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Application of Cut-Glue Approximation in Analytical Solution of the Problem of Nonlinear Control Design

  • A. R. GaidukEmail author
  • R. A. Neydorf
  • N. V. Kudinov
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 260)

Abstract

To create control systems for various objects, their mathematical models are used. They are obtained, often experimentally, by approximating arrays of numerical data. With significant non-linearity of the data, they are approximated at separate sites. However, such a fragmentary model of a nonlinear object as a whole is not analytical, which excludes the use of most methods for the synthesis of nonlinear controls. In such a situation, there is the prospect of applying the Cut-Glue approximation method, which allows us to obtain a common object model as a single analytical function. The chapter considers the theory and application of this method to the synthesis of nonlinear control. The mathematical model obtained by the Cut-Glue approximation method is reducing to a quasilinear form, which makes it possible to find a nonlinear control by an analytical method.

Keywords

Experimental data Fragmentary approximation Analytical function Multiplicative isolation function  Additive union Mathematical model Quasilinear form Analytical synthesis Control system 

Notes

Acknowledgements

The Russian Fund of Basic Research (grant No. 18-08-01178\19) supported this research.

References

  1. 1.
    Dyshlyuk, E.N., Kotlyarov, R.V., Pachkin, S.G.: Methods of structural and parametric identification of control objects on the example of the furnace emulator EP10. Equip. Technol. Food Prod. 47(4), 159–165 (2017).  https://doi.org/10.21179/2074-9414-2017-4-159-165
  2. 2.
    Wu, C.F.J., Hamada, M.S.: Experiments: Planning, Analysis, and Optimization, 743 p. Wiley (2009)Google Scholar
  3. 3.
    Voevoda, A.A., Bobobekov, K.M., Troshina, G.V.: The parameters identification of the automatic control system with the controller. J. Phys.: Conf. Ser. 1210(art. 012021) (2019)Google Scholar
  4. 4.
    Neidorf, R.A., Neydorf, A.R.: Modeling of essentially non-linear technical systems by the method of multiplicatively additive fragmentary approximation of experimental data. Inf. Space 1, 47–57 (2019)Google Scholar
  5. 5.
    Neydorf, R., Neydorf, A., Vučinić, D.: Cut-glue approximation method for strongly nonlinear and multidimensional object dependencies modeling. Ad. Struct. Mater. 72, 155–173 (2018)CrossRefGoogle Scholar
  6. 6.
    Neydorf, R., Neydorf, A.: Technology of cut-glue approximation method for modeling strongly nonlinear multivariable objects. Theoretical Bases and Prospects of Practical Application. SAE Technical Paper 2016–01-2035 (2016).  https://doi.org/10.4271/2016-01-2035
  7. 7.
    Neydorf, R., Yarakhmedov, O., Polyakh, V., Chernogorov, I., Vucinic, D.: Cut-glue approximation based on particle swarm sub-optimization for strongly nonlinear parametric dependencies of mathematical models. In: Improved Performance of Materials. Design and Experimental Approaches, pp. 185–196. Springer (2018)Google Scholar
  8. 8.
    Gaiduk, A.R.: Algebraic synthesis of nonlinear stabilizing controls. In: Synthesis of Algorithms of Complex Systems, vol. 7, pp. 15–19. TRTI Publisher, Taganrog (1989)Google Scholar
  9. 9.
    Gaiduk, A.R., Stojković, N.M.: Analytical design of quasilinear control systems. Facta Univ. Ser.: Autom. Control Robot 13(2), 73–84 (2014)Google Scholar
  10. 10.
    Gaiduk, A.R.: Theory and Methods of Analytical Synthesis of Automatic Control Systems (Polynomial Approach), p. 415. Phizmatlit Publisher, Moscow (2012)Google Scholar
  11. 11.
    Loran, P.-J.: Approximation and Optimization, p. 496. World Publisher, Moscow (1975)Google Scholar
  12. 12.
    Insung, I., Naylor, B.: Piecewise linear approximations of digitized space curves with applications. Computer Science Technical Reports, Report Number 90–1036, Paper 37 (1990)Google Scholar
  13. 13.
    Pinheiro, A.M.G., Ghanbari, M.: Piecewise approximation of contours through scale-space selection of dominant points. IEEE Trans. Image Process 19(6) (2010)Google Scholar
  14. 14.
    Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and their Applications. Academic Press, New York (1967)zbMATHGoogle Scholar
  15. 15.
    De Boor, C.: A Practical Guide to Splines. Springer (1978)Google Scholar
  16. 16.
    Micula, G., Micula, S.: Handbook of Splines. Kluwer Academic Publishers, Dordrecht, Boston, London (1999)CrossRefGoogle Scholar
  17. 17.
    Rawlings, J.O., Pantula, S.G., Dickey D.A.: Applied Regression Analysis: A Research Tool, 2nd edn., 659 p (1998)Google Scholar
  18. 18.
    Bates, D.M., Watts, D.G.: Nonlinear Regression Analysis and Its Applications. Wiley, New York (1988)CrossRefGoogle Scholar
  19. 19.
    Drapper, N.R., Smith, H.: Applied Regression Analysis, vol. 1. Wiley, New York (1981)Google Scholar
  20. 20.
    Drapper, N.R., Smith, H.: Applied Regression Analysis, vol. 2. Wiley, New York (1981)Google Scholar
  21. 21.
    Totik, V.: Orthogonal polynomials. Surv. Approx. Theory 1, 70–125 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Khrushchev, S.: Orthogonal Polynomials and Continued Fractions From Euler’s Point of View. Atilim University, Turkey: Cambridge University Press. www.cambridge.org/9780521854191 (2008)
  23. 23.
    Powell, M.J.D.: The theory of radial basis function approximation. Adv. Num. Anal. II. OUP, Oxford (1992)Google Scholar
  24. 24.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press. http://catdir.loc.gov/catdir/samples/cam033/2002034983 (2003)
  25. 25.
    Neydorf, R., Sigida, Y., Voloshin, V., Chen, Y.: Stability analysis of the MAAT feeder airship during ascent and descent with wind disturbances. SAE Technical Paper 2013–01-2111 (2013).  https://doi.org/10.4271/2013-01-2111
  26. 26.
    Voloshin, V., Chen, Y., Neydorf, R., Boldyreva, A.: Aerodynamic characteristics study and possible improvements of MAAT feeder airships. SAE Technical Paper 2013-01-2112 (2013).  https://doi.org/10.4271/2013-01-2112
  27. 27.
    Ku, Y.H., Puri N.N.: On Lyapunov Functions of Order Nonlinear Systems. Journal Franklin Institution (1963)Google Scholar
  28. 28.
    Ku, Y.H.: Lyapunov function of a fourth-Order system. IEEE Trans. Autom. Control, 9, 276–278 (1064)Google Scholar
  29. 29.
    Chowdhary, G., Yucelen, T., Muhlegg, M., Johnson, E.: Concurrent learning adaptive control of linear systems with exponentially convergent bounds. Int. J. Adapt. Control Sig. Process. 27(4), 280–301 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhu, Y., Hou, Z.: Controller dynamic linearisation-based model-free adaptive control framework for a class of non-linear system. IET Control Theory Appl. 9(7), 1162–172 (2015)Google Scholar
  31. 31.
    Shao, Z., Zheng, C., Efimov, D., Perruquetti, W.: Identification, estimation and control for linear systems using measurements of higher order derivatives. J. Dyn. Syst. Measur. Control Am. Soc. Mech. Eng. 139(12), 1–6 (2017)Google Scholar
  32. 32.
    Jayawardhana, R.N., Ghosh, B.K.: Kalman filter based iterative learning control for discrete time MIMO systems. In: Proceedings of 30th Chinese Control and Decision Conference, pp. 2257–2264 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Southern Federal UniversityTaganrogRussia
  2. 2.Don State Technical UniversityRostov-on-DonRussia

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