Skip to main content
SpringerLink
Log in

Selected Engineering Applications of Fractional-Order Calculus

  • Chapter
  • First Online:
Fractional Dynamical Systems: Methods, Algorithms and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 402))

Abstract

In this chapter several examples of using fractional-order calculus in selected engineering applications are presented. It is shown that some real systems can be better mathematically described with fractional-order differential equations. The chapter focuses on ladder network structures with fractional-order elements to model both electrical and nonelectrical systems with distributed parameters. Examples of modeling of supercapacitors, batteries, a chain of vehicles functioning in adaptive cruise control mode, and thermal processes inside buildings are provided. The effectiveness of the proposed modeling approach is verified by both simulation and experimental results.

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

Access this chapter

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
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

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrawal, O.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29(1–4), 145–155 (2002)

    Article  MathSciNet  Google Scholar 

  2. Aoki, Y., Sen, M., Paolucci, S.: Approximation of transient temperatures in complex geometries using fractional derivatives. Heat Mass Transf. 44(7), 771–777 (2008). ISSN 0947-7411. https://doi.org/10.1007/s00231-007-0305-0

  3. Bauer, W., Baranowski, J.: Fractional PI\(^\lambda \)D controller design for a magnetic levitation system. Electronics 9(12), 2135 (2020)

    Article  Google Scholar 

  4. Bauer, W., Mitkowski, W., Zagorowska, M.: RC-ladder networks with supercapacitors. Arch. Electr. Eng. (2018)

    Google Scholar 

  5. Bohaienko, V., Bulavatsky, V.: Fractional-fractal modeling of filtration-consolidation processes in saline saturated soils. Fractal Fract. 4(4), 59 (2020)

    Article  Google Scholar 

  6. Burke, A.: Ultracapacitor technologies and application in hybrid and electric vehicles. Int. J. Energy Res. 34(2), 133–151 (2010)

    Article  Google Scholar 

  7. Clarke, J.: Energy Simulation in Building Design, 2 edn. Butterworth-Heinemann (2001). ISBN 9780750650823. http://books.google.com/books?id=ksNIQ4kx6UIC

  8. Crabb, J.A., Murdoch, N., Penman, J.M.: A simplified thermal response model. Build. Serv. Eng. Res. Technol. 8(1), 13–19 (1987)

    Article  Google Scholar 

  9. Diethelm, K., Ford, N., Freed, A.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Article  MathSciNet  Google Scholar 

  10. Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16(3–4), 231–253 (1997)

    Article  MathSciNet  Google Scholar 

  11. Dlugosz, M., Skruch, P.: The application of fractional-order models for thermal process modelling inside buildings. J. Build. Phys. 39(5), 440–451 (2016)

    Article  Google Scholar 

  12. Gabano, J.D., Poinot, T.: Fractional modelling applied to heat conductivity and diffusivity estimation. Physica Scripta 2009(T136), 014015 (2009). http://stacks.iop.org/1402-4896/2009/i=T136/a=014015

  13. Gouda, M., Danaher, S., Underwood, C.: Low-order model for the simulation of a building and its heating system. Build. Serv. Eng. Res. Technol. 21(3), 199–208 (2000)

    Article  Google Scholar 

  14. Huseynov, I., Ahmadovay, A., Ojo, G., Mahmudov, N.: A natural extension of Mittag-Leffler function associated with a triple infinite series. arXiv preprint arXiv:2011.03999 (2020)

  15. Ilin, W., Kuznyetsow, Y.: Tridiagonal matrices and their applications. Science, Moscow (1985)

    Google Scholar 

  16. Kaczorek, T.: Positive fractional linear systems. Pomiary Automatyka Robotyka 2, 91–112 (2011)

    Google Scholar 

  17. Kaczorek, T.: Selected Problems of Fractional Systems Theory, vol. 411. Springer (2011)

    Google Scholar 

  18. Lakeb, A., Kaisserli, Z., Bouagada, D.: Computing the \(h_2\)-norm of a fractional-order system using the state-space linear model. Kragujevac J. Math. 47(4), 531–538 (2023)

    Google Scholar 

  19. Ljung, L.: System Identification Toolbox. The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 (2013)

    Google Scholar 

  20. Lopes, A., Machado, J.: A review of fractional order entropies. Entropy 22(12), 1374 (2020)

    Article  MathSciNet  Google Scholar 

  21. Mitkowski, W.: Uniform spatial networks RC. Arch. Electrotech. 22(2), 398–405 (1973)

    Google Scholar 

  22. Mitkowski, W.: Finite-dimensional approximations of distributed RC networks. Bull. Pol. Acad. Sci. Tech. Sci. 263–269 (2014)

    Google Scholar 

  23. Mitkowski, W., Bauer, W.: Comparison of non-integer PID, PD and PI controllers for DC motor. In: Advanced, Contemporary Control, pp. 904–913. Springer (2020)

    Google Scholar 

  24. Mitkowski, W., Obraczka, A.: Simple identification of fractional differential equation. In: Solid State Phenomena, vol. 180, pp. 331–338. Trans Tech Publications (2012)

    Google Scholar 

  25. Mitkowski, W., Oprzedkiewicz, K.: Modelling and control of heat conduction processes. In: Automatic Control, Robotics, and Information Processing, pp. 767–789. Springer (2021)

    Google Scholar 

  26. Mitkowski, W., Skruch, P.: Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Pol. Acad. Sci. Tech. Sci. 61(3) (2013)

    Google Scholar 

  27. Obraczka, A., Mitkowski, W.: The comparison of parameter identification methods for fractional, partial differential equation. In: Solid State Phenomena, vol. 210, pp. 265–270. Trans Tech Publications (2014)

    Google Scholar 

  28. Oprzedkiewicz, K.: Non integer order, state space model of heat transfer process using Atangana-Baleanu operator. Bull. Pol. Acad. Sci. Tech. Sci. 43–50 (2020)

    Google Scholar 

  29. Oprzedkiewicz, K., Dziedzic, K., Wieckowski, L.: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator. Bull. Pol. Acad. Sci. Tech. Sci. 905–914 (2019)

    Google Scholar 

  30. Oprzedkiewicz, K., Podsiadlo, M., Dziedzic, K.: Integer order vs fractional order temperature models in the forced air heating system

    Google Scholar 

  31. Pellegrino, E., Pezza, L., Pitolli, F.: A collocation method based on discrete spline quasi-interpolatory operators for the solution of time fractional differential equations. Fractal Fract. 5(1), 5 (2021)

    Article  Google Scholar 

  32. Pillai, R.: On Mittag-Leffler functions and related distributions. Ann. Inst. Stat. Math. 42(1), 157–161 (1990)

    Article  MathSciNet  Google Scholar 

  33. Podlubny, I.: Fractional Differential Equation. Rinehart, and Winston, San Diego, USA, SHolt (1999)

    Google Scholar 

  34. Sierociuk, D., Dzielinski, A., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 371, 2013 (1990)

    MATH  Google Scholar 

  35. Sierociuk, D., Petráš, I.: Modeling of heat transfer process by using discrete fractional-order neural networks. In: 2011 16th International Conference on Methods and Models in Automation and Robotics, MMAR 2011, pp. 146–150 (2011)

    Google Scholar 

  36. Skruch, P., Dlugosz, M., Mitkowski, W.: Stability analysis of a series of cars driving in adaptive cruise control mode. In: Trends in Advanced Intelligent Control, Optimization and Automation, vol. 577, pp. 168–177. Springer (2017)

    Google Scholar 

  37. Tepljakov, A., Petlenkov, E., Belikov, J.: Fomcom: a matlab toolbox for fractional-order system identification and control. Int. J. Microelectron. Comput. Sci. 2(2), 51–62 (2011)

    Google Scholar 

  38. Tufenkci, S., Senol, B., Matusu, R., Alagoz, B.: Optimal V-plane robust stabilization method for interval uncertain fractional order PID control systems. Fractal Fract. 5(1), 3 (2021)

    Article  Google Scholar 

  39. Underwood, C., Yik, F.: Modelling Methods for Energy in Buildings. Blackwell Science (2004). ISBN 0-632-05936-2. http://books.google.com/books?id=bzdV3iSRhsUC

  40. Weilbeer, M.: Efficient Numerical Methods for Fractional Differential Equations and Their Analytical Background. Papierflieger (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059, Cracow, Poland

    Wojciech Mitkowski, Marek Długosz & Paweł Skruch

Authors
  1. Wojciech Mitkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marek Długosz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paweł Skruch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paweł Skruch .

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Piotr Kulczycki

  2. Institute of Control and Computation Engineering, University of Zielona Góra, Zielona Góra, Poland

    Józef Korbicz

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mitkowski, W., Długosz, M., Skruch, P. (2022). Selected Engineering Applications of Fractional-Order Calculus. In: Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds) Fractional Dynamical Systems: Methods, Algorithms and Applications. Studies in Systems, Decision and Control, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-030-89972-1_12

Download citation

Publish with us

Policies and ethics

Search

Navigation