arXiv:2604.14905v2 Announce Type: replace Abstract: This paper studies the data-driven synthesis of linear quadratic integral (LQI) controllers for continuous-time systems. The objective is to achieve optimal state-feedback control with integral action for reference tracking using only measured data. To this end, we derive a data-driven closed-loop parameterization of the augmented dynamics that incorporates the integral state while relying solely on input-state-output measurements of the underlying system. Based on this parameterization, a data-driven convex optimization problem is formulated whose solution yields the optimal linear quadratic regulator (LQR) feedback gain for the augmented system without explicit knowledge of the system matrices. In addition, a projected policy gradient flow, which is a preconditioned version of the model-based policy gradient flow, is derived to compute the optimal controller within the space of stabilizing gains. The proposed approach enables data-driven optimal tracking control while avoiding explicit state augmentation in the data collection phase. For time-varying gains, as they arise in an online implementation, the direct, incremental, and feedforward realizations of the LQI controller are compared, and the equilibrium required for feedforward is obtained from the same data. The method is illustrated on a distributed generation unit (DGU) in a DC microgrid, including an online implementation with closed-loop data under load changes, and by a Monte Carlo study on random systems with noisy data.