On the Use of the Harmonic Linearizaiton Method in the Automatic Control Theory

The method of harmonic linearization (harmonic balance), first proposed by N. M. Krylov and N. N. Bogolyubov for the approximate investigation of nonlinear vibrations, has been developed and received wide practical application to problems in the theory of automatic control. Recently, some doubt has been expressed on the legitimacy of application of the method to these problems, and assertions were made on the absence in them of a small parameter of any kind. Nevertheless, the method gives practical, acceptable results and is a simple and powerful means in engineering computations. Hence, the importance of questions arises as to its justification. The underlying principle of the method is the replacement of the given nonlinear equation by a linear equation. In establishing the method, a small parameter is considered whose presence makes it possible to speak, with some degree of approximation, of the solution of this new equation to the solution of the given nonlinear equation. In an article by the author, certain considerations were given on the presence of the small parameter, but this question has not as yet received a final answer. In the present report, a somewhat different approach to the problem is applied that permits: (a) establishing, in the clearest manner, the form of the presence of the small parameter in nonlinear problems of control theory, solvable by the method of harmonic linearization; (b) connecting it with previous intuitive physical concepts (with the "filter property") and extending the class of problems possessing this property; and (c) discussing various generalizations of the method.