An autoregressive (AR) Gaussian time series is observed in the presence of additive white Gaussian noise. It is assumed that some of the parameters of the system are unknown. The goal is to construct an adaptive Kalman filter and describe the errors of approximation in the asymptotic framework of large samples. This filter is realized in several steps. First, the model of observations and the corresponding Kalman filter are introduced. Then, the asymptotic properties of the estimators obtained by the method of moments are described. In the next step, these estimators are used to construct the One-step MLE-process. The final step involves substituting the MLE-process into the Kalman filtering equations. The question of the asymptotic efficiency of this approximation is also discussed. The asymptotic properties of the MME, MLE, and BE estimators are described as well.