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Quadrature Based Approximations of Non-integer Order Integrator on Finite Integration Interval

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Theory and Applications of Non-integer Order Systems

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 407))

Abstract

Implementation of non-integer order systems is the subject of an ongoing research. In this paper we consider the approximation of non-integer order integrator with the use of diffusive realization of pseudo differential operator. We propose a transformation of variables allowing easier approximation with use of quadratures. We then analyze the convergence and discuss the consequences of reduction in the integration interval.

Work realised in the scope of project titled “Design and application of non-integer order subsystems in control systems”. Project was financed by National Science Centre on the base of decision no. DEC-2013/09/D/ST7/03960.

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References

  1. Oustaloup, A.: La Commande CRONE: Commande Robuste D’ordre Non Entier. Hermes (1991)

    Google Scholar 

  2. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I, Fundam. Theory, 47(1), 25–39 (2000)

    Google Scholar 

  3. Vinagre, B.M., Chen, Y.Q., Petráš, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Vinagre, B., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Frac. Calc. Appl. Anal. 3(3), 231–248 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-order Systems and Controls. Fundamentals and Applications. Advances in Industrial Control. Springer, London (2010)

    Google Scholar 

  7. Baranowski, J., Bauer, W., Zagórowska, M., Dziwiński, T., Piątek, P.: Time-domain Oustaloup approximation. In: IEEE 20th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 116–120 (2015)

    Google Scholar 

  8. Baranowski, J., Bauer, W., Zagórowska, M.: Stability properties of discrete time-domain Oustaloup approximation. In Domek, S., Dworak, P. (eds.) Theoretical Developments and Applications of Non-Integer Order Systems, vol. 357 of Lecture Notes in Electrical Engineering, pp. 93–103. Springer International Publishing (2016)

    Google Scholar 

  9. Tseng, C.C.: Design of FIR and IIR fractional order Simpson digital integrators. Signal Process. 87(5), 1045–1057 (2007)

    Article  MATH  Google Scholar 

  10. Leulmi, F., Ferdi, Y.: Improved digital rational approximation of the operator S\(^{\alpha }\) using second-order s-to-z transform and signal modeling. Circuits Syst. Signal Process. 34(6), 1869–1891 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferdi, Y.: Computation of fractional order derivative and integral via power series expansion and signal modelling. Nonlinear Dyn. 46(1), 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Maione, G.: Laguerre approximation of fractional systems. Electron. Lett. 38(20), 1234–1236 (2002)

    Article  Google Scholar 

  13. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Synthesis of fractional Laguerre basis for system approximation. Automatica 43(9), 1640–1648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bania, P., Baranowski, J.: Laguerre polynomial approximation of fractional order linear systems. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds.) Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, vol. 257 of the series Lecture Notes in Electrical Engineering, pp. 171–182. Springer (2013)

    Google Scholar 

  15. Zagórowska, M., Baranowski, J., Bania, P., Piątek, P., Bauer, W., Dziwiński, T.: Impulse response approximation method for “fractional order lag”. In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Noninteger-order Systems - 6th Conference on Non-Integer Order Caculus and its Applications, vol. 320 of the series Lecture Notes in Electrical Engineering, pp. 113–122. Springer, Heidelberg (2014)

    Google Scholar 

  16. Zagórowska, M., Baranowski, J., Bania, P., Bauer, W., Dziwiński, T., Piątek, P.: Parametric optimization of PD controller using Laguerre approximation. In: IEEE 20th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 104–109, (2015)

    Google Scholar 

  17. Zagórowska, M.: Parametric optimization of non-integer order PD\(^\mu \) controller for delayed system. In: Domek, S., Dworak, P. (eds.) Theoretical Developments and Applications of Non-Integer Order Systems, vol. 357 of Lecture Notes in Electrical Engineering, pp. 259–270. Springer, Heidelberg (2016)

    Google Scholar 

  18. Baranowski, J., Bauer, W., Zagórowska, M., Piątek, P.: On digital realizations of non-integer order filters. Circuits Syst. Signal Process (2016)

    Google Scholar 

  19. Montseny, G.: Diffusive representation of pseudo-differential time-operators. ESAIM: Proc. 5, 159–175 (1998)

    Google Scholar 

  20. Heleschewitz, D., Matignon, D.: Diffusive realisations of fractional integrodifferential operators: Structural analysis under approximation. In: IFAC conference Systems Structure and Control, vol. 2, Nantes, France pp. 243–248 (1998)

    Google Scholar 

  21. Trigeassou, J., Maamri, N., Sabatier, J., Oustaloup, A.: State variables and transients of fractional order differential systems. Comput. Math. Appl. 64(10), 3117–3140 (2012) (Advances in FDE, III)

    Google Scholar 

  22. Liang, S., Peng, C., Liao, Z., Wang, Y.: State space approximation for general fractional order dynamic systems. Int. J. Syst. Sci. 45(10), 2203–2212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grabowski, P.: Stabilization of wave equation using standard/fractional derivative in boundary damping. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds.) Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, vol. 257 of the series Lecture Notes in Electrical Engineering, pp. 101–121. Springer International Publishing, Heidelberg (2013)

    Google Scholar 

  24. Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50(1), 67–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Jerzy Baranowski

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  1. Jerzy Baranowski
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Correspondence to Jerzy Baranowski .

Editor information

Editors and Affiliations

  1. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Artur Babiarz

  2. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Adam Czornik

  3. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Jerzy Klamka

  4. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Michał Niezabitowski

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Baranowski, J. (2017). Quadrature Based Approximations of Non-integer Order Integrator on Finite Integration Interval. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-319-45474-0_2

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  • DOI: https://doi.org/10.1007/978-3-319-45474-0_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45473-3

  • Online ISBN: 978-3-319-45474-0

  • eBook Packages: EngineeringEngineering (R0)

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