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Numerical simulation of the fractional-order control system

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Abstract

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractionalorder dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.

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References

  1. W. M. Ahmad and J. C. Sprott,Chaos in fractional-order autonomous nonlinear systems, Chaos Solitons and Fractals16 (2003), 339–351.

    Article  MATH  Google Scholar 

  2. W. M. Ahmad and J. C. Sprott,Stabilization of generalized fractional order chaos systems using state feedback control, Chaos Solitons and Fractals22 (2004), 141–150.

    Article  MATH  Google Scholar 

  3. C. Bai and Z. Fang,The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Journal of Computational and Applied Mathematics150 (2004), 611–621.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Basu and D. P. Acharya,On quadratic fractional generalized solid bi-criterion, J. Appl. Math. & Computing2 (2002), 131–144.

    Article  MathSciNet  Google Scholar 

  5. K. Diethelm and J. F. Neville,Numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders, http://www.ma.man.ac.uk/MCCM/MCCM.html.

  6. A. M. A. El-Sayed and M. A. E. Aly,Continuation theorem of fractional order evolutionary integral equations, J. Appl. Math. & Computing2 (2002), 525–534.

    MathSciNet  Google Scholar 

  7. R. Gorenflo, F. Mainardi and D. Moretti,Time fractional diffusion: a discrete random walk approach, Journal of Nonlinear Dynamics29 (2000), 129–143.

    Article  MathSciNet  Google Scholar 

  8. Y. Hu and F. Liu,Numerical Methods for a Fractional-Order Control System, Journal of Xiamen University (NATURAL Science)44(3) (2005), 313–317.

    MATH  MathSciNet  Google Scholar 

  9. R. Lin and F. Liu,Analysis of fractional-order numerical method for the fractional relaxation equation, Computational Mechanics, WCCM VI in conjuncton with APCOM’04, (2004) ID-362.

  10. F. Liu, V. Anh and I. Turner,Numerical Solution of the Space Fractional Fokker-Planck Equation, Journal of Computational and Applied Mathematics166 (2004), 209–219.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Liu, V. Anh and I. Turner,Numerical Solution of the Space Fractional Fokker-Planck Equation, ANZIAM J.45 (2004), 461–473.

    MathSciNet  Google Scholar 

  12. F. Liu, S. Shen, V. Anh and I. Turner,Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J.46(E) (2005), 101–117.

    MathSciNet  Google Scholar 

  13. I. Podlubny,Fractional Differential Equations, Academic, Press, New York, 1999.

    MATH  Google Scholar 

  14. Z. B. Wang, G. Y. Cao and X. Zhu,Application of Fractional Calculus in System Modeling, Journal of ShangHa JiaTong University (in Chinese)38 (2004), 802–805.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, 361005, Xiamen, China

    F. Liu

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, 4001, Brisbane, Qld., Australia

    F. Liu

  3. Department of Mathematics, Jimei University, 361021, Xiamen, Fujian, P. R. China

    X. Cai

  4. School of Mathematical Science, Xiamen University, 361005, Xiamen, China

    X. Cai

Authors
  1. X. Cai
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  2. F. Liu
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Correspondence to F. Liu.

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Cai, X., Liu, F. Numerical simulation of the fractional-order control system. J. Appl. Math. Comput. 23, 229–241 (2007). https://doi.org/10.1007/BF02831971

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  • DOI: https://doi.org/10.1007/BF02831971

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