As the beams at the interaction points (IPs) of circular colliders are pushed toward smaller sizes, the correction of the magnetic field errors in high- regions become increasingly important, but also challenging. This paper presents an algorithm developed to compute automatically local corrections. This algorithm has been successfully used in the LHC and in simulations of the HL-LHC to establish tolerances for the magnetic errors. The limitations of the current measurement technique (K-modulation) are studied, together with alternative techniques for control: computing the minimum near the IP using the betatron phase measured with new instrumentation and locating the beam waist via luminosity scans. This push toward smaller beam sizes also requires large -functions in the arcs that enhance local errors currently negligible. Experimental results of a way of correcting this new type of local errors using orbit bumps in sextupoles is also presented. These studies forecast a drastic change in the LHC commissioning strategy to be applied in the HL-LHC for needing luminosity measurements in intermediate stages.