Synchronization of chaotic fractional-order systems via active sliding mode controller

Physica A: Statistical Mechanics and its Applications

Volumn 387, Issue 1, 2008, Pages 57-70

  Tavazoei, Mohammad Saleh ^a   Haeri, Mohammad ^a  

^a SHARIF UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Active sliding mode controller; Chaos synchronization; Chaotic fractional order systems

Indexed keywords

CHAOTIC SYSTEMS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; SYNCHRONIZATION;

ACTIVE SLIDING MODE CONTROLLER; CHAOS SYNCHRONIZATION; CHAOTIC FRACTIONAL-ORDER SYSTEMS; FRACTIONAL DIFFERENTIAL EQUATIONS;

SLIDING MODE CONTROL;

EID: 35748971445     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2007.08.039     Document Type: Article

References (63)

1. D. del-Castillo-Negrete, Fractional calculus: basic theory and applications (Part I), in: Lectures Presented at the Institute of Mathematics UNAM, Mexico, August 2005.
2. Bagley R.L., and Calico R.A. Fractional order state equations for the control of visco-elastically damped structures. J. Guidance Control Dyn. 14 (1991) 304-311
3. Friedrich Ch., and Braun H. Linear viscoelastic behavior of complex behavior of complex polymeric materials: a fractional calculus approach. Acta Polym. 46 (1995) 385-390
4. Rossikhin Y.A., and Shitikova M.V. Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system. Acta Mech. 120 (1997) 109-125
5. Heaviside O. Electromagnetic Theory (1971), Chelsea, New York
6. Engheta N. On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44 4 (1996) 554-566
7. Le Méhauté A. Les Géométries Fractales (1990), Eidtions Hermès, Paris, France
8. de Levie R. Fractals and rough electrodes. J. Electroanal. Chem. 281 (1990) 1-21
9. Westerlund S. Dead matter has memory! (capacitor model). Phys. Scrip. 43 2 (1991) 174-179
10. Arkhincheev V.E. Anomalous diffusion in inhomogeneous media: Some exact results. Model. Meas. Contr. A 26 2 (1993) 11-29
11. El-Sayed A.M.A. Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35 2 (1996) 311-322
12. Ichise M., Nagayanagi Y., and Kojima T. An analog simulation of non-integer order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33 (1971) 253-265
13. Kaplan T., Gray L.J., and Liu S.H. Self-affine fractal model for a metal-electrolyte interface. Phys. Rev. B 35 10 (1987) 5379-5381
14. Le Méhauté A., and Crepy G. Introduction to transfer and motion in fractal media: the geometry of kinetics. Solid State Ionics 9 & 10 (1983) 17-30
15. Nakagawa M., and Sorimachi K. Basic characteristics of a fractance device. IEICE Trans. Fundamentals E75-A 12 (1992) 1814-1819
16. Oldham K.B., and Zoski C.G. Analogue instrumentation for processing polarographic data. J. Electroanal. Chem. 157 (1983) 27-51
17. Chen G., and Friedman G. An RLC interconnect model based on Fourier analysis. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24 2 (2005) 170-183
18. Jenson V.G., and Jeffreys G.V. Mathematical Methods in Chemical Engineering. second ed (1977), Academic Press, New York
19. Cole K.S. Electric conductance of biological systems. Proc. Cold Spring Harbor Symp. Quant. Biol. (1993), Cold Spring Harbor, New York 107-116
20. Anastasio T.J. The fractional-order dynamics of brainstem vestibule-oculumotor neurons. Biol. Cybern. 72 (1994) 69-79
21. A. Oustaloup, From fractality to non integer derivation through recursivity, a property common to these two concepts: a fundamental idea for a new process control strategy, in: Proceeding of the 12th IMACS World Congress, Paris, July 1988, vol. 3, pp. 203-208.
22. Laskin N. Fractional market dynamics. Physica A 287 (2000) 482-492
23. In: Hilfer R. (Ed). Applications of Fractional Calculus in Physics (2000), World Scientific Pub Co, Singapore
24. Mandelbrot B., and Van Ness J.W. Fractional brownian motions, fractional noises and applications. SIAM Rev. 10 4 (1968) 422-437
25. Oustaloup A. La Dérivation Non Entière: Théorie, Synthèse et Applications (1995), Editions Hermès, Paris, France
26. μ-controllers. IEEE Trans. Automat. Control 44 1 (1999) 208-214
27. H. Linares, Ch. Baillot, A. Oustaloup, Ch. Ceyral, Generation of a fractal ground: Application in robotics, in: International Congress in IEEE-SMC CESA'96 IMACS Multiconf., Lille, July 1996.
28. Duarte F.B.M., and Macado J.A.T. Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators. Nonlinear Dyn. 29 (2002) 315-342
29. Hartley T.T., Lorenzo C.F., and Qammer H.K. Chaos in a fractional order Chua's system. IEEE Trans. CAS-I 42 (1995) 485-490
30. P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proceedings ECCTD, Budapest, 1997, pp. 1259-1262.
31. Ahmad W.M., and Sprott J.C. Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons Fractals 16 (2003) 339-351
32. Lu J.G., and Chen G. A note on the fractional-order Chen system. Chaos, Solitons Fractals 27 3 (2006) 685-688
33. Lu J.G. Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354 4 (2006) 305-311
34. Li C., and Chen G. Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A: Stat. Mech. Appl. 341 (2004) 55-61
35. Lu J.G. Chaotic dynamics and synchronization of fractional-order Arneodo's systems. Chaos, Solitons Fractals 26 4 (2005) 1125-1133
36. L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton-Leipnik system with fractional order, Chaos, Solitons Fractals, 2006, doi:10.1016/j.chaos.2006.06.013.
37. Ott E., Grebogi C., and Yorke J.A. Controlling chaos. Phys. Rev. Lett. 64 (1990) 1196-1199
38. D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application multi-conference, vol. 2, IMACS, in: IEEE-SMC Proceedings, Lille, France, July 1996, pp. 963-968.
39. Deng W., and Li C.P. Chaos synchronization of the fractional Lu system. Physica A 353 (2005) 61-72
40. Deng W., and Li C. Synchronization of chaotic fractional Chen system. J. Phys.Soc. Jpn 74 6 (2005) 1645-1648
41. Li C.P., Deng W.H., and Xu D. Chaos synchronization of the Chua system with a fractional order. Physica A 360 (2006) 171-185
42. Podlubny I. Fractional Differential Equations (1999), Academic Press, New York
43. Charef A., Sun H.H., Tsao Y.Y., and Onaral B. Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37 9 (1992) 1465-1470
44. Diethelm K. An algorithm for the numerical solution of differential equations of fractional order. Elec. Trans. Numer. Anal. 5 (1997) 1-6
45. Diethelm K., Ford N.J., and Freed A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29 (2002) 3-22
46. Diethelm K., Ford N.J., and Freed A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms 36 (2004) 31-52
47. Li C., and Peng G. Chaos in Chen's system with a fractional order. Chaos Solitons Fractals 22 (2004) 443-450
48. Diethelm K., and Ford N.J. Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229-248
49. Bai E.W., and Lonngren K.E. Synchronization of two Lorenz systems using active control. Chaos, Solitons Fractals 8 (1997) 51-58
50. Agiza H.N., and Yassen M.T. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys. Lett. A 278 (2001) 191-197
51. Ho M.Ch., and Hung Y.Ch. Synchronization of two different systems by using generalized active control. Phys. Lett. A 301 (2002) 424-428
52. Zhang H., Ma X.K., and Liu W.Z. Synchronization of chaotic systems with parametric uncertainty using active sliding mode control. Chaos, Solitons Fractals 21 (2004) 1249-1257
53. Yau H.T. Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Solitons Fractals 22 (2004) 341-347
54. Lin J.S.H., Yan J.J., and Liao T.L. Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity. Chaos, Solitons Fractals 24 (2005) 371-381
55. Haeri M., and Emadzadeh A.A. Synchronizing different chaotic systems using active sliding mode control. Chaos, Solitons Fractals 31 1 (2007) 119-129
56. Tavazoei M.S., and Haeri M. Determination of active sliding mode controller parameters in synchronizing different chaotic systems. Chaos, Solitons Fractals 32 (2007) 583-591
57. Haeri M., Tavazoei M.S., and Naseh M.R. Synchronization of uncertain chaotic systems using active sliding mode control. Chaos, Solitons Fractals 33 (2007) 1230-1239
58. Ahmed E., El-Sayed A.M.A., and El-Saka H.A.A. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325 (2007) 542-553
59. Lü J., and Chen G. A new chaotic attractor coined. Int. J. Bifurcation Chaos 12 3 (2002) 659-661
60. Lorenz E.N. Deterministic non-periodic flows. J. Atmos. Sci. 20 1 (1963) 130-141
61. Chen G., and Ueta T. Yet another attractor. Int. J. Bifurcation Chaos 9 (1999) 1465-1466
62. Li C., and Peng G. Chaos in Chen's system with a fractional order. Chaos, Solitons Fractals 22 (2004) 443-450
63. Li C., and Peng G. Chaos in the fractional order Chen system with and its control. Chaos, Solitons Fractals 22 (2004) 549-554