This report deals with the problem of designing an adaptive observer for estimating unknown periodical disturbances. This is very practical problem because in the area of control of servomechanisms such disturbances are always encountered. When the disturbance cannot be directly measured or eliminated at the source it is necessary to perform a prediction. When a periodical disturbance is present the frequencies appear as unknown parameters and they have to be identified. In order to identify the unknown parameters, it is necessary to transform the composite system model, which contains the models of the controlled system and the disturbances, into observable canonical form. In addition, an inverse transformation is required to calculate the estimates of the present disturbances. In this report, firstly, a review of an adaptive observer for estimation of unknown periodical disturbances is presented. Later a calculation of the disturbance estimate is derived using the algebraic programming system REDUCE. The proposed method here allows to perform all the necessary transformations and to obtain the disturbance estimation without using the transformation matrix. The calculations of these transformations are complicated and, hitherto, there is no simple method to perform them. The results of disturbance estimation are illustrated by two examples.