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Synchronization of Integral and Fractional Order Chaotic Systems pp 219–236Cite as

An Observer for a Class of Incommensurate Fractional-Order Systems

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Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter, we present a new observer-model-free type of synchronization for a certain class of incommensurate fractional-order systems. We apply our proposals to the master–slave synchronization scheme, where the unknown dynamics are considered the master system, and we propose an observer structure as a slave system that estimates the unknown state variables. For solving this problem, we introduce a new incommensurate fractional algebraic observability (IFAO) property, which is used as the main ingredient in the design of the slave system. Some numerical results show the effectiveness of the suggested approach.

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Notes

  1. 1.

    Equation (14.16) is nonautonomous, but the Lipschitz condition ensures a unique solution [10].

References

  1. T. Zhou, C. Li, Synchronization of fractional-order differential systems. Physica D 212, 111–125 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. L. Pecora, T. Carrol, Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. C. Li, J. Yan, The synchronization of three fractional differential systems. Chaos Solitons Fractals 32, 751–757 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  4. J.G. Lu, Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos Solitons Fractals 27, 519–525 (2006)

    Article  ADS  MATH  Google Scholar 

  5. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, 367–386 (2002)

    MATH  MathSciNet  Google Scholar 

  6. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Dover Books on Mathematics (Dover Publications, New York, 2006)

    Google Scholar 

  7. I. Podlubny, Fractional Differential Equations, 9th edn. (Academic, San Diego, 1999)

    MATH  Google Scholar 

  8. J.J. de Espindola, J. Neto, E. Lopes, A generalized fractional derivative approach to viscoelastic material properties measurement. Appl. Math. Comput. 164, 493–506 (2005)

    Article  MATH  Google Scholar 

  9. K.S. Miller, S.G. Samko, Completely monotonic functions. Integr. Transforms Spec. Funct. 12 (4), 389–402 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  11. M.S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional-order systems. Physica D 237, 2628–2637 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. R. Caponetto, G. Dongola, L. Fortuna, I. Petrás, Fractional Order Systems. World Scientific Series on Nonlinear Science, Series A, vol. 72 (World Scientific Publishing, Singapore, 2010), pp. 53–75

    Google Scholar 

  13. W.C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36(5), 1305–1314 (2008)

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Control Automatico, CINVESTAV-IPN, Mexico, D.F., Mexico

    Rafael Martínez-Guerra, Claudia A. Pérez-Pinacho & Gian Carlo Gómez-Cortés

Authors
  1. Rafael Martínez-Guerra
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  2. Claudia A. Pérez-Pinacho
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  3. Gian Carlo Gómez-Cortés
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Martínez-Guerra, R., Pérez-Pinacho, C.A., Gómez-Cortés, G.C. (2015). An Observer for a Class of Incommensurate Fractional-Order Systems. In: Synchronization of Integral and Fractional Order Chaotic Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-15284-4_14

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