Data-Driven LQR with Uncertainty Refinement and Domain Randomization---Part~I: Certified Uncertainty Refinement

Overview & Synopsis

Learning controllers from trajectories corrupted by process and measurement noise is difficult when the system matrices and the noise covariances are unknown: noisy regressors bias least squares, and model uncertainty cannot be quantified without the noise covariance, which can only be estimated from residuals of the uncertain model. This paper, Part~I of a two-part study, resolves both obstacles with a finite-sample framework that certifies, and iteratively refines, uncertainty sets for the model, the noise in the collected data, and the noise covariance. A closed-loop parametrization based on delayed instrumental variables (IVs) removes the bias and assigns to every candidate model exactly one noise sequence that reproduces the data. Learning this noise breaks the circular dependence: the noise sequences give the covariance estimate directly, models whose noise violates the prior or the statistical certificates are discarded, and the surviving models yield a tighter covariance certificate. Several delayed instruments further tighten the model set and give a certified test of data consistency. We prove finite-sample containment of the true model, noise, and covariance, recursive nesting, and convergence at the IV rate. Part~II uses these certificates for domain-randomized linear quadratic regulator synthesis.