The paper presents an adaptive controller for discrete-time systems, with parameters modelled as a Gauss-Markov process with an unknown noise covariance matrix. The cost function adopted in the optimisation of the control performance is the sum of the output variances up to M steps ahead in time. Optimal predictors are used to estimate the future outputs y(k + i), i = 1, 2,.... M, that are needed in the solution of the optimisation problem that yields the value of the control signal at a given time k. The estimates for the system parameters are obtained using a Kalman filter, together with an algorithm to tune the covariance matrix in real time. The adaptation mechanism reduces the risk of divergence of the Kalman filter, as shown by simulation results that illustrate the actual performance of the new controller under uncertainty in the noise covariance matrix.