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The learning gain, for a selected learning algorithm, is derived based on minimizing the trace of the input error covariance matrix for linear time-varying systems. It is shown that, if the product of the input/output coupling matrices is a full-column rank, then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances. However, the state error covariance matrix converges uniformly to zero in presence of measurement noise. Moreover, it is shown that, if a certain condition is met, then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm. The proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance. The application of this algorithm to a class of nonlinear systems is also considered. A numerical example is included to illustrate the performance of the algorithm.