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In this study, the convergence of non-linear extended state observer (ESO) for a class of multi-input multi-output non-linear systems with uncertainty is studied. The unknown part that comes from either the system itself or the external disturbance is considered as an augmented state. The state variable and augmented state are estimated simultaneously through the ESO. It is shown that with the pertinent choice of non-linear functions for observer, the error between the state and observer can be as small as desired when the high-gain tuning parameter is sufficiently small. The current control for permanent-magnet synchronous motor is applied.