Data-Driven Synthesis of Robust Positively Invariant Sets from Noisy Data

This paper develops a method to construct robust positively invariant (RPI) tube sets from finite noisy input-state data of an unknown linear time-invariant (LTI) system, yielding tubes that can be directly embedded in tube-based robust data-driven predictive control. Data-consistency uncertainty sets are constructed under process/measurement noise with polytopic/ellipsoidal bounds. In the measurement-noise case, we provide a deterministic and data-consistent procedure to certify the induced residual bound from data. Based on these sets, a robustly stabilizing state-feedback gain is certified via a common quadratic contraction, which in turn enables constructive polyhedral/ellipsoidal RPI tube computation. Numerical examples quantify the conservatism induced by noisy data and the employed certification step.