Fractional Calculus and Applied Analysis

, Volume 17, Issue 1, pp 136–153 | Cite as

Robust stability bounds of uncertain fractional-order systems

  • YingDong Ma
  • Jun-Guo Lu
  • WeiDong Chen
  • YangQuan Chen
Research Paper

Abstract

This paper considers the robust stability bound problem of uncertain fractional-order systems. The system considered is subject either to a two-norm bounded uncertainty or to a infinity-norm bounded uncertainty. The robust stability bounds on the uncertainties are derived. The fact that these bounds are not exceeded guarantees that the asymptotical stability of the uncertain fractional-order systems is preserved when the nominal fractional-order systems are already asymptotically stable. Simulation examples are given to demonstrate the effectiveness of the proposed theoretical results.

Key Words and Phrases

fractional-order system linear matrix inequality robust stability bound uncertainty 

MSC 2010

Primary 26A33 Secondary 34A08, 34D10, 93C73, 93D09, 93D21 

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References

  1. [1]
    H.S. Ahn and Y. Q. Chen, Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44, No 11 (2008), 2985–2988.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    H.S. Ahn, Y.Q. Chen, and I. Podlubny, Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, No 1 (2007), 27–34.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh, and R.R. Nigmatullin, Newtonian law with memory. Nonlinear Dynamics 60, No 1–2 (2010), 81–86.CrossRefMATHGoogle Scholar
  4. [4]
    D. Cafagna, Fractional calculus: A mathematical tool from the past for present engineers. IEEE Industrial Electronics Magazine 1, No 2 (2007), 35–40.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Y.Q. Chen, H.S. Ahn, and I. Podlubny, Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing 86, No 10 (2006), 2611–2618.CrossRefMATHGoogle Scholar
  6. [6]
    H. Delavari, D. Baleanu, and J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynamics 67, No 4 (2012), 2433–2439.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    K. Diethelm, N.J. Ford, and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29, No 1–4 (2002), 3–22.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    C. Farges, M. Moze, and J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica 46, No 10 (2010), 1730–1734.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    C. Farges, J. Sabatier, and M. Moze, Fractional order polytopic systems: Robust stability and stabilization. Advances in Difference Equations 2011, No 1 (2011), 1–10.CrossRefMathSciNetGoogle Scholar
  10. [10]
    S.E. Hamamci, An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Trans. Autom. Control 52, No 10 (2007), 1964–1969.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Z. Jiao and Y.Q. Chen, Impulse response of a generalized fractional second order filter. Fract. Calc. Appl. Anal. 15, No 1 (2012), 97–116; DOI: 10.2478/s13540-012-0007-2; at http://link.springer.com/article/10.2478/s13540-012-0007-2.MATHMathSciNetGoogle Scholar
  12. [12]
    Z. Jiao and Y.Q. Chen, Stability analysis of fractional-order systems with double noncommensurate orders for matrix case. Fract. Calc. Appl. Anal. 14, No 3 (2011), 436–453; DOI: 10.2478/s13540-011-0027-3; at http://link.springer.com/article/10.2478/s13540-011-0027-3.MATHMathSciNetGoogle Scholar
  13. [13]
    P.P. Khargonekar, I.R. Petersen, and K.M. Zhou, Robust stabilization of uncertain linear systems: Quadratic stabilizability and H control theory. IEEE Trans. Autom. Control 35, No 3 (1990), 356–361.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    Y.H. Lan, and Y. Zhou, LMI-based robust control of fractional-order uncertain linear systems. Computers and Mathematics with Appl. 62, No 3 (2011), 1460–1471.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    C. Li, and J.C. Wang, Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0 < α < 1 case. J. of the Franklin Institute 349, No 7 (2012), 2406–2419.CrossRefGoogle Scholar
  16. [16]
    J.G. Lu, and G.R. Chen, Robust stability and stabilization of fractional-order interval systems: An LMI approach. IEEE Trans. Autom. Control 54, No 6 (2009), 1294–1299.CrossRefGoogle Scholar
  17. [17]
    J.G. Lu, and Y.Q. Chen, Robust stability and stabilization of fractionalorder interval systems with the fractional order α: The 0 < α < 1 case. IEEE Trans. Autom. Control 55, No 1 (2010), 152–158.CrossRefGoogle Scholar
  18. [18]
    J.A.T. Machado (Guest Editor), Special issue on fractional calculus and applications. Nonlinear Dynamics 29, No 1-4 (2002), 1–385.CrossRefMathSciNetGoogle Scholar
  19. [19]
    R.L. Magin, Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Appl. 59, No 5 (2010), 1586–1593.Google Scholar
  20. [20]
    R.L. Magin, W.G. Li, M.P. Velasco, J. Trujillo, D.A. Reiter, A. Morgenstern, and R.G. Spencer, Anomalous NMR relaxation in cartilage matrix components and native cartilage: Fractional-order models. J. of Magnetic Resonance 210, No 2 (2011), 184–191.CrossRefGoogle Scholar
  21. [21]
    R.L. Magin, M.D. Ortigueira, I. Podlubny, and J. Trujillo, On the fractional signals and systems. Signal Processing 91, No 3 (2011), 350–371.CrossRefMATHGoogle Scholar
  22. [22]
    D. Matignon, Stability results for fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC Lille, France (1996), 963–968.Google Scholar
  23. [23]
    C.A. Monje, Y.Q. Chen, B.M. Vinagre, D.Y. Xue, and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications. Springer-Verlag, 2010.CrossRefGoogle Scholar
  24. [24]
    R.R. Nigmatullin, A.A. Arbuzov, F. Salehli, A. Giz, I. Bayrak, and H. Catalgil-Giz, The first experimental confirmation of the fractional kinetics containing the complex-power-law exponents: Dielectric measurements of polymerization reactions. Physica B: Condensed Matter 388, No 1–2 (2007), 418–434.CrossRefGoogle Scholar
  25. [25]
    R.R. Nigmatullin and S.O. Nelson, Recognition of the fractional kinetics in complex systems: Dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz. Signal Processing 86, No 10 (2006), 2744–2759.CrossRefMATHGoogle Scholar
  26. [26]
    M.D. Ortigueira, An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits and Systems Magazine 8, No 3 (2008), 19–26.CrossRefGoogle Scholar
  27. [27]
    A. Oustaloup, B. Mathieu, and P. Lanusse, The CRONE control of resonant plants: Application to a flexible transmission. European J. of Control 1, No 2 (1995), 113–121.CrossRefGoogle Scholar
  28. [28]
    I. Petras, Y.Q. Chen, B.M. Vinagre, and I. Podlubny, Stability of linear time invariant systems with interval fractional orders and interval coefficients. In: Second IEEE Int. Conference on Computation Cybernetics Viena Austria (2005), 341–346.Google Scholar
  29. [29]
    I. Petras and R.L. Magin, Simulation of drug uptake in a two compartmental fractional model for a biological system. Commun. Nonlinear Sci. Numer. Simulation 16, No 12 (2011), 4588–4595.CrossRefMATHGoogle Scholar
  30. [30]
    I. Podlubny, Fractional Differential Equations. Academic Press, London (1999).MATHGoogle Scholar
  31. [31]
    J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, 2007.CrossRefGoogle Scholar
  32. [32]
    J. Sabatier, C. Farges, M. Merveillaut, and L. Feneteau, On observability and pseudo state estimation of fractional order systems. European Journal of Control 18, No 3 (2012), 260–271.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    J. Sabatier, C. Farges, and J.C. Trigeassou, Fractional systems state space description: some wrong ideas and proposed solutions. J. on Vibration and Control, To appear, DOI: 10.1177/1077546313481839.Google Scholar
  34. [34]
    J. Sabatier, M. Merveillaut, R. Malti, and A. Oustaloup, How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simulation 15, No 5 (2010), 1318–1326.CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    J. Sabatier, M. Moze, and C. Farges, LMI stability conditions for fractional order systems. Computers and Mathematics with Appl. 59, No 5 (2010), 1594–1609.CrossRefMATHMathSciNetGoogle Scholar
  36. [36]
    N. Tan, Ö. F. Özgüven, and M.M. Özyetkin, Ro, Robust stability analysis of fractional order interval polynomials. ISA Transactions 48, No 2 (2009), 166–172.CrossRefGoogle Scholar
  37. [37]
    M.S. Tavazoei and M. Haeri, Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach. Automatica 46, No 1 (2010), 94–100.CrossRefMATHMathSciNetGoogle Scholar
  38. [38]
    M.S. Tavazoei and M. Haeri, A note on the stability of fractional order systems. Mathematics and Computers in Simulation 79, No 5 (2009), 1566–1576.CrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    J.A. Tenreiro Machado, Analysis and design of fractional-order digital control systems. Systems Analysis Modelling Simulation 27, No 2–3 (1997), 107–122.MATHGoogle Scholar
  40. [40]
    J.C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, State variables and transients of fractional order differential systems. Computers and Mathematics with Applications 64, No 10 (2012), 3117–3140.CrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    J.C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, Transients of fractional order integrator and derivatives. Signal, Image and Video Processing 6, No 3 (2012), 359–372.CrossRefGoogle Scholar
  42. [42]
    B.M. Vinagre and V. Feliu, Optimal fractional controllers for rational order systems: a special case of the wiener-hopf spectral factorization method. IEEE Trans. Autom. Control 52, No 12 (2007), 2385–2389.CrossRefMathSciNetGoogle Scholar
  43. [43]
    D.J. Wang, and X.L. Gao, H∞ design with fractional-order PDµ controllers. Automatica 48, No 5 (2012), 974–977.CrossRefMATHGoogle Scholar
  44. [44]
    R.K. Yedavalli and Z. Liang, Reduced conservatism in stability robustness bounds by state transformation. IEEE Trans. Autom. Control 31, No 9 (1986), 863–866.CrossRefMATHGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  • YingDong Ma
    • 1
  • Jun-Guo Lu
    • 1
  • WeiDong Chen
    • 1
  • YangQuan Chen
    • 2
  1. 1.Department of AutomationShanghai Jiao Tong University and Key Laboratory of System Control and Information Processing Ministry of Education of ChinaShanghaiP.R. China
  2. 2.Mechatronics, Embedded Systems and Automation (MESA) LabSchool of Engineering, University of CaliforniaMercedUSA

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