Skip to main content
SpringerLink
Log in
Book cover

Cyber-Physical Systems: Industry 4.0 Challenges pp 117–132Cite as

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

  • Chapter
  • First Online:

Part of the book series: Studies in Systems, Decision and Control ((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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  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. Wu, C.F.J., Hamada, M.S.: Experiments: Planning, Analysis, and Optimization, 743 p. Wiley (2009)

    Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  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. 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. 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. 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. Gaiduk, A.R.: Theory and Methods of Analytical Synthesis of Automatic Control Systems (Polynomial Approach), p. 415. Phizmatlit Publisher, Moscow (2012)

    Google Scholar 

  11. Loran, P.-J.: Approximation and Optimization, p. 496. World Publisher, Moscow (1975)

    Google Scholar 

  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. 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. Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and their Applications. Academic Press, New York (1967)

    MATH  Google Scholar 

  15. De Boor, C.: A Practical Guide to Splines. Springer (1978)

    Google Scholar 

  16. Micula, G., Micula, S.: Handbook of Splines. Kluwer Academic Publishers, Dordrecht, Boston, London (1999)

    Book  Google Scholar 

  17. Rawlings, J.O., Pantula, S.G., Dickey D.A.: Applied Regression Analysis: A Research Tool, 2nd edn., 659 p (1998)

    Google Scholar 

  18. Bates, D.M., Watts, D.G.: Nonlinear Regression Analysis and Its Applications. Wiley, New York (1988)

    Book  Google Scholar 

  19. Drapper, N.R., Smith, H.: Applied Regression Analysis, vol. 1. Wiley, New York (1981)

    Google Scholar 

  20. Drapper, N.R., Smith, H.: Applied Regression Analysis, vol. 2. Wiley, New York (1981)

    Google Scholar 

  21. Totik, V.: Orthogonal polynomials. Surv. Approx. Theory 1, 70–125 (2005)

    MathSciNet  MATH  Google Scholar 

  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. Powell, M.J.D.: The theory of radial basis function approximation. Adv. Num. Anal. II. OUP, Oxford (1992)

    Google Scholar 

  24. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press. http://catdir.loc.gov/catdir/samples/cam033/2002034983 (2003)

  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. 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. Ku, Y.H., Puri N.N.: On Lyapunov Functions of Order Nonlinear Systems. Journal Franklin Institution (1963)

    Google Scholar 

  28. Ku, Y.H.: Lyapunov function of a fourth-Order system. IEEE Trans. Autom. Control, 9, 276–278 (1064)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. 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 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Southern Federal University, Taganrog, Russia

    A. R. Gaiduk

  2. Don State Technical University, Rostov-on-Don, Russia

    R. A. Neydorf & N. V. Kudinov

Authors
  1. A. R. Gaiduk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. A. Neydorf
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. V. Kudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. R. Gaiduk .

Editor information

Editors and Affiliations

  1. Volgograd State Technical University, Volgograd, Russia

    Alla G. Kravets

  2. Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    Alexander A. Bolshakov

  3. Volgograd State Technical University, Volgograd, Russia

    Maxim V. Shcherbakov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gaiduk, A.R., Neydorf, R.A., Kudinov, N.V. (2020). Application of Cut-Glue Approximation in Analytical Solution of the Problem of Nonlinear Control Design. In: Kravets, A., Bolshakov, A., Shcherbakov, M. (eds) Cyber-Physical Systems: Industry 4.0 Challenges. Studies in Systems, Decision and Control, vol 260. Springer, Cham. https://doi.org/10.1007/978-3-030-32648-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-32648-7_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-32647-0

  • Online ISBN: 978-3-030-32648-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation