×

Approximation of a fractional order model by an integer order model: a new approach taking into account approximation error as an uncertainty. (English) Zbl 1365.34019

Summary: In order to solve some analysis or control problems for fractional order models, integer order approximations are often used. However, in many works, approximation error is not taken into account, leading to results that cannot be guaranteed for the initial fractional order model. The objective of the paper is thus to provide a new methodology that takes into account approximation error and leads to rewriting the fractional order model as an uncertain integer order model.

MSC:

34A08 Fractional ordinary differential equations
93C23 Control/observation systems governed by functional-differential equations
26A33 Fractional derivatives and integrals
93B40 Computational methods in systems theory (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anastasio TJ, Biological Cybernetics 79 (5) pp 377– (1998) · Zbl 0918.92003 · doi:10.1007/s004220050487
[2] Aoun M, Nonlinear Dynamics 38 (1) pp 117– (2004) · Zbl 1134.65300 · doi:10.1007/s11071-004-3750-z
[3] Barbosa RS, Acta Polytechnica Hungarica 3 (4) pp 5– (2006)
[4] Battaglia JL, International Journal of Thermal Science 39 (3) pp 374– (2000) · doi:10.1016/S1290-0729(00)00220-9
[5] Bertrand N, Communications in Nonlinear Science and Numerical Simulation 15 (5) pp 1327– (2010) · doi:10.1016/j.cnsns.2009.05.066
[6] Bonnet C, Automatica 38 (7) pp 1133– (2002) · Zbl 1007.93065 · doi:10.1016/S0005-1098(01)00306-5
[7] Caputo M, International Geophysical Journal 13 (5) pp 529– (1967) · doi:10.1111/j.1365-246X.1967.tb02303.x
[8] Djouambi A, International Journal of Applied Mathematics and Computer Science 17 (4) pp 455– (2007) · Zbl 1234.93049 · doi:10.2478/v10006-007-0037-9
[9] Farges C, Mechatronics 23 (7) pp 772– (2013) · doi:10.1016/j.mechatronics.2013.06.005
[10] Farges C, Automatica 46 (10) pp 1730– (2010) · Zbl 1204.93094 · doi:10.1016/j.automatica.2010.06.038
[11] Liang S, International Journal of Systems Science 45 (10) pp 2203– (2014) · Zbl 1317.93104 · doi:10.1080/00207721.2013.766773
[12] Lu JG, IEEE Transactions on Automatic Control 55 (1) pp 152– (2010) · Zbl 1368.93506 · doi:10.1109/TAC.2009.2033738
[13] Malti R, Automatica 47 (11) pp 2425– (2011) · Zbl 1228.93064 · doi:10.1016/j.automatica.2011.08.021
[14] Matignon D, ESAIM Proceedings 5 (1) pp 145– (1998) · Zbl 0920.34010 · doi:10.1051/proc:1998004
[15] Miller K, An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[16] Montseny G, ESAIM Proceedings 5 (1) pp 159– (1998) · Zbl 0916.93022 · doi:10.1051/proc:1998005
[17] DOI: 10.1016/j.automatica.2013.02.066 · Zbl 1360.93620 · doi:10.1016/j.automatica.2013.02.066
[18] Oustaloup A, La dérivation non entière: Théorie, synthèse et applications (1995)
[19] DOI: 10.1016/j.automatica.2013.04.012 · Zbl 1364.93222 · doi:10.1016/j.automatica.2013.04.012
[20] Petras I, Envirautom 4 (1) pp 83– (1999)
[21] Podlubny I, Fractional Differential Equations (1999)
[22] Rodrigues S, Journal of Power Sources 87 pp 12– (2000) · doi:10.1016/S0378-7753(99)00351-1
[23] DOI: 10.3166/ejc.18.260-271 · Zbl 1264.93030 · doi:10.3166/ejc.18.260-271
[24] Samko S, Fractional Integrals and Derivatives: Theory and Applications (1993)
[25] Sommacal L, Advances in Fractional Calculus Theoretical Developments and Applications in Physics and Engineering pp 271– (2007) · Zbl 1125.92021 · doi:10.1007/978-1-4020-6042-7_19
[26] Vinagre BM, Fractional Calculus and Applied Analysis 3 (3) pp 231– (2000)
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.