Nonlinear Dynamics

, Volume 66, Issue 1–2, pp 141–152 | Cite as

Stabilization of fractional order systems using a finite number of state feedback laws

  • Saeed Balochian
  • Ali Khaki Sedigh
  • Mohammad Haeri
Original Paper


In this paper, the stabilization of linear time-invariant systems with fractional derivatives using a limited number of available state feedback gains, none of which is individually capable of system stabilization, is studied. In order to solve this problem in fractional order systems, the linear matrix inequality (LMI) approach has been used for fractional order systems. A shadow integer order system for each fractional order system is defined, which has a behavior similar to the fractional order system only from the stabilization point of view. This facilitates the use of Lyapunov function and convex analysis in systems with fractional order 1<q<2. To this end, an extremum-seeking method is used for obtaining Lyapunov function and defining a suitable sliding sector in order to enable switching between available control gains for system stabilization. Consequently, using the LMI approach in fractional order systems, necessary and sufficient conditions are provided for stabilization of systems with fractional order 1<q<2 using a limited number of available state feedback gains which lead to variable structure control.


Fractional order hybrid system Linear matrix inequality (LMI) Variable structure control 


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  1. 1.
    Edwards, C., Spurgeon, S.K.: Sliding Mode Control Theory and Applications. Taylor & Francis, London (1998) Google Scholar
  2. 2.
    Utkin, V.I.: Variable structure systems with sliding modes. IEEE Trans. Autom. Cont. 22, 212–222 (1977) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    DeCarlo, R.A., Drakunov, S., Li, X.: A unifying characterization of robust sliding mode control: A Lyapunov approach. ASME J. Dyn. Syst. 122, 708–718 (2000) CrossRefGoogle Scholar
  4. 4.
    Furuta, K.: Sliding mode control of a discrete system. Syst. Control. Lett. 14, 145–152 (1990) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Furuta, K., Pan, Y.: Variable structure control with sliding sector. Automatica 36, 211–228 (2000) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Pan, Y., Suzuki, S., Furuta, K.: Variable structure control for hybrid system with sliding sector. In: Proceedings of the 6th Asian-Pacific Conference on Control and Measurement, Chengdu, China, pp. 350–353 (2004) Google Scholar
  7. 7.
    Pan, Y., Furuta, K.: Hybrid control with sliding sector. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005) Google Scholar
  8. 8.
    Pan, Y., Ozguner, U.: Sliding mode extremum seeking control for linear quadratic dynamic game. In: Proceedings of the American Control Conference, vol. 1, pp. 614–619 (2004) Google Scholar
  9. 9.
    Ahn, H.S., Chen, Y.Q.: Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44(11), 2985–2988 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ahn, H.S., Chen, Y.Q., Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, 27–34 (2007) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Banavar, R.N.: Extremum seeking loops with quadratic functions: Estimation and control. Int. J. Control 76, 1475–1482 (2003) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Nonlinear Non-integer Order Circuits and Systems – An Introduction. World Scientific Series on Nonlinear Science, vol. 38. Singapore, World Scientific (2000) CrossRefGoogle Scholar
  13. 13.
    Baleanu, D., Güvenç, Z.B., Machado, J.A.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  14. 14.
    Machado, J.A.T.: Variable structure control of manipulators with joints having flexibility and backlash. Syst. Anal. Model. Simul. 23(1–2), 93–101 (1996) MATHGoogle Scholar
  15. 15.
    Matignon, D.: Stability properties for generalized fractional differential systems. In: Systèmes différentiels fractionnaires – modèles, méthodes et applications. ESAIM: Proc., vol. 5 (1998) Google Scholar
  16. 16.
    Pan, Y., Ozguner, U.: Stability and performance improvement of extremum seeking control with sliding mode. Int. J. Control 60, 968–985 (2003) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering. Springer, London (2007) MATHGoogle Scholar
  18. 18.
    Ferreira, N.M.F., Machado, J.A.T., Galhano, A.M., Cunha, J.B.: Fractional order position/force robot control. In: Proceedings of the IFAC Workshop on Fractional Differentiation and Its Applications, Bordeaux, France, July 19–21 (2004) Google Scholar
  19. 19.
    Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. Trans. ASME 51, 294–298 (1984) MATHCrossRefGoogle Scholar
  20. 20.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004) MATHCrossRefGoogle Scholar
  21. 21.
    Pommier, V., Sabatier, J., Lanusse, P., Oustaloup, A.: Crone control of a nonlinear hydraulic actuator. Control Eng. Pract. 10(4), 391–402 (2002) CrossRefGoogle Scholar
  22. 22.
    Lurie, B.J.: Three-parameter tunable tiltintegral-derivative (TID) controller. US Patent US5371670 (1994) Google Scholar
  23. 23.
    Bouafoura, M.K., Braiek, N.B.: PIλ Dμ controller design for integer and fractional plants using piecewise orthogonal functions. Commun. Nonlinear Sci. Numer. Simulat. 15(5), 1267–1278 (2009) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Raynaud, H.F., Zergaïnoh, A.: State-space representation for fractional order controllers. Automatica 36(7), 1017–1021 (2000) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Si-Ammour, A., Djennoune, S., Bettayeb, M.: A sliding mode control for linear fractional systems with input and state delays. Commun. Nonlinear Sci. Numer. Simulat. 14, 2310–2318 (2009) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A, 387(1), 57–70 (2008) CrossRefGoogle Scholar
  27. 27.
    El-Khezali, R., Ahmad, W.H.: Variable structure control of fractional time-delay systems. In: Proceedings of the 2nd IFAC, Workshop on Fractional Differentiation and its Applications, Porto, Portugal, July 19–21 (2006) Google Scholar
  28. 28.
    Hotzel, R., Fliess, M.: On linear system with a fractional derivation: Introductory theory and examples. Math. Comput. Simul. 45, 385–395 (1998) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Ortigueira, M.D.: An introduction to the fractional continuous-time linear systems: The 21st century systems. IEEE Circuits Syst. Mag., 8(3), 19–26 (2008) CrossRefGoogle Scholar
  30. 30.
    Vinagre, B.M., Monje, C.A., Calderon, A.J.: Fractional order systems and fractional order actions. In: IEEE Conference on Decision and Control, Tutorial Workshop #2, Las Vegas, NV, December 9 (2002) Google Scholar
  31. 31.
    Pan, Y., Furuta, K.: Variable Structure Control Systems Using Sectors for Switching Rule Design. Springer, Berlin (2006) Google Scholar
  32. 32.
    El-Khazali, R.: On the state space modeling of fractional systems. In: 2nd IFAC Workshop on Fractional Differentiation and Applications (FDA’06), Porto, Portugal, July 19–21 (2006) Google Scholar
  33. 33.
    El-Khazali, R., Ahmad, W., Al-Assaf, Y.: Sliding mode control of fractional chaotic systems. In: First IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, pp. 495–500, July 19–21 (2004) Google Scholar
  34. 34.
    El-Khazali, R., Ahmad, W., Al-Assaf, Y.: Sliding mode control of generalized fractional chaotic systems. Int. J. Bifurc. Chaos. 16(10), 1–13 (2006) MathSciNetGoogle Scholar
  35. 35.
    Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008) MATHGoogle Scholar
  36. 36.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, IMACS, IEEE-SMC, vol. 2, pp. 963–968, Lille, France, July (1996) Google Scholar
  37. 37.
    Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59(5), 1594–1609 (2010) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Oustaloup, A.: La Derivation non entiere: theorie, synthese et applications. Editions Hermes, Paris (1994) Google Scholar
  39. 39.
    Machado, J.A.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal. Modell. Simul. 27(2–3), 107–122 (1997) MATHGoogle Scholar
  40. 40.
    Poinot, T., Trigeassou, J.C.: A method for modeling and simulation of fractional systems. Signal Process. 83, 2319–2333 (2003) MATHCrossRefGoogle Scholar
  41. 41.
    Sabatier, J., Merveillaut, M., Malti, R., Oustaloup, A.: How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear. Sci. Numer. Simulat. 15(5), 1318–1326 (2010) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Saeed Balochian
    • 1
  • Ali Khaki Sedigh
    • 2
  • Mohammad Haeri
    • 3
  1. 1.Department of Electrical Engineering, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Electrical and Computer EngineeringK.N. Toosi University of TechnologyTehranIran
  3. 3.Department of Electrical EngineeringSharif University of TechnologyTehranIran

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