Adaptive control with selective memory updates the parameter estimates only when there is new information present. The information content increases and the estimator eventually stops. In this paper we use the Fisher information matrix and a variance estimator to measure the information content. The parameter estimates are updated only when the Information Matrix and/or the estimated variance of the prediction error increases. This gives an algorithm for adaptive control which converges to a linear time invariant controller. The resulting controller is robust with respect to bounded disturbances and small model/plant mismatch. Knowledge about overbounding sets for the parameter estimates or the disturbances are not needed. Simulation results using theoretical test cases illustrate near optimal performance for the converged parameters.

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