Skip to main content
SpringerLink
Log in

Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling

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

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

  • 554 Accesses

Abstract

The chapter presents an overview of some particular derivative and difference operators of fractional variable order, their properties, equivalent forms, and applications. When fundamental properties of a system or its structure are changing in time, a variation of the system’s order may be observed. In such a case, time-dependent variable order operators are taken into consideration. Recently, cases where order is time-varying, have began to be studied extensively. In order to give a deeper insight into fractional variable order calculus, alternative, intuitive descriptions of some particular variable order operators, in the form of equivalent switching schemes, is provided. According to such a schematic interpretation of variable order operators analysis of variable order systems can be simpler and more effective than on the basis of purely analytical definitions. Based on those switching schemes it is possible to categorize fractional order derivatives according to their behaviour and intrinsic properties. Thanks to this schematic description and duality property between chosen variable order operators, analytical solutions of variable order linear differential equations can be effectively derived. Examples of applications of these operators to automatic control and modelling of the heat transfer process in specific grid-holes and two-dimensional fractal-like structure media, of which the geometry is changing in time, are presented.

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

References

  1. Dabiri, A., Moghaddam, B.P., Tenreiro Machado, J.A.: Optimal variable-order fractional PID controllers for dynamical systems. J. Comput. Appl. Math. (2018)

    Google Scholar 

  2. Dzieliński, A., Sarwas, G., Sierociuk, D.: Time domain validation of ultracapacitor fractional order model. In: 2010 49th IEEE Conference on Decision and Control (CDC), pp. 3730–3735 (2010)

    Google Scholar 

  3. Dzieliński, A., Sierociuk, D.: Fractional order model of beam heating process and its experimental verification. In: Baleanu, D., Guvenc, Z.B., Machado, J.A.T. (eds.) New Trends in Nanotechnology and Fractional Calculus Applications, pp. 287–294. Springer, Netherlands (2010)

    Google Scholar 

  4. Dzieliński, A., Sarwas, G., Sierociuk, D.: Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. 2011, 11 (2011)

    Article  MathSciNet  Google Scholar 

  5. Garrappa, R., Mainardi, F., Maione, G.: Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19(5), 1105–1160 (2016)

    Article  MathSciNet  Google Scholar 

  6. Giusti, A.: On infinite order differential operators in fractional viscoelasticity. Fract. Calc. Appl. Anal. 20(4), 854–867 (2017)

    Article  MathSciNet  Google Scholar 

  7. Harris, P.A., Garra, R.: Nonlinear heat conduction equations with memory: physical meaning and analytical results. J. Math. Phys. 58(6), 063501 (2017)

    Article  MathSciNet  Google Scholar 

  8. Heymans, Nicole, Podlubny, Igor: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006). Jun

    Article  Google Scholar 

  9. Ikeda, F., Toyama, R., Toyama, S.: Anti-windup controller design by fractional calculus for linear control systems with input saturation. In: 2009 ICCAS-SICE, pp. 3317–3320 (2009)

    Google Scholar 

  10. Kosztolowicz, T.: Subdiffusion in a system with a thick membrane. J. Membr. Sci. 320(1–2), 492–499 (2008)

    Article  Google Scholar 

  11. Li, Z., Wang, H., Xiao, R., Yang, S.: A variable-order fractional differential equation model of shape memory polymers. Chaos, Solitons Fractals 102, 473–485 (2017). Future Directions in Fractional Calculus Research and Applications

    Google Scholar 

  12. Lorenzo, C.F., Hartley, T.T.: Initialization in fractional order systems. In: The European Control Conference, Porto, Portugal, pp. 1471–1476 (2001)

    Google Scholar 

  13. Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)

    Article  MathSciNet  Google Scholar 

  14. Lorenzo, C.F., Hartley, T.T.: Initialization of fractional differential equations: theory and application. In: The ASME 2007 International Design Engineering Technical Conferences, DETC2007-34814 Las Vegas, USA (2007)

    Google Scholar 

  15. Macias, M., Sierociuk, D.: An alternative recursive fractional variable-order derivative definition and its analog validation. In: Proceedings of International Conference on Fractional Differentiation and Its Applications, Catania, Italy (2014)

    Google Scholar 

  16. Malesza, W., Macias, M.: Numerical solution of fractional variable order linear control system in state-space form. Bull. Pol. Acad. Sci.: Tech. Sci. 65(5) (Special Section on Multilevel Converters), 715–724 (2017)

    Google Scholar 

  17. Malesza, W., Macias, M., Sierociuk, D.: Analytical solution of fractional variable order differential equations. J. Comput. Appl. Math. 348, 214–236 (2019)

    Article  MathSciNet  Google Scholar 

  18. Malesza, W., Macias, M., Sierociuk, D.: Matrix approach and analog modeling for solving fractional variable order differential equations. In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Non-integer-Order Systems, pp. 71–80. Springer International Publishing, Cham (2015)

    Google Scholar 

  19. Metzler, Ralf, Klafter, Joseph: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  20. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  21. Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer (2010)

    Google Scholar 

  22. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press (1974)

    Google Scholar 

  23. Ortigueira, M.D., Tenreiro Machado, J.A.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015). Fractional PDEs

    Google Scholar 

  24. Ostalczyk, P., Rybicki, T.: Variable-fractional-order dead-beat control of an electromagnetic servo. J. Vib. Control (2008)

    Google Scholar 

  25. Padula, F., Visioli, A., Pagnoni, M.: On the anti-windup schemes for fractional-order PID controllers. In: Proceedings of 2012 IEEE 17th International Conference on Emerging Technologies Factory Automation (ETFA 2012), pp. 1–4 (2012)

    Google Scholar 

  26. Pandey, S., Soni, N.K., Pandey, R.K.: Fractional order integral and derivative (FOID) controller with anti-windup for temperature profile control. In: 2015 2nd International Conference on Computing for Sustainable Global Development (INDIACom), pp. 1567–1573 (2015)

    Google Scholar 

  27. Pandey, S., Dwivedi, P., Junghare, A.: Anti-windup fractional order PI\(^\lambda \)-PD\(^\mu \) controller design for unstable process: a magnetic levitation study case under actuator saturation. Arab. J. Sci. Eng. (2017)

    Google Scholar 

  28. Pandey, S., Dwivedi, P., Junghare, A.S.: A novel 2-DOF fractional-order PI\(^\alpha \)-D\(^\mu \) controller with inherent anti-windup capability for a magnetic levitation system. AEU-Int. J. Electron. Commun. 79, 158–171 (2017)

    Google Scholar 

  29. Podlubny, I.: Fractional Differential Equations. Academic Press (1999)

    Google Scholar 

  30. Sabatier, Jocelyn, Merveillaut, Mathieu, Malti, Rachid, Oustaloup, Alain: How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simul. 15(5), 1318–1326 (2010)

    Article  MathSciNet  Google Scholar 

  31. Sakrajda, P., Sierociuk, D.: Modeling heat transfer process in grid-holes structure changed in time using fractional variable order calculus. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds.) Theory and Applications of Non-integer Order Systems, pp. 297–306. Springer International Publishing, Cham (2017)

    Google Scholar 

  32. Samko, S.G., Kilbas, A.A., Maritchev, O.I.: Fractional Integrals and Derivative. Theory and Applications. Gordon & Breach Science Publishers (1987)

    Google Scholar 

  33. Sheng, H., Sun, H., Coopmans, C., Chen, Y., Bohannan, G.W.: Physical experimental study of variable-order fractional integrator and differentiator. In: Proceedings of the 4th IFAC Workshop Fractional Differentiation and its Applications FDA’10 (2010)

    Google Scholar 

  34. Sierociuk, D., Dzieliński, 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., Sarwas, G., Twardy, M.: Resonance phenomena in circuits with ultracapacitors. In: Proceedings of 12th International Conference on Environment and Electrical Engineering (EEEIC), pp. 197–202 (2013)

    Google Scholar 

  36. Sierociuk, D., Twardy, M.: Duality of variable fractional order difference operators and its application in identification. Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 809–815 (2014)

    Google Scholar 

  37. Sierociuk, Dominik, Malesza, Wiktor, Macias, Michal: Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model. 39(13), 3876–3888 (2015). https://doi.org/10.1016/j.apm.2014.12.009

    Article  MathSciNet  MATH  Google Scholar 

  38. Sierociuk, Dominik, Malesza, Wiktor, Macias, Michal: On the recursive fractional variable-order derivative: equivalent switching strategy, duality, and analog modeling. Circuits Syst. Signal Process. 34(4), 1077–1113 (2015). Apr

    Article  MathSciNet  Google Scholar 

  39. Sierociuk, Dominik, Malesza, Wiktor, Macias, Michal: Numerical schemes for initialized constant and variable fractional-order derivatives: matrix approach and its analog verification. J. Vib. Control 22(8), 2032–2044 (2016)

    Article  MathSciNet  Google Scholar 

  40. Sierociuk, Dominik, Skovranek, Tomas, Macias, Michal, Podlubny, Igor, Petras, Ivo, Dzielinski, Andrzej, Ziubinski, Pawel: Diffusion process modeling by using fractional-order models. Appl. Math. Comput. 257, 2–11 (2015)

    Google Scholar 

  41. Sun, HongGuang, Zhang, Yong, Baleanu, Dumitru, Chen, Wen, Chen, YangQuan: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  Google Scholar 

  42. Trigeassou, J.C., Maamri, N.: Initial conditions and initialization of linear fractional differential equations. Signal Process. 91(3), 427–436 (2011). Advances in Fractional Signals and Systems

    Google Scholar 

  43. Valerio, D., Sa da Costa, J.: Variable-order fractional derivatives and their numerical approximations. Signal Process. 91(3, SI), 470–483 (2011)

    Google Scholar 

  44. Yin, Deshun, Pengfei, Qu.: Variable-order fractional MSD function to describe the evolution of protein lateral diffusion ability in cell membranes. Phys. A: Stat. Mech. Appl. 492, 707–714 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electrical Engineering, Institute of Control and Industrial Electronics, Warsaw University of Technology, Koszykowa 75, 00-662, Warszawa, Poland

    Andrzej Dzieliński, Dominik Sierociuk, Wiktor Malesza, Michał Macias, Michał Wiraszka & Piotr Sakrajda

Authors
  1. Andrzej Dzieliński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Sierociuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wiktor Malesza
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michał Macias
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Michał Wiraszka
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Piotr Sakrajda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrzej Dzieliński .

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

Dzieliński, A., Sierociuk, D., Malesza, W., Macias, M., Wiraszka, M., Sakrajda, P. (2022). Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling. 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_4

Download citation

Publish with us

Policies and ethics

Search

Navigation