Skip to main content
SpringerLink
Log in
Book cover

Sliding Modes after the First Decade of the 21st Century pp 169–197Cite as

Second-Order Sliding Mode Approaches to Control and Estimation for Fractional Order Dynamics

  • Chapter

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 412))

Abstract

This chapter outlines some results concerning the application of second-order sliding-mode techniques in the framework of control and estimation problems for some classes of fractional-order systems (FOS). Concerning the control problems, a second-order sliding mode control approach is developed to stabilize a class of linear uncertain multivariable fractional-order dynamics. Concerning estimation and observation problems, two main results are illustrated. A method for reconstructing in finite time an external disturbance acting on a known FOS is presented, and, as a second instance, a method for estimating the discrete state of a switched FOS is discussed. Both the schemes make use of second-order sliding mode observers. The method for discrete state reconstruction in switched FOS find useful application in the framework of fault detection, as shown in the experimental section part. Key point of all the approaches herein presented is the use of fractional-order sliding manifolds. Simple controller/observer tuning formulas are constructively developed along the paper by Lyapunov analysis. Simulation and experimental results confirm the expected performance.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.P.: A general formulation and solution for fractional optimal control problems. Nonlinear Dynamics 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrawal, O.P., Baleanu, D.: A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems. Journal of Vibrations and Control 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atanacković, T.M., Pilipović, S., Zorica, D.: A diffusion wave equation with two fractional derivatives of different order. Journal of Physics A 40, 5319–5333 (2007)

    Article  MATH  Google Scholar 

  5. Atanacković, T.: On Distributed Derivative Model of a Viscoelastic Body. C. R. Mecanique 331, 687–692 (2003)

    Article  MATH  Google Scholar 

  6. Bartolini, G., Ferrara, A., Levant, A., Usai, E.: On Second Order Sliding Mode Controllers. In: Young, K.D., Ozguner, U. (eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control. LNCIS, vol. 247, pp. 329–350. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  7. Bartolini, G., Fridman, L., Pisano, A., Usai, E.: Modern Sliding Mode Control Theory. In: New Perspectives and Applications. LNCIS, vol. 375. Springer, Heidelberg (2008)

    Google Scholar 

  8. Barbosa, R.S., Tenreiro Machado, J.A., Silva, M.F.: Time-Domain Design of Fractional Differintegrators using Least Squares. Signal Processing 86, 2567–2581 (2006)

    Article  MATH  Google Scholar 

  9. Bettayeb, M., Djennoune, S.: A note on the controllability and the observability of fractional dynamical systems. In: Proc. 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, pp. 506–511 (2006)

    Google Scholar 

  10. Calderon, A.J., Vinagre, B.M., Felix, V.: On Fractional Sliding Mode Control. In: 7th Portuguese Conference on Automatic Control (CONTROLO 2006), Lisbon, Portugal (2006)

    Google Scholar 

  11. Chen, Y., Ahn, H.S., Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing 86, 2611–2618 (2006)

    Article  MATH  Google Scholar 

  12. Chen, Y., Vinagre, B.M., Podlubny, I.: On fractional order disturbance observers. In: Proc. of The First IFAC Symposium on FOS, Chicago (US), pp. 1–8 (2003)

    Google Scholar 

  13. Dzieliski, A., Sierociuk, D.: Observer for discrete fractional order systems. In: Proc. 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, pp. 524–529 (2006)

    Google Scholar 

  14. Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)

    MATH  Google Scholar 

  15. Dadras, S., Momeni, H.R.: Control of a fractional-order economical system via sliding mode. Physica A 389(12), 2434–2442 (2010)

    Article  MathSciNet  Google Scholar 

  16. Debnath, L.: A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technl. 35(4), 487–501 (2004)

    Article  MathSciNet  Google Scholar 

  17. Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4+ T-cells. Mathematical and Computer Modelling 50(3), 386–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Edwards, C., Spurgeon, S.K.: Sliding Mode Control: Theory And Applications. Taylor and Francis, Abington (1998)

    Google Scholar 

  19. Efe, M.O., Kasnakog̃lu, C.: A Fractional Adaptation Law for Sliding Mode Control. International Journal of Adaptive Control and Signal Processing 22, 968–986 (2008)

    Article  MathSciNet  Google Scholar 

  20. Efe, M.O.: Fractional Order Sliding Mode Controller Design for Fractional Order Dynamic Systems. In: Gven, Z.B., Baleanu, D., Tenreiro Machado, J.A. (eds.) New Trends in Nanotechnology and Fractional Calculus Applications (2009); manuscript presented during Fractional Differentiation and Its Applications, Ankara, Turkey (November 2008)

    Google Scholar 

  21. Emelyanov, S.V., Korovin, S.K., Levant, A.: Higher-order sliding modes in control systems. Differential Equations 29(11), 1627–1647 (1993)

    MathSciNet  Google Scholar 

  22. Emelyanov, S.V., Korovin, S.K.: Control complex and uncertain systems: new types of feed back. Springer Communication and Control Engineering Series, vol. 302 (2000)

    Google Scholar 

  23. Hartley, T.T., Lorenzo, C.F.: Dynamics and Control of Initialized Fractional-Order Systems. Nonlinear Dynamics 29, 201–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jeličić, Z.D., Petrovački, N.: Optimality conditions and a solution scheme for fractional optimal control problems. Structural and Multidisciplinary Optimization 38, 571–581 (2009)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  26. Ladaci, S., Charef, A.: On fractional adaptive control. Nonlinear Dynamics 43(4), 365–378 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58, 1247–1263 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76, 924–941 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Luo, Y., Chen, Y.Q.: Fractional order proportional-derivative controller for a class of fractional order systems. Automatica 45(10), 2446–2450 (2009)

    Article  MATH  Google Scholar 

  30. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, New Yark (2006)

    Google Scholar 

  31. Manabe, S.: The non-integer integral and its application to control systems. Electrotechnical Journal of Japan 6(3-4), 83–87 (1961)

    Google Scholar 

  32. Melchior, P., Cugnet, M., Sabatier, J., Poty, A., Oustaloup, A.: Flatness Control of a Fractional Thermal System. In: Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.) Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Heidelberg (2007)

    Google Scholar 

  33. Molinari, B.P.: A strong contollability and observability in linear multivariable control. IEEE Transaction on Automatic Control 21(5), 761–764 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Moreno, J.A., Osorio, M.: A Lyapunov approach to second-order sliding mode controllers and observers. In: 47th IEEE Conference on Decision and Control, Cancun, MX (2008)

    Google Scholar 

  35. Mozyrska, D., Bartosiewicz, Z.: Observability Concepts for Nonlinear Discrete-Time Fractional Order Control Systems. In: Proc. 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey (2008)

    Google Scholar 

  36. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, London (1974)

    MATH  Google Scholar 

  37. Oustaloup, A., Moreau, X., Nouillant, M.: The CRONE Suspension. Control Engineering Practice 4(8), 1101–1108 (1996)

    Article  Google Scholar 

  38. Petráš, I., Podlubny, I., O’Leary, P., Dorcak, L., Vinagre, B.: Analogue Realizations of Fractional-Order Controllers. In: FBERG, TU Kosice, Kosice (2002)

    Google Scholar 

  39. Polyakov, A., Poznyak, A.: Lyapunov function design for finite- time convergence analysis: ”Twisting” controller for second-order slidingmode realization. Automatica 45(2), 444–448 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pisano, A., Rapaic, M., Jelecic, Z., Usai, E.: Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. International Journal of Robust and Nonlinear Control (2010), doi:10.1002/rnc.1565 (published online on April 9, 2010)

    Google Scholar 

  41. Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)

    MATH  Google Scholar 

  42. Podlubny, I.: Fractional Order Systems and PI λ D μ-Controllers. IEEE Transactions on Automatic Control 44(1), 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Physical Review E 53(2), 1890–1899 (1995)

    Article  MathSciNet  Google Scholar 

  44. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus - Theoretical Developments and Applications in Physics and Engineering. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  45. Scalas, E.: On the Application of Fractional Calculus in Finance and Economics. Plenary Lecture paper. In: Proc. 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey (2008)

    Google Scholar 

  46. Si-Ammour, A., Djennoune, S., Bettayeb, M.: A sliding mode control for linear fractional systems with input and state delays. Communications in Nonlinear Science and Numerical Simulation 14(5), 2310–2318 (2009)

    Article  MathSciNet  Google Scholar 

  47. Sira-Ramírez, H., Feliu-Battle, V.: A Generalized PI Sliding Mode and PWM Control of Switched Fractional Systems. In: Bartolini, G., Fridman, L., Pisano, A., Usai, E. (eds.) Modern Sliding Mode Control Theory. New Perspectives and Applications. LNCIS, vol. 375, pp. 215–236. Springer, Heidelberg (2008)

    Google Scholar 

  48. Tavazoei, M.H., Haeri, M.: A note on the stability of fractional order systems. Mathematics and Computers in Simulation 79, 1566–1576 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)

    MATH  Google Scholar 

  50. Young, K.D., Özgüner, U., Utkin, V.I.: A Control Engineer’s Guide to Sliding Mode Control. IEEE Trans. Control Systems Technology 7, 328–342 (1999)

    Article  Google Scholar 

  51. Vinagre, B.M., Chen, Y.Q., Petráš, I.: Two Direct Tustin Discretization Methods for Fractional-Order Differentiator/Integrator. Journal of the Franclin Institute 34, 349–362 (2003)

    Article  Google Scholar 

  52. Vinagre, B.M., Petracs, I., Podlubny, I., Chen, Y.Q.: Using fractional order adjustment rules and fractional order reference models in model reference adaptive control. Nonlinear Dyn. 29(1-4), 269–279 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Elettrica ed Elettronica (DIEE), Universitá degli studi di Cagliari, Cagliari, Italy

    A. Pisano & E. Usai

  2. Automation and Control Systems Department, University of Novi Sad, Novi Sad, Serbia

    M. Rapaić

Authors
  1. A. Pisano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Rapaić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Usai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Ingenieria de Control y Robotica, Division de Ingenieria Electrica, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Instituto de Ingenieria, 04510, Mexico, D.F., Mexico

    Leonid Fridman

  2. Instituto de Ingenieria, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, 04510, Mexico, D.F., Mexico

    Jaime Moreno

  3. Instituto de Ingenieria, Departamento de Ingenieria de Control y Robotica, Division de Ingenieria Electrica, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria,, 04510, Mexico, D.F., Mexico

    Rafael Iriarte

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pisano, A., Rapaić, M., Usai, E. (2011). Second-Order Sliding Mode Approaches to Control and Estimation for Fractional Order Dynamics. In: Fridman, L., Moreno, J., Iriarte, R. (eds) Sliding Modes after the First Decade of the 21st Century. Lecture Notes in Control and Information Sciences, vol 412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22164-4_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22164-4_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22163-7

  • Online ISBN: 978-3-642-22164-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation