The infinite state approach: Origin and necessity

Computers and Mathematics with Applications

Volumn 66, Issue 5, 2013, Pages 892-907

  Trigeassou, J C ^a   Maamri, N ^b   Oustaloup, A ^a  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b UNIVERSITÉ DE POITIERS   (France)

Author keywords

FDS initialization; FDS modeling; Fractional derivatives; Fractional integrator; History of analog computing; Infinite state approach

Indexed keywords

ANALOG COMPUTING; FDS INITIALIZATION; FRACTIONAL DERIVATIVES; FRACTIONAL INTEGRATORS; INFINITE STATE;

ORDINARY DIFFERENTIAL EQUATIONS; SYSTEM THEORY;

ALGEBRA;

EID: 84881477754     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2012.11.020     Document Type: Article

References (37)

1. C.F. Lorenzo, T.T. Hartley, Initialization in fractional order systems, in: Proceedings of the European Control Conference, ECC'01, Porto, Portugal, 2001, pp. 1471-1476.
2. M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, 1969.
3. J.C. Trigeassou, N. Maamri, A. Oustaloup, State variables, initial conditions and transients of fractional order derivatives and systems, Plenary talk, FDA'12, Nanjing, China, May 2012.
4. J.C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup State variables and transients of fractional order differential systems Computers and Mathematics with Applications 64 10 2012 3117 3140
5. J.C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup Transients of fractional order integrator and derivatives Signal, Image and Video Processing 6 3 2012 359 372 (Special issue: "fractional systems and signals")
6. M.D. Ortigueira, F.J. Coito, Initial conditions: what are we talking about?, in: 3rd IFAC Workshop, FDA'08, Ankara, Turkey, 5-7 November 2008.
7. J. Sabatier, M. Merveillaut, R. Malti, and A. Oustaloup How to impose physically coherent initial conditions to a fractional system? Communications in Non Linear Science and Numerical Simulation 15 5 2010
8. W. Thomson (Lord Kelvin) Mechanical integration of the general linear differential equation of any order with variable coefficients Proceedings of the Royal Society 24 1876 271 275
9. T.T. Hartley, C.F. Lorenzo, The error incurred in using the Caputo derivative Laplace transform, in: Proceedings of the ASME IDET-CIE Conferences San Diego, California, USA, 2009.
10. C. Lorenzo, T. Hartley, Time-varying initialization and Laplace transform of the Caputo derivative: with order between zero and one, in: Proceedings of IDET/CIE FDTA'2011 Conference, Washington DC, USA, August 2011.
11. J.C. Trigeassou, N. Maamri, The initial conditions of Riemman-Liouville and Caputo derivatives: an integrator interpretation, in: FDA'2010 Conference, Badajoz, Spain, October 2010.
12. H. Poincaré Mémoire sur les courbes limites définies par une equation différentielle Journal de Mathématiques 3 1882 251 296 Série 8
13. L. Levine Methods for Solving Engineering Problems Using Analog Computers 1964 McGraw Hill New-York
14. D.M. Wiberg State Space and Linear Systems Schaum's Outline Series 1971 McGraw-Hill New-York
15. T. Kailath Linear Systems 1980 Prentice Hall Inc. Englewood Cliffs
16. J.C. Trigeassou, A. Benchellal, N. Maamri, and T. Poinot A frequency approach to the stability of fractional differential equations Transactions on Systems, Signals and Devices 4 1 2009 1 26
17. O.P. Agrawal, and P. Kumar Comparison of five numerical schemes for fractional differential equations J. Sabatier, Advances in Fractional Calculus 2007 Springer 43 60
18. K. Diethelm An investigation of some non classical methods for the numerical approximation of Caputo-type fractional derivatives Numerical Algorithm 47 2008 361 390
19. K. Diethelm The analysis of fractional differential equations Numerical Solution of Fractional Equations Lecture Notes in Mathematics 2010 Springer-Verlag 195 225 Appendix C
20. I. Petras Fractional Order Non Linear Systems: Modelling, Analysis and Simulation 2011 Springer-Verlag
21. M.D. Ortigueira Fractional Calculus for Scientists and Engineers 2011 Springer Science
22. C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, and V. Feliu Fractional Order Systems and Control 2010 Springer-Verlag London
23. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 John Wiley and Sons New-York
24. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Academic Press New-York
25. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego
26. J.C. Trigeassou, et al. Modelling and identification of a non integer order system, in: ECC'99 European Control Conference, KARLSRUHE, Germany, 1999.
27. J.C Trigeassou, and N. Maamri Initial conditions and initialization of linear fractional differential equations Signal Processing 91 3 2011 427 436
28. J.C. Trigeassou, A. Oustaloup, Fractional integration: a comparative analysis of fractional integrators, in: IEEE SSD'11, Sousse, Tunisia, 2011.
29. D. Heleschewitz, D. Matignon, Diffusive realizations of fractional integro-differential operators: structural analysis under approximation, in: Conference IFAC, System, Structure and Control. vol. 2, Nantes, France, July 1998, pp. 243-248.
30. D. Helechewitz, Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels sous representation diffusive, Thèse de doctorat, ENST Paris, 18 décembre 2000.
31. G. Montseny, Diffusive representation of pseudo differential time operators, in: Proceedings ESSAIM, vol. 5, 1998, pp. 159-175.
32. Y. Li, Y.Q. Chen, and I. Podlubny Mittag Leffler stability of fractional order non linear dynamic systems Automatica 45 2009 1965 1969
33. J.C Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup A Lyapunov approach to the stability of fractional differentiel equations Signal Processing 91 3 2011 437 445
34. A. Oustaloup La Dérivation Non Entière: Théorie, Synthèse et Applications 1995 Hermès Paris
35. L. Owens Vannevar Bush and the differential analyser: the text and context of an early computer Technology and Culture 27 1 1986 63 95
36. J.R. Ragazzini, R.H. Randall, F.A. Russell, Analysis of problems in dynamics by electronic circuits, in: Proceedings of IRE, vol. 35(5), May 1947, pp. 444-452.
37. J.W. Harris, and H. Stocker Handbook of Mathematics and Computational Science 1998 Springer-Verlag New-York