Influence Functions for Data Attribution in Linear System Identification and LQR Control

When a controller is designed from an identified model, its performance ultimately depends on the trajectories used for identification, but pinpointing which ones help or hurt remains an open problem. We bring influence functions, a data attribution tool from machine learning, into this setting by chaining two closed form sensitivity analyses across a regularized least squares identification and an infinite horizon LQR pipeline. On the identification side, the quadratic loss admits an exact leave one trajectory out parameter shift and a reusable first order approximation with a Neumann series error bound. On the control side, we implicitly differentiate through the DARE via its discrete Lyapunov structure and compress the cost gradient to a single adjoint Lyapunov solve. The resulting scores track true LOTO retraining with Pearson correlations above 0.99 and speedups of 7 to 60 times on linear systems of dimension 2 to 10.