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The performance of set-membership normalized least mean-square (SM-NLMS) adaptive filters deteriorates significantly in the presence of impulsive noise or interference. To solve this problem a new robust SM-NLMS (RSM-NLMS) algorithm is proposed. In addition, a framework to achieve robust performance in other algorithms of the set-membership (SM) family is developed. The proposed RSM-NLMS algorithm is compared with the conventional SM-NLMS and the robust normalized least mean-square (RNLMS) algorithms in impulsive-noise environments. Simulation results show that (1) the proposed RSM-NLMS algorithm has similar robustness with respect to impulsive noise as the RNLMS algorithm, (2) the RSM-NLMS and the conventional SM-NLMS algorithms offer reduced steady-state misalignment for the same convergence speed as compared to the RNLMS algorithm, and (3) the amount of computation is significantly reduced in the RSM-NLMS algorithm as it takes fewer weight updates to converge than the SM-NLMS and the RNLMS algorithms.