Stability theory for adaptive systems: Methods 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 a (global) average. 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.