The search for exoplanets is pushing adaptive optics (AO) systems on ground-based telescopes to their limits. One of the major limitations at small angular separations, exactly where exoplanets are predicted to be, is the servo-lag of the AO systems. The servo-lag error can be reduced with predictive control where the control is based on the future state of the atmospheric disturbance. We propose to use a linear data-driven integral predictive controller based on subspace methods that are updated in real time. The new controller only uses the measured wavefront errors and the changes in the deformable mirror commands, which allows for closed-loop operation without requiring pseudo-open loop reconstruction. This enables operation with non-linear wavefront sensors such as the pyramid wavefront sensor. We show that the proposed controller performs near-optimal control in simulations for both stationary and non-stationary disturbances and that we are able to gain several orders of magnitude in raw contrast. The algorithm has been demonstrated in the lab with MagAO-X, where we gain more than two orders of magnitude in contrast.