Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

Fractional Calculus and Applied Analysis

Volumn 14, Issue 3, 2011, Pages 436-453

  Jiao, Zhuang ^a,b   Chen, Yangquan ^b  

^a TSINGHUA UNIVERSITY   (China)

^b UTAH STATE UNIVERSITY   (United States)

Author keywords

bounded input bounded output stability; double noncommensurate orders; fractional order systems; numerical inverse Laplace transform technique

Indexed keywords

EID: 80051682888     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-011-0027-3     Document Type: Article

References (32)

1. M.J. Ablowitz, A.S. Fokas, Complex Variables: Introduction and Applications (2nd Ed). Cambridge University Press, Cambridge (2003).
2. K. Adolfsson M. Enelund 2003 Fractional derivative viscoelasticity at large deformations Nonlinear Dynamics 33 3 301 321 1065.74015 10.1023/A:1026003130033
3. H.S. Ahn Y.Q. Chen 2008 Necessary and sufficient stability condition of fractional-order interval linear systems Automatica 44 11 2985 2988 1152.93455 10.1016/j.automatica.2008.07.003
4. R.L. Bagley P.J. Torvik 1984 On the appearance of the fractional derivative in the behavior of real materials ASME Journal of Applied Mechanics 51 2 294 298 1203.74022 10.1115/1.3167615 (Pubitemid 14583152)
5. C. Bonnet J.R. Partington 2002 Analysis of fractional delay systems of retarded and neutral type Automatica 38 7 1133 1138 1007.93065 10.1016/S0005-1098(01)00306-5 (Pubitemid 34524669)
6. C. Bonnet J.R. Partington 2007 Stabilization of some fractional delay systems of neutral type Automatica 43 12 2047 2053 1138.93049 10.1016/j.automatica.2007.03.017 (Pubitemid 350148069)
7. R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: Modeling and Control Applications. World Scientific, New Jersey (2010).
8. Y.Q. Chen K.L. Moore 2002 Discretization schemes for fractional-order differentiators and integrators IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49 3 363 367 10.1109/81.989172 (Pubitemid 34466606)
9. Y.Q. Chen H.S. Ahn I. Podlubny 2006 Robust stability check of fractional order linear time invariant systems with interval uncertainties Signal Processing 86 10 2611 2618 1172.94385 10.1016/j.sigpro.2006.02.011 (Pubitemid 44026624)
10. B. Gross, E.P. Braga, Singularities of Linear System Functions. Elsevier Publishing, New York (1961).
11. T.T. Hartley C.F. Lorenzo 2003 Fractional-order system identification based on continuous order-distributions Signal Processing 83 11 2287 2300 1145.93433 10.1016/S0165-1684(03)00182-8
12. N. Heymans 2004 Fractional calculus description of non-linear viscoelastic behaviour of polymers Nonlinear Dynamics 38 1-4 221 231 1142.74312 10.1007/s11071-004-3757-5 (Pubitemid 40406781)
13. C. Hwang Y.C. Cheng 2006 A numerical algorithm for stability testing of fractional delay systems Automatica 42 5 825 831 1137.93375 10.1016/j. automatica.2006.01.008
14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Inc., New York (2006).
15. R.C. Koeller 2007 Toward an equation of state for solid materials with memory by use of the half-order derivative Acta Mechanica 191 3-4 125 133 1117.74005 10.1007/s00707-006-0411-y (Pubitemid 46993918)
16. H.S. Li Y. Luo Y.Q. Chen 2009 A fractional order proportional and derivative (fopd) motion controller: Tuning rule and experiments IEEE Transactions on Control Systems Technology 18 2 1 5
17. Y. Li H. Sheng Y. Q. Chen 2011 Analytical impulse response of a fractional second order filter and its impulse response invariant discretization Signal Processing 91 3 498 507 1203.94039 10.1016/j.sigpro.2010.01.017
18. J.G. Lu G.R. Chen 2009 Robust stability and stabilization of fractionalorder interval systems: An LMI approach IEEE Transactions on Automatic Control 54 6 1294 1299 10.1109/TAC.2009.2013056
19. J.G. Lu Y.Q. Chen 2010 Robust stability and stabilization of fractionalorder interval systems with the fractional order α: The 0 < α < 1 case IEEE Transactions on Automatic Control 55 1 152 158 10.1109/TAC.2009.2033738
20. B.N. Lundstrom M.H. Higgs W.J. Spain A.L. Fairhall 2008 Fractional differentation by neocortical pyramidal neurons Nature Neuroscience 11 11 1335 1342 10.1038/nn.2212
21. D. Matignon, Stability results on fractional differential equations with applications to control processing. In: Proceedings of the multiconference on computational engineering in systems and application, Lille, France (1996), 963-968.
22. D. Matignon, B. D'Andrea-Novel. Some results on controllability and observability of finite-dimensional fractional differential systems. In: Proceedings of the multiconference on computational engineering in systems and application, Lille, France (1996), 952-956.
23. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York (1993).
24. C.A. Monje, Y.Q. Chen, B.M. Vinagre, D.Y. Xue, V. Feliu, Fractionalorder Systems and Controls: Fundamentals and Applications. Springer- Verlag, London (2010).
25. K.B. Oldham and J. Spanier. The Fractional Calculus. Academic Press, New York and London (1974).
26. A. Oustaloup B. Mathieu P. Lanusse 1995 The CRONE control of resonant plants: application to a flexible transmission European Journal of Control 1 2 113 121
27. I. Podlubny 1999 Fractional-order systems and PIλDμ Controllers IEEE Transactions on Automatic Control 41 1 208 214 10.1109/9.739144
28. I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).
29. A.G. Radwan A.S. Elwakil A.M. Soliman 2008 Fractional-order sinusoidal oscillators: design procedure and practical examples IEEE Transactions on Circuits and Systems I: Regular Papers 55 7 2051 2063 10.1109/TCSI.2008.918196
30. Y.A. Rossikhin M.V. Shitikova 2010 Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results Applied Mechanics Reviews 63 1 010801-1 010801-52
31. J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus - Theoretical Developments and Applications in Physics and Engineering. Springer, Netherlands (2007).
32. J. Sabatier, C. Farges, J.C. Trigeassou, A stability test for non commensurate fractional order system. In: Proceedings of the 4th IFAC Workshop fractional differentiation and its applications, Badajoz, Spain (2010), 1-6.