Nonlinear Dynamics

, Volume 92, Issue 3, pp 1379–1393 | Cite as

On dynamic sliding mode control of nonlinear fractional-order systems using sliding observer

  • Ali Karami-Mollaee
  • Hamed Tirandaz
  • Oscar Barambones
Original Paper


In this study, a new fractional-order dynamic sliding mode control (FDSMC) for a class of nonlinear systems is presented. In FDSMC, an integrator is placed before the input control signal of the plant, in order to remove the chattering. However, in FDSMC method, the dimension of the resulted system (integrator plus the system) is bigger than the primary system. As a result, the model of the plant is needed to be known completely, in order to stabilize the system. Then, a sliding observer is proposed to extract an appropriate model for the unknown nonlinear system. Then, the chattering free controller can be obtained such that the closed-loop system has the desired properties. Lyapunov theory is used to verify the stability problem of the presented both observer and controller. For practical applications consideration, we have not applied the upper bound of the system dynamic either in controller or in observer. The effectiveness of the proposed method in comparison to the conventional fractional sliding mode control (FSMC) method is addressed. We have utilized a same observer in both control approach, in order to have a valid comparison. The simplicity of the proposed FDSMC method in concept and also in realization can be seen with comparison of the relevant equations. In addition, it is clear that the FDSMC can remove chattering completely, while chattering is the main challenge of the conventional FSMC. Finally, the validity of the proposed method is shown by some simulation examples based on the Arneodo chaotic system.


Fractional-order dynamic sliding mode control (FDSMC) Fractional-order sliding mode control (FSMC) Nonlinear system Sliding observer Arneodo chaotic system 


  1. 1.
    Utkin, V., Lee, H.: Chattering problem in sliding mode control systems, In: International Workshop on Variable Structure Systems, 2006. VSS’06, pp. 346–350. IEEE (2006)Google Scholar
  2. 2.
    Fuyang, C., Zhang, K., Jiang, B., Wen, C.: Adaptive sliding mode observer based robust fault reconstruction for a helicopter with actuator fault. Asian J. Control 18(4), 1558–1565 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hung, J.Y., Gao, W., Hung, J.C.: Variable structure control: a survey. IEEE Trans. Ind. Electron. 40(1), 2–22 (1993)CrossRefGoogle Scholar
  4. 4.
    Karami-Mollaee, A., Pariz, N., Shanechi, H.: Position control of servomotors using neural dynamic sliding mode. J. Dyn.Syst. Meas. Control 133(6), 061014 (2011)CrossRefGoogle Scholar
  5. 5.
    Tagne, G., Talj, R., Charara, A.: Immersion and invariance vs sliding mode control for reference trajectory tracking of autonomous vehicles, In: Control Conference (ECC), 2014 European, pp. 2888–2893. IEEE (2014)Google Scholar
  6. 6.
    Fuh, C.-C.: Variable-thickness boundary layers for sliding mode control. J. Mar. Sci. Technol. 16(4), 288–294 (2008)Google Scholar
  7. 7.
    Chen, H.-M., Renn, J.-C., Su, J.-P.: Sliding mode control with varying boundary layers for an electro-hydraulic position servo system. Int. J. Adv. Manuf. Technol. 26(1), 117–123 (2005)CrossRefGoogle Scholar
  8. 8.
    Zhang, X.: Sliding mode-like fuzzy logic control with adaptive boundary layer for multiple-variable discrete nonlinear systems. J. Intell. Syst. 25(2), 209–220 (2016)Google Scholar
  9. 9.
    Allamehzadeh, H., Cheung, J.Y.: Optimal fuzzy sliding mode control with adaptive boundary layer. WSEAS Trans. Syst. 3(5), 1887–1892 (2004)Google Scholar
  10. 10.
    Cucuzzella, M., Incremona, G.P., Ferrara, A.: Design of robust higher order sliding mode control for microgrids. IEEE J. Emerg. Sel. Top. Circuits Syst. 5(3), 393–401 (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Nonaka, R., Yamashita, Y., Tsubakino, D.: General scheme for design of higher-order sliding-mode controller. In: American Control Conference (ACC), 2015, pp. 5176–5181. IEEE (2015)Google Scholar
  12. 12.
    Koshkouei, A.J., Burnham, K.J., Zinober, A.S.: Dynamic sliding mode control design. IEE Proc. Control Theory Appl. 152(4), 392–396 (2005)CrossRefGoogle Scholar
  13. 13.
    Moldoveanu, F.: Sliding mode controller design for robot manipulators. Bull. Transilv. Univ. Brasov Eng. Sci. Ser. I 7(2), 97 (2014)Google Scholar
  14. 14.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58(6), 1247–1263 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Plestan, F., Glumineau, A., Laghrouche, S.: A new algorithm for high-order sliding mode control. Int. J. Robust Nonlinear Control 18(4–5), 441–453 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gao, J., Cai, Y.: Higher order sliding mode control with fast transient performance. In: 2015 Seventh International Conference on Measuring Technology and Mechatronics Automation (ICMTMA), pp. 524–528. IEEE (2015)Google Scholar
  18. 18.
    Hadri-Hamida, A.: Higher-order sliding mode control scheme with an adaptation low for uncertain power dc–dc converters. J. Control Autom. Electr. Syst. 26(2), 125–133 (2015)CrossRefGoogle Scholar
  19. 19.
    Shtessel, Y.B., Fridman, L., Zinober, A.: Higher order sliding modes. Int. J. Robust Nonlinear Control 18(4–5), 381–384 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yang, Y., Qin, S., Jiang, P.: A modified super-twisting sliding mode control with inner feedback and adaptive gain schedule. Int. J. Adapt. Control Signal Process. 31(3), 398–416 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Perruquetti, W., Barbot, J.-P.: Sliding Mode Control in Engineering. CRC Press, Boca Raton (2002)CrossRefGoogle Scholar
  22. 22.
    Chen, M.-S., Hwang, Y.-R., Tomizuka, M.: A state-dependent boundary layer design for sliding mode control. IEEE Trans. Autom. Control 47(10), 1677–1681 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chen, M.-S., Chen, C.-H., Yang, F.-Y.: An ltr-observer-based dynamic sliding mode control for chattering reduction. Automatica 43(6), 1111–1116 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kilbsa, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol 204. Elsevier, New York, NY, USA (2006)Google Scholar
  25. 25.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, San Diego (1998)zbMATHGoogle Scholar
  26. 26.
    Rapaić, M.R., Jeličić, Z.D.: Optimal control of a class of fractional heat diffusion systems. Nonlinear Dyn. 62(1), 39–51 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Efe, M.Ö., Kasnakoglu, C.: A fractional adaptation law for sliding mode control. Int. J. Adapt. Control Signal Process. 22(10), 968–986 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Efe, M.Ö.: Fractional order sliding mode controller design for fractional order dynamic systems. In: Baleanu D., Guvenc Z., Machado J. (eds.) New Trends in Nanotechnology and Fractional Calculus Applications, 463–470. Springer, Dordrecht (2010)Google Scholar
  29. 29.
    Ni, J., Liu, L., Liu, C., Hu, X.: Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems. Nonlinear Dyn. 89, 2065–2083 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Muoz-Vzquez, A.J., Parra-Vega, V., Snchez-Orta, A.: Fractional integral sliding modes for robust tracking of nonlinear systems. Nonlinear Dyn. 87(2), 895–901 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jakovljević, B., Pisano, A., Rapaić, M., Usai, E.: On the sliding-mode control of fractional-order nonlinear uncertain dynamics. Int. J. Robust Nonlinear Control 26(4), 782–798 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bettayeb, M., Djennoune, S.: Design of sliding mode controllers for nonlinear fractional-order systems via diffusive representation. Nonlinear Dyn. 84(2), 593–605 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Dadras, S., Momeni, H.R.: Fractional-order dynamic output feedback sliding mode control design for robust stabilization of uncertain fractional-order nonlinear systems. Asian J. Control 16(2), 489–497 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang, D., Cao, L., Tang, S.: Fractional-order sliding mode control for a class of uncertain nonlinear systems based on lqr. Int. J. Adv. Robot. Syst. 14(2), 1729881417694290 (2017)Google Scholar
  35. 35.
    Fuyang, C., Lei, W., Zhang, K., Tao, G., Jiang, B.: A novel nonlinear resilient control for a quadrotor UAV via backstepping control and nonlinear disturbance observer. Nonlinear Dyn. 85(2), 1281–1295 (2016)CrossRefzbMATHGoogle Scholar
  36. 36.
    Ren, W., Kumar, P.: Stochastic adaptive prediction and model reference control. IEEE Trans. Autom. Control 39(10), 2047–2060 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, Y.-T., Kung, T.-T., Chang, K.-M., Chen, S.-Y.: Observer-based adaptive sliding mode control for pneumatic servo system. Precis. Eng. 37(3), 522–530 (2013)CrossRefGoogle Scholar
  38. 38.
    Xia, Y., Zhu, Z., Fu, M.: Back-stepping sliding mode control for missile systems based on an extended state observer. IET Control Theory Appl. 5(1), 93–102 (2011)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Butt, Q.R., Bhatti, A.I., Mufti, M.R., Rizvi, M.A., Awan, I.: Modeling and online parameter estimation of intake manifold in gasoline engines using sliding mode observer. Simul. Model. Pract. Theory 32, 138–154 (2013)CrossRefGoogle Scholar
  40. 40.
    Davila, J., Fridman, L., Levant, A.: Second-order sliding-mode observer for mechanical systems. IEEE Trans. Autom. Control 50(11), 1785–1789 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Xiong, Y., Saif, M.: Sliding mode observer for nonlinear uncertain systems. IEEE Trans. Autom. Control 46(12), 2012–2017 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Benchaib, A., Rachid, A., Audrezet, E., Tadjine, M.: Real-time sliding-mode observer and control of an induction motor. IEEE Trans. Ind. Electron. 46(1), 128–138 (1999)CrossRefGoogle Scholar
  43. 43.
    Liu, C.: Circuit Theory and Applications for Fractional-orderChaotic Systems. Xian Jiaotong University Press, Xian (2011)Google Scholar
  44. 44.
    Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Slotine, J.-J.E., Li, W., et al.: Applied Nonlinear Control, vol. 199. Prentice-Hall, Englewood Cliffs (1991)zbMATHGoogle Scholar
  46. 46.
    Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability. Comput. Math. Appl. 59, 181021 (2010)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs (1991)zbMATHGoogle Scholar
  48. 48.
    Lu, J.G.: Chaotic dynamics and synchronization of fractional-order arneodos systems. Chaos Solitons Fractals 26(4), 1125–1133 (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Electrical and Computer Engineering FacultyHakim Sabzevari UniversitySabzevarIran
  2. 2.Automatic Control and System Engineering DepartmentUniversity of the Basque Country, UPV/EHUVitoriaSpain

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