×

Multivariable fractional system approximation with initial conditions using integral state space representation. (English) Zbl 1189.93032

Summary: We consider the approximation of general multivariable non commensurate fractional systems by integer order state space models. This work contains two main contributions. First, a new state space representation using the fractional integral operator is introduced. Second, the approximate model carries explicitly the initial conditions of the system. Two examples are given to illustrate the accuracy of the approximation.

MSC:

93B15 Realizations from input-output data
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations

Software:

CRONE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1974), A Wiley Interscience Publication
[2] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York and London · Zbl 0428.26004
[3] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (1993), Gordon and Breach Science Publishers · Zbl 0818.26003
[5] M. Axtell, E.M. Bise, Fractional calculus applications in control systems, in: Proc. of the IEEE Nat. Aerospace and Electronics Conf., New York, 1990, pp. 563-566; M. Axtell, E.M. Bise, Fractional calculus applications in control systems, in: Proc. of the IEEE Nat. Aerospace and Electronics Conf., New York, 1990, pp. 563-566
[6] Manabe, S., The non-integer integral and its application to control systems, ETJ of Japan, 06, 83-87 (1961)
[7] Oustaloup, A., Fractional order sinusoidal oscillators: Optimization and their use in highly linear FM modulation, IEEE Transactions on Circuits and Systems, CAS-28, 369-383 (1981)
[8] Chen, Y. Q.; Moore, K. L., Discretization schemes for fractional-order differentiators and integrators, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 49, 363-367 (2002) · Zbl 1368.65035
[9] Gorenflo, R., Fractional calculus: Some numerical methods, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag: Springer Verlag Vienna, New York)
[10] Vinagre, B. M.; Podlubny, I.; Hernandez, A.; Feliu, V., Some approximations of fractional order operators used in control theory and applications, Fractional Calculus & Applied Analysis, 03, 231-248 (2000) · Zbl 1111.93302
[11] Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F., Frequency-band complex non integer differentiator: Characterization and synthesis, IEEE Transactions on Circuits and Systems, 47, 25-40 (2000)
[12] A. Dzielinski, D. Sierociuk, Stability of discrete fractional order state-space systems, in: Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, 19-21 July 2006; A. Dzielinski, D. Sierociuk, Stability of discrete fractional order state-space systems, in: Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, 19-21 July 2006 · Zbl 1334.93172
[13] R. Mansouri, Contribution à l’analyse et la synthèse des systèmes d’ordre fractionnaire par la représentation d’état. Ph.D. Thesis, Université de Tizi-Ouzou, Algeria, 2008; R. Mansouri, Contribution à l’analyse et la synthèse des systèmes d’ordre fractionnaire par la représentation d’état. Ph.D. Thesis, Université de Tizi-Ouzou, Algeria, 2008
[14] Mansouri, R.; Bettayeb, M.; Djennoune, S., Multivariable fractional order system approximation using derivative representation, International Journal of Applied Mathematics, 20, 7, 983-1003 (2007) · Zbl 1132.26315
[15] D. Matignon, Représentations en variables d’état de modèles de guides d’ondes avec dérivation fractionnaire. Ph.D. Thesis, Université de Paris-Sud, Orsay, 1998; D. Matignon, Représentations en variables d’état de modèles de guides d’ondes avec dérivation fractionnaire. Ph.D. Thesis, Université de Paris-Sud, Orsay, 1998
[16] Poinot, T.; Trigeassou, J. C., A method for modelling and simulation of fractional systems, Signal Processing, 83, 2319-2333 (2003) · Zbl 1145.94372
[17] Raynaud, A.; Zerganoh, R., State-space representation for fractional order controllers, Automatica, 36, 1017-1021 (2000) · Zbl 0964.93024
[18] C.F. Lorenzo, T.T. Hartley, Initialization of fractional differential equations: Theory and application, in: Proceedings of the ASME 2007 International Design Engineering Technical Conferences, DETC2007, 04-07 September, Las Vegas, 19-21 July 2007; C.F. Lorenzo, T.T. Hartley, Initialization of fractional differential equations: Theory and application, in: Proceedings of the ASME 2007 International Design Engineering Technical Conferences, DETC2007, 04-07 September, Las Vegas, 19-21 July 2007
[19] J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, On a representation of fractional order systems: Interests for the initial condition problem. in: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 05-07 November 2008; J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, On a representation of fractional order systems: Interests for the initial condition problem. in: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 05-07 November 2008
[20] M.D. Ortigueira, F.J. Coito, Initial conditions: What are we talking about? in: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 05-07 November 2008; M.D. Ortigueira, F.J. Coito, Initial conditions: What are we talking about? in: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 05-07 November 2008
[21] Caputo, M., Linear models of dissipation whose \(q\) is almost frequency independent, Geophysical Journal of the Royal Astronomical Society, 2, 13, 529-539 (1967)
[22] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Van Mill, Jan, Theory and Application of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam) · Zbl 1092.45003
[23] Oustaloup, A., La commande CRONE: du scalaire au multivariable (1999), Hermes: Hermes Paris · Zbl 0936.93004
[24] Oustaloup, A., La Dérivation non Entière: Théorie, synthèse et application (1995), Hermes: Hermes Paris · Zbl 0864.93004
[25] Mansouri, R.; Bettayeb, M.; Djennoune, S., Non integer order system approximation by an integer reduced model, Fractional Systems. Fractional Systems, Journal Europèen des systèmes Automatisés, 42/6-8, 689-700 (2008), (special issue)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.