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

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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. ... continued below

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Popov, E. P. January 1, 1957.

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  • Main Title: On the Use of the Harmonic Linearizaiton Method in the Automatic Control Theory
  • Series Title: NACA Technical Memorandums

Description

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.

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  • Doklady Akademii Nauk (SSSR); Volume 106; No. 2; 211-214

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  • URL: http://hdl.handle.net/2060/20050019630 External Link
  • Report No.: NACA-TM-1406
  • Center for AeroSpace Information Number: 20050019630
  • Archival Resource Key: ark:/67531/metadc64471

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National Advisory Committee for Aeronautics Collection

The National Advisory Committee for Aeronautics (NACA) was a U.S. federal agency founded on March 3, 1915 to undertake, promote, and institutionalize aeronautical research. On October 1, 1958 the agency was dissolved, and its assets and personnel transferred to the newly created National Aeronautics and Space Administration (NASA).

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Creation Date

  • January 1, 1957

Added to The UNT Digital Library

  • Nov. 17, 2011, 5:13 p.m.

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  • Feb. 6, 2017, 4:15 p.m.

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Popov, E. P. On the Use of the Harmonic Linearizaiton Method in the Automatic Control Theory, report, January 1, 1957; (digital.library.unt.edu/ark:/67531/metadc64471/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.