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Time-domain simulation of MIMO fractional systems

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Abstract

The paper proposes an original method for time-domain simulation of MIMO fractional systems based on their rational approximation using Oustaloup’s method. Both commensurate and incommensurate fractional systems are tackled. The main contribution is to propose straightforward formulae allowing to approximate a pseudo-state-space representation of a fractional system with a state-space representation of a rational one whatever the dimension of the original system and whatever the number of poles and zeros used in the approximation.

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Notes

  1. \(\lfloor . \rfloor \) is the floor operator.

  2. The physical model involves hyperbolic sin and cos functions [33].

References

  1. Sabatier, J., Aoun, M., Oustaloup, A., Grégoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery sate charge estimation. Signal Process. 86(10), 2645–2657 (2006)

    Article  MATH  Google Scholar 

  2. Oldham, K.B., Spanier, J.: Diffusive transport to planar, cylindrical and spherical electrodes. Electroanal. Chem. Interfacial Electrochem. 41, 351–358 (1973)

    Article  Google Scholar 

  3. Battaglia, J., Le Lay, L., Batsale, J., Oustaloup, A., Cois, O.: Heat flux estimation through inverted non integer identification models. Int. J. Therm. Sci. 39(3), 374–389 (2000)

    Article  Google Scholar 

  4. Moreau, X., Ramus-Serment, C., Oustaloup, A.: Fractional differentiation in passive vibration control. J. Nonlinear Dyn. 29, 343–362 (2002)

    Article  MATH  Google Scholar 

  5. Xu, B., Chen, D., Zhang, H., Wang, F.: Modeling and stability analysis of a fractional-order francis hydro-turbine governing system. Chaos Solitons Fract. 75, 50–61 (2015)

    Article  MathSciNet  Google Scholar 

  6. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dyn. 38(1–4), 117–131 (2004)

    Article  MATH  Google Scholar 

  7. Podlubny, I.: Fractional-order systems and pi-lambda d-mu-controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Orsingher, E., Beghin, L.: Time-fractional telegraph equations and telegraph processes with brownian time. Probab. Theory Relat. Fields 128, 141–160 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Oustaloup, A.: La dérivation non-entière: théorie, synthèse et applications. Hermès, Paris (1995)

    Google Scholar 

  10. Huang, F., Liu, F.: The time fractional diffusion equation and the advection-dispersion equation. J Aust. Math. Soc. 46, 317–330 (2005)

  11. Schneider, W., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338, 1364–1377 (2008)

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

    MathSciNet  MATH  Google Scholar 

  15. Aoun, M., Amairi, M., M.N., A.: Simulation method of fractional systems based on the discrete-time approximation of the Caputo fractional derivatives. In: 8th International Multi-Conference on Systems, Signals and Devices. Gabes, Tunisia (2011)

  16. Hwang, C., Leu, J., Tsay, S.: A note on time-domain simulation of feedback fractional-order systems. IEEE Trans. Autom. Control 47(4), 625–631 (2002)

    Article  MathSciNet  Google Scholar 

  17. Rapaic, M., Pisano, A., Jelicic, Z.: Trapezoidal rule for numerical evaluation of fractional order integrals with applications to simulation and identification of fractional order systems. In: IEEE International Conference on Control Applications. Dubrovnik, Croatia (2012)

  18. Al-Alaoui, M.: Novel IIR differentiator from the Simpson integration rule. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41(2), 186–187 (1994)

    Article  MATH  Google Scholar 

  19. Tabak, D.: Digitalization of control systems. Comput. Aided Des. 3, 13–18 (1971)

    Article  Google Scholar 

  20. Trigeassou, J.C., Poinot, T., Lin, J., Oustaloup, A., Levron, F.: Modeling and identification of a non integer order system. In: ECC. Karlsruhe, Germany (1999)

  21. Poinot, T., Trigeassou, J.C.: A method for modelling and simulation of fractional systems. Signal Process. 83, 2319–2333 (2003)

    Article  MATH  Google Scholar 

  22. Krajewski, W., Viaro, U.: A method for the integer-order approximation of fractional-order systems. J. Franklin Inst. 351(1), 555–564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tavazoei, M., Haeri, M.: Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach. Automatica 46(1), 94–100 (2010). doi:10.1016/j.automatica.2009.09.016

    Article  MathSciNet  MATH  Google Scholar 

  24. Mansouri, R., Bettayeb, M., Djennoune, S.: Multivariable fractional system approximation with initial conditions using integral state-space representation. Comput. Math. Appl. 59, 1842–1851 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mansouri, R., Bettayeb, M., Djennoune, S.: Comparison between two approximation methods of state-space fractional systems. Signal Process. 91, 461–469 (2011)

    Article  MATH  Google Scholar 

  26. Bouafoura, M., Moussi, O., Braiek, N.: A fractional state-space realization method with block pulse basis. Signal Process. 91, 492–497 (2011)

    Article  MATH  Google Scholar 

  27. Khanra, M., Pal, J., Biswas, K.: Reduced order approximation of MIMO fractional order systems. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 451–458 (2013)

    Article  MathSciNet  Google Scholar 

  28. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  29. Matignon, D., d’Andréa Novel, B.: Some results on controllability and observability of finite-dimensional fractional differential systems. In: IEEE-CESA’96, SMC IMACS ulticonference. Lille, France, pp. 952–956 (1996)

  30. Matignon, D.: Stability properties for generalized fractional differential systems. In: ESAIM : Proceedings, Fractional Differential Systems: Models, Methods and Applications, vol. 5, pp. 145–158 (1998)

  31. Oustaloup, A.: Systèmes asservis linéaires d’ordre fractionnaire. Masson, Paris (1983)

    Google Scholar 

  32. Malti, R., Thomassin, M.: Differentiation similarities in fractional pseudo-state-space representations and the subspace-based methods. Fract. Calc. Appl. Anal. 16, 273–287 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Maillet, D., André, S., Batsale, J.C., Degiovanni, A., Moyne, C.: Solving the Heat Equation Through Integral Transforms. Wiley, New York (2000)

    MATH  Google Scholar 

  34. Victor, S., Melchior, P., Oustaloup, A.: Robust path tracking using flatness for fractional linear mimo systems: a thermal application. Comput. Math. Appl. 59, 1667–1678 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. IMS UMR 5218 CNRS, University of Bordeaux, Talence, France

    Elena Ivanova, Rachid Malti & Xavier Moreau

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  1. Elena Ivanova
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  2. Rachid Malti
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  3. Xavier Moreau
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Correspondence to Elena Ivanova.

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Ivanova, E., Malti, R. & Moreau, X. Time-domain simulation of MIMO fractional systems. Nonlinear Dyn 84, 2057–2068 (2016). https://doi.org/10.1007/s11071-016-2628-1

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