Abstract
In this paper, a multivariable version of the Grünwald-Letnikov noncommensurate fractional-order difference (FD) is defined and approximated with a powerful Grünwald-Letnikov-Laguerre (GLL) combination of finite fractional difference (FFD) and finite Laguerre-based difference (FLD) to obtain finite fractional/Laguerre-based difference (FFLD). The multivariable FFLD is effectively employed to model noncommensurate fractional-order state-space LTI MIMO systems.
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References
Podlubny, I.: Fractional Differential Equations. Academic Press, Orlando (1999)
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 (2010)
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, Chicago, IL, vol. 340, pp. 349–362 (2003)
Maione, G.: On the Laguerre rational approximation to fractional discrete derivative and integral operators. IEEE Trans. Autom. Control 58(6), 1579–1585 (2013)
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)
Gao, Z.: Improved Oustaloup approximation of fractional-order operators using adaptive chaotic particle swarm optimization. J. Syst. Eng. Electron. 23(1), 145–153 (2012)
Gao, Z., Liao, X.: Rational approximation for fractional-order system by particle swarm optimization. Nonlinear Dyn. 67(2), 1387–1395 (2012)
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)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
Ditzian, Z.: Fractional derivatives and best approximation. Acta Mathematica Hungarica 81(4), 323–348 (1998)
Tseng, C.C.: Design of variable and adaptive fractional order FIR differentiators. Sig. Process. 86(10), 2554–2566 (2006)
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)
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)
Stanisławski, R.: New Laguerre filter approximators to the Grünwald-Letnikov fractional difference. Math. Probl. Eng. 2012, 1–21 (2012). Article ID: 732917
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 (2015). Article ID: 512104
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, August 2015
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. Discrete Dyn. Nature Soc. 2016, 1–9 (2016). Article ID: 9590687
Latawiec, K.J., Stanisławski, R., Łukaniszyn, M., Rydel, M., Szkuta, B.R.: FFLD-based modeling of fractional-order state space LTI MIMO systems. In: International Conference on Applied Physics, System Science and Computers (APSAC2016). Lecture Notes in Electrical Engineering, vol. 428. Springer (2017). https://doi.org/10.1007/978-3-319-53934-8_36
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. Polish Acad. Sci. Tech. Sci. 61(2), 353–361 (2013)
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. Polish Acad. Sci. Tech. Sci. 61(2), 362–370 (2013)
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Latawiec, K.J., Stanisławski, R., Łukaniszyn, M., Rydel, M., Szkuta, B.R. (2019). Grünwald-Letnikov-Laguerre Modeling of Discrete-Time Noncommensurate Fractional-Order State Space LTI MIMO Systems. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_7
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