Stability of the filter whose gain is constructed on the basis of several leading singular modes of the system dynamics, is studied. It is shown that to ensure a stability of the filter, it is sufficient to construct the projection subspace from all unstable and neutral singular modes. The question on the modification of the structure of the gain in filter for adaptation purpose is also addressed. Simple numerical examples as well as experiments for assimilating the simulated data in the system of differential equations are presented to illustrate the theory and the efficiency of the adaptive filter.