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FFLD-Based Modeling of Fractional-Order State Space LTI MIMO Systems

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Applied Physics, System Science and Computers (APSAC 2017)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 428))

Abstract

This paper introduces a multivariable version of the Grünwald-Letnikov fractional-order difference (FD) and approximates it with a powerful combination of finite fractional difference (FFD) and finite Laguerre-based difference (FLD) to yield finite fractional/Laguerre-based difference (FFLD). The multivariable FFLD is effectively used to model fractional-order state-space LTI MIMO systems.

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References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, Orlando, FL (1999)

    MATH  Google Scholar 

  2. Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Series on Advances in Industrial Control. Springer, London, UK (2010)

    Google Scholar 

  3. Chen, Y., Vinagre, B., Podlubny, I.: A new discretization method for fractional order differentiators via continued fraction expansion. In: Proceedings of DETC’2003, ASME Design Engineering Technical Conferences. vol. 340, pp. 349–362, Chicago, IL (2003)

    Google Scholar 

  4. Maione, G.: On the Laguerre rational approximation to fractional discrete derivative and integral operators. IEEE Trans. Autom. Control 58(6), 1579–1585 (2013)

    Article  MathSciNet  Google Scholar 

  5. Baeumer, B., Kovacs, M., Sankaranarayanan, H.: Higher order Grünwald approximations of fractional derivatives and fractional powers of operators. Trans. Am. Math. Soc. 367(2), 813–834 (2015)

    Article  MATH  Google Scholar 

  6. Gao, Z.: Improved Oustaloup approximation of fractional-order operators using adaptive chaotic particle swarm optimization. J. Sys. Eng. Electron. 23(1), 145–153 (2012)

    Article  Google Scholar 

  7. Gao, Z., Liao, X.: Rational approximation for fractional-order system by particle swarm optimization. Nonlinear Dyn. 67(2), 1387–1395 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Khanra, M.: Rational approximation of fractional operator—a comparative study. In: International Conference on Power, Control and Embedded Systems (ICPCES), Allahabad, India, pp. 1–5 (2010)

    Google Scholar 

  9. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ditzian, Z.: Fractional derivatives and best approximation. Acta Math. Hung. 81(4), 323–348 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tseng, C.C.: Design of variable and adaptive fractional order FIR differentiators. Signal Process. 86(10), 2554–2566 (2006)

    Article  MATH  Google Scholar 

  12. Stanisławski, R., Latawiec, K.J.: Normalized finite fractional differences—the computational and accuracy breakthroughs. Int. J. Appl. Math. Comput. Sci. 22(4), 907–919 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Stanisławski, R., Latawiec, K.J.: Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions. In: Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 411–414 (2010)

    Google Scholar 

  14. Stanisławski, R.: New Laguerre filter approximators to the Grünwald-Letnikov fractional difference. Math. Probl. Eng. 2012, 1–21 (2012) Article ID: 732917

    Google Scholar 

  15. Stanisławski, R., Latawiec, K.J., Łukaniszyn, M.: A comparative analysis of Laguerre-based approximators to the Grünwald-Letnikov fractional-order difference. Math. Probl. Eng. 2015, 1–10, Article ID: 512104 (2015)

    Google Scholar 

  16. Stanisławski, R., Latawiec, K.J., Łukaniszyn, M., Gałek, M.: Time-domain approximations to the Grünwald-Letnikov difference with application to modeling of fractional-order state space systems. In: 20th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, pp. 579–584, Aug 2015

    Google Scholar 

  17. Stanisławski, R., Latawiec, K.J.: Fractional-order discrete-time Laguerre filters—a new tool for modeling and stability analysis of fractional-order LTI SISO systems. Discret. Dyn. Nat. Soc. 2016, 1–9, Article ID: 9590687 (2016)

    Google Scholar 

  18. Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for asymptotic stability. Bull. Pol. Acad. Sci. Tech. Sci. 61(2), 353–361 (2013)

    Google Scholar 

  19. Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems. Bull. Pol. Acad. Sci. Tech. Sci. 61(2), 362–370 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical, Control and Computer Engineering, Opole University of Technology, ul. Prószkowska 76, 45-758, Opole, Poland

    Krzysztof J. Latawiec, Rafał Stanisławski, Marian Łukaniszyn, Marek Rydel & Bogusław R. Szkuta

Authors
  1. Krzysztof J. Latawiec
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  2. Rafał Stanisławski
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  3. Marian Łukaniszyn
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  4. Marek Rydel
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  5. Bogusław R. Szkuta
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Corresponding author

Correspondence to Krzysztof J. Latawiec .

Editor information

Editors and Affiliations

  1. Ag. Spyridon, University of Applied Sciences Ag. Spyridon, Athens, Greece

    Klimis Ntalianis

  2. Faculty of Mathematics, Al.I. Cuza University Faculty of Mathematics, Iasi, Romania

    Anca Croitoru

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Latawiec, K.J., Stanisławski, R., Łukaniszyn, M., Rydel, M., Szkuta, B.R. (2018). FFLD-Based Modeling of Fractional-Order State Space LTI MIMO Systems. In: Ntalianis, K., Croitoru, A. (eds) Applied Physics, System Science and Computers. APSAC 2017. Lecture Notes in Electrical Engineering, vol 428. Springer, Cham. https://doi.org/10.1007/978-3-319-53934-8_36

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  • DOI: https://doi.org/10.1007/978-3-319-53934-8_36

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53933-1

  • Online ISBN: 978-3-319-53934-8

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