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An improved set-membership affine-projection (AP) adaptive-filtering algorithm is proposed. The new algorithm uses two error bounds that are estimated during the learning phase and by this means significantly reduced steady-state misalignment is achieved as compared to those in the conventional AP and set-membership AP algorithms while achieving similar convergence speed and re-adaptation capability. In addition, the proposed algorithm offers robust performance with respect to the error bound, projection order, impulsive-noise interference, and in tracking abrupt changes in the underlying system. These features of the proposed algorithm are demonstrated through extensive simulation results in system-identification and echo-cancellation applications.