From Model-Based to Data-Driven Linear Matrix Inequalities and back: a Routine for Robust Design Certificates

This paper presents a systematic data-driven framework for the robust control of Linear Parameter-Varying (LPV) systems with unknown system matrices. Bypassing traditional system identification, we propose a direct conversion procedure that translates standard model-based Linear Matrix Inequalities (LMIs) into equivalent data-driven conditions. The proposed method leverages the Matrix S-Lemma to characterize the set of all systems consistent with a finite set of noisy input-state data, effectively replacing the explicit state-space dependencies with data-dependent constraints. This formulation provides a robust guarantee for the solution to satisfy the original performance specifications for the true underlying system. The effectiveness of the procedure is validated through numerical simulations on two problem examples: the design of an $\mathcal{H}_\infty$ optimal controller, and the stabilization of input-saturated systems using a sector condition. Comparative results demonstrate that the data-driven solution rigorously satisfies the model-based conditions, providing a sufficient guarantee for the original design requirements despite data uncertainty.