On two time-scales in linear systems with special reference to adaptive control

Abstract : A comparison is made of various methods presented in the literature for treating Linear systems with time-varying parameters. These variations are considered to arise both from small-amplitude signals of arbitrary frequency and also from signals which are much slower than the dominant signals of the system. Two time-domain approaches and one frequency-domain approach are shown to be equivalent, resulting in identical truncated series solutions, expressed in terms of a single time variable. The method is extended to the case where the slow variations are expressed as functions of a second-time-variable. The result is shown to be equivalent to a two time-scale approach described in the literature. A general unified approach is suggested encompassing all the above methods, whether for single- or double-time-scale series solutions. (Author)