatch with the true dynamics is decomposed into tangential and normal components with respect to a local chart. The correction is learned from data through a bounded dictionary and separate ridge regressions with asymmetric regularization. Unlike generic residual learning, the method preserves the linear backbone while providing an interpretable geometric description of model-form error. A chart-selection strategy is also included, allowing canonical or data-driven coordinates. The approach is validated on a synthetic benchmark and a vehicle-dynamics case, where it improves prediction accuracy and shows that the most suitable geometric representation is problem-dependent.