Stability theory for adaptive systems: Method of averaging and persistency of excitation

A method of averaging is developed for the stability analysis of linear differential equations with small time-varying coefficients which do not necessarily possess an average value. The technique is then applied to determine the stability of a linear equation which arises in the study of adaptive systems where the adaptive parameters are slowly varying. The stability conditions are stated in the frequency domain, which shows the relation between persistent excitation and unmodeled dynamics.