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Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

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Abstract

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

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References

  1. Svante W and Lars E, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1994, 1(5): 826–839.

    Article  Google Scholar 

  2. Cao X, Abhirup D, Fahad A B, et al., Fractional-order model of the disease Psoriasis: A control based mathematical approach, Journal of Systems Science and Complexity, 2016, 29(6): 1565–1584.

    Article  MathSciNet  Google Scholar 

  3. Monje C A, Chen Y, Vinagre B M, et al., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer-Verlag, London, UK, 2010.

    Book  Google Scholar 

  4. West B J, Fractional Calculus View of Complexity: Tomorrow's Science, CRC Press, New York, 2016.

    Book  Google Scholar 

  5. Podlubny I, Fractional-order systems and PI λ Dμ-controllers, IEEE Transactions on Automatic Control, 1999, 44(1): 208–214.

    Article  MathSciNet  Google Scholar 

  6. Oustaloup A, Cois O, Lanusse P, et al., The CRONE aproach: Theoretical developments and major applications, IFAC Proceedings Volumes, 2006, 39(11): 324–354.

    Article  Google Scholar 

  7. Matignon D, Stability properties for generalized fractional differential systems, ESAIM: Proceedings, 1998, 5: 145–158.

    Article  MathSciNet  Google Scholar 

  8. Liang S, Wang S, and Wang Y, Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees, Journal of the Franklin Institute, 2017, 354(1): 83–104.

    Article  MathSciNet  Google Scholar 

  9. Farges C, Moze M, and Sabatier J, Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica, 2010, 46(10): 1730–1734.

    Article  MathSciNet  Google Scholar 

  10. Vinagre B M and Feliu V, Optimal fractional controllers for rational order systems: A special case of the Wiener-Hopf spectral factorization method, IEEE Transactions on Automatic Control, 2007, 52(12): 2385–2389.

    Article  MathSciNet  Google Scholar 

  11. Lu J and Chen Y, Robust stability and stabilization of fractional-order interval systems with the fractional order: The 0 α 1 case, IEEE Transactions on Automatic Control, 2010, 55(1): 152–158.

    Article  MathSciNet  Google Scholar 

  12. Padula F, Alcántara S, Vilanova R, et al., H ∞ control of fractional linear systems, Automatica, 2013, 47(7): 2276–2280.

    Article  Google Scholar 

  13. Liang S, Peng C, Liao Z, et al., State space approximation for general fractional order dynamic systems, International Journal of Systems Science, 2014, 45(10): 2203–2212.

    Article  MathSciNet  Google Scholar 

  14. Khalil H K, Nonlinear Systems, 3rd Edition, Prentice Hall, New Jersey, 2002.

    MATH  Google Scholar 

  15. Li Y, Chen Y, and Podlubny I, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 2009, 45(8): 1965–1969.

    Article  MathSciNet  Google Scholar 

  16. Aguila-Camacho N, Duarte-Mermoud M A, and Gallegos J A, Lyapunov functions for fractional order systems, Communications in Nonlinear Science and Numerical Simulation, 2014, 19(9): 2951–2957.

    Article  MathSciNet  Google Scholar 

  17. Sabatier J, Merveillaut M, Malti R, et al., How to impose physically coherent initial conditions to a fractional system?, Communications in Nonlinear Science and Numerical Simulation, 2010, 15(5): 1318–1326.

    Article  MathSciNet  Google Scholar 

  18. Trigeassou J C, Maamri N, Sabatier J, et al., State variables and transients of fractional order differential systems, Computers and Mathematics with Applications, 2012, 64(10): 3117–3140.

    Article  MathSciNet  Google Scholar 

  19. Sabatier J, Farges C, and Trigeassou J C, Fractional systems state space description: Some wrong ideas and proposed solutions, Journal of Vibration and Control, 2014, 20(7): 1076–1084.

    Article  MathSciNet  Google Scholar 

  20. Montseny G, Diffusive representation of pseudo-differential time-operators, ESAIM: Proceedings, 1998, 5: 159–175.

    Article  MathSciNet  Google Scholar 

  21. Trigeassou J C, Maamri N, Sabatier J, et al., A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 2011, 91(3): 437–445.

    Article  Google Scholar 

  22. Hartley T T, Trigeassou J C, Lorenzo C F, et al., Energy storage and loss in fractional-order systems, Journal of Computational and Nonlinear Dynamics, 2015, 10(6): 061006.

    Article  Google Scholar 

  23. Trigeassou J C, Maamri N, and Oustaloup A, Lyapunov stability of commensurate fractional order systems: A physical interpretation, Journal of Computational and Nonlinear Dynamics, 2016, 11(5): 051007.

    Article  Google Scholar 

  24. Trigeassou J C, Maamri N, and Oustaloup A, Lyapunov stability of noncommensurate fractional order systems: An energy balance approach, Journal of Computational and Nonlinear Dynamics, 2016, 11(4): 041007.

    Article  Google Scholar 

  25. Bonnet C and Partington J R, Coprime factorizations and stability of fractional differential systems, Systems & Control Letters, 2000, 41(3): 167–174.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Shu Liang

  2. Institute of Information Spreading Engineering, Changchun University of Technology, Changchun, 130012, China

    Yinshan Liang

Authors
  1. Shu Liang
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  2. Yinshan Liang
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Corresponding author

Correspondence to Shu Liang.

Additional information

This research was supported by Fundamental Research Funds for the China Central Universities of USTB under Grant No. FRF-TP-17-088A1.

This paper was recommended for publication by Editor ZHAO Yanlong.

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Liang, S., Liang, Y. Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems. J Syst Sci Complex 32, 1544–1559 (2019). https://doi.org/10.1007/s11424-019-7049-z

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  • DOI: https://doi.org/10.1007/s11424-019-7049-z

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