Influence Functions for Data Attribution in System Identification and LQR Control

When a controller is designed from an identified model, its performance ultimately depends on the trajectories used for identification, yet 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 linear quadratic regulator (LQR) pipeline. On the identification side, the quadratic loss admits an exact leave-one-trajectory-out (LOTO) parameter shift; a reusable first-order approximation follows with a Neumann-series error bound. On the control side, we implicitly differentiate through the discrete algebraic Riccati equation (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, at 7× to 60× the speed, on linear systems of dimension 2 to 10, while a nonlinear benchmark confines the method to nominal, surrogate-level attribution.