FAST RATIONAL INTERPOLATION OF TRANSFER FUNCTIONS OF LINEAR DYNAMIC SYSTEMS WITH DISTRIBUTED PARAMETERS
DOI:
https://doi.org/10.15588/1607-3274-2020-4-5Keywords:
Rational interpolation, linear dynamic system, transmission function, distributed parameter system, discrete Fourier transform.Abstract
Contex. Fast method of rational interpolation of the transfer function of linear dynamical systems with distributed parameters is described, the values of which can be found by numerical methods or by calculating the transcendental functions of the Laplace integral transform variable. The method makes it possible to determine explicitly the transfer function and, in particular, the characteristic equation of such a degree, which is sufficient to meet the accuracy requirements when calculating the root quality criteria for the dynamics of automatic control systems.
Objective. According to the proposed method, rational interpolation is reduced to solving a system of linear equations, the order of which is much lower (more than twice) the order of similar systems used for rational interpolation of functions by known methods. The properties of this system are such that its solution can be obtained by special fast methods of the quadratic order of complexity.
Method. An iterative algorithm for calculating the transfer function coefficients of a linear dynamic system with distributed parameters is carried out using the methods of complex variable functions theory using the discrete Laplace transform. The proposed approach made it possible to significantly speed up the calculations by decomposing the system of linear equations with respect to the coefficients of the transfer function to a system of approximately half the order, which allows a quick solution by the methods of Trench, Berlekamp-Massey, or Euclid.
Results. An example of the practical use of an iterative algorithm for rational interpolation and calculation with a given accuracy of the root quality criteria for the dynamics of a support with gas lubrication is considered.
Conclusions. The method allows to define explicitly the characteristic equation of such a degree, which is sufficient to meet the accuracy requirements when calculating the root quality criteria for the dynamics of automatic control systems. Rational interpolation is reduced to solving a system of linear equations, the order of which is much lower (more than twice) the order of similar systems used for rational interpolation of functions by known methods. The properties of the system are such that its solution can be obtained by special fast methods of the quadratic order of complexity.
References
Fraleigh J. B., Beauregard R. A. Linear Algebra. AddisonWesley, 1995, 608 p.
Riley K. F., Hobson M. P., Bence S. J. Mathematical methods for physics and engineering. Cambridge University Press, 2010, 455 p.
Middlebrook R. D. Input filter considerations in design and application of switching regulators, IEEE Industry Applications Society Annual Meeting, 1976, pp. 366–382.
Carrol J. An input impedance stability criterion allowing more flexibility for multiple loads which are independently designed, Naval Air Warfare Center, Aircraft Division, Indianapolis, B/812, 1992.
Wildrick C. M., Lee F. C., Cho B. H., Choi B. A method of defining the load impedance specification for a stable distributed power system, IEEE Transactions on Power Electronics, 1995, pp. 280–285.
Kodnyanko V. A. Stability of energy-saving adaptive radial hydrostatic bearing with limitation of the output flow of lubricant, Journal of the Siberian Federal University. Technics and technology, Krasnoyarsk, 2011, Vol. 6, No. 4, pp. 907–914.
Bradie B.A. Friendly Introduction to Numerical Analysis, Upper Saddle River, New Jersey: Pearson Prentice Hall, 2006.
Smith S.W. Chapter 8: The Discrete Fourier Transform, The Scientist and Engineer's Guide to Digital Signal Processing, San Diego, Calif.: California Technical Publishing, 1999.
Golub G. H., Van Loan C. F. Matrix computations, John Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.
Heinig G., Rost K., Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations, Structured Matrices in Mathematics, Computer Science, and Engineering, II, Contemp. Math., Amer. Math. Soc., Providence, RI, 2001, vol. 281, pp. 247– 264.
Beale E. M. L. Cycling in the dual simplex algorithm, Naval Research Logistics Quarterly, 1955, 2 (4), pp. 269–276. DOI: 10.1002/nav.3800020406.
Voevodin V. V., Tyrtyshnikov E. E. Toeplitz matrices and their applications, Computing Methods in Applied Sciences and Engineering, North-Holland, Amsterdam, 1984, pp. 7585.
Trench W. F. An algorithm for the inversion of finite Hankel matrices, SIAMJ. Appl. Math, 13, 1965, pp. 1102–1107.
Zohar S. Toeplitz matrix inversion: The algorithm of W. F. Trench, J. Assoc. Comput, Mach.16, 1967, pp. 592–601.
Petrov O. A. A fast algorithm for solving systems of equations with a Toeplitz matrix, Infocommunication technologies, 2006, Vol. 4, No. 1, pp. 57–59.
Rahman Q. I., Schmeisser G. Analytic theory of polynomials, London Mathematical Society Monographs, New Series, 26, Oxford: Oxford University Press, 2002.
Besekersky V. A., Popov E. P. Theory of automatic control systems, Saint Petersburg, Profession, 2003,752 p.
Kodnyanko V. A. Numerical calculation of the static characteristics of a single-row slotted gas-static suspension, Problems of mechanical engineering and reliability of machines, 2002, No. 2, pp. 17–19.
Constantinescu V. N. Gas Lubrication, New York: American Society of Mechanical Engineers, 1969.
Muir T. A treatise on the theory of determinants, Dover Publications, 1960, pp. 516–525.
Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. Rational Function Interpolation and Extrapolation, Numerical Recipes in C. Cambridge: Cambridge University Press, 1994.
Blahut R.E. Fast Algorithms for Signal Processing, Cambridge University Press, 2010, 469 p. DOI: 10.1017/CBO9780511760921.
Downloads
How to Cite
Issue
Section
License
Copyright (c) 2020 V. A. Kodnyanko
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Creative Commons Licensing Notifications in the Copyright Notices
The journal allows the authors to hold the copyright without restrictions and to retain publishing rights without restrictions.
The journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles.
The journal allows to reuse and remixing of its content, in accordance with a Creative Commons license СС BY -SA.
Authors who publish with this journal agree to the following terms:
-
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License CC BY-SA that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
-
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
-
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
