We propose the correlation-locking optimization scheme (CLOSE), a real-time adaptive filtering technique for adaptive optics (AO) systems controlled with integrators. CLOSE leverages the temporal autocorrelation of modal signals in the controller telemetry and drives the gains of the integral command law in a closed servo-loop. This supervisory loop is configured using only a few scalar parameters and automatically controls the modal gains to closely match transfer functions achieving minimum variance control. This optimization is proven to work throughout the range of noise and seeing conditions relevant to the AO system. This technique was designed while preparing the high-order AO systems for extremely large telescopes, in particular to tackle the optical gain (OG) phenomenon. This is a sensitivity reduction induced by on-sky residuals and is a prominent issue with pyramid wavefront sensors (PWFS). CLOSE follows upon the linear modal compensation approach to OG that was previously demonstrated to substantially improve AO correction with high-order PWFS systems. Operating on modal gains through multiplicative increments, CLOSE naturally compensates for the recurring issue of unaccounted sensitivity factors throughout the AO loop. We present end-to-end simulations of the MICADO instrument single-conjugate AO to demonstrate the performances and capabilities of CLOSE. We demonstrate that a single configuration provides an efficient and versatile optimization of the modal integrator while accounting for OG compensation and while providing significant robustness to transient effects impacting the PWFS sensitivity.