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The iterative learning control (ILC) has been demonstrated to be capable of considerably improving the tracking performance of systems which are affected by the iteration-independent disturbance. However, the achievable performance is greatly degraded when iteration-dependent, stochastic disturbances are presented. This paper considers the robustness of the ILC algorithm for the nonlinear system in presence of stochastic measurement disturbances. The robust convergence of the P-type ILC algorithm is firstly addressed, and then an improved ILC algorithm with a decreasing gain is proposed. Theoretical analyses show that the proposed algorithm can guarantee that the tracking error of the nonlinear system tends to zero in presence of measurement disturbances. The analysis is also supported by a numerical example.