Nonlinear Dynamics

, Volume 85, Issue 1, pp 633–643

# Sliding mode control with a second-order switching law for a class of nonlinear fractional order systems

• Yuquan Chen
• Yiheng Wei
• Hua Zhong
• Yong Wang
Original Paper

## Abstract

This article investigates a novel sliding model control approach for a class of nonlinear fractional order systems. In particular, the sliding surface with additional nonlinear part is designed by a Lyapunov-like function and one can achieve much more satisfying system performances by selecting suitable parameters. Moreover, a second-order switching law is generated from the commonly used adaptive switching law. Its properties are carefully discussed, and it is proven that the reaching time of the sliding surface can be guaranteed nonsensitive to the initial conditions with the second-order switching law, while the adaptive switching law cannot even guarantee the finite-time convergence. Then the stability of the closed-loop control system is rigorously analyzed via indirect Lyapunov method. Finally, simulation examples are presented to illustrate the effectiveness of the proposed control method.

## Keywords

Fractional order systems Sliding mode control Second-order switching law Indirect Lyapunov method Frequency distributed model

## Notes

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

## References

1. 1.
Vinagre, B.M., Feliu, V.: Modeling and control of dynamic system using fractional calculus: application to electrochemical processes and flexible structures. In: 41st IEEE Conference on Decision and Control, vol. 1, pp. 214–239 (2002)Google Scholar
2. 2.
Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926–935 (2013)
3. 3.
Wilkie, K.P., Drapaca, C.S., Sivaloganathan, S.: A nonlinear viscoelastic fractional derivative model of infant hydrocephalus. Appl. Math. Comput. 217(21), 8693–8704 (2011)
4. 4.
Aghababa, M.P.: Control of non-linear non-integer-order systems using variable structure control theory. Trans. Inst. Meas. Control 36(3), 425–432 (2014)
5. 5.
Efe, M.Ö.: Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38(6), 1561–1570 (2008)
6. 6.
Wei, Y.H., Chen, Y.Q., Liang, S., Wang, Y.: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 165, 395–402 (2015)
7. 7.
Yang, J., Li, S.H., Yu, X.H.: Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans. Ind. Electron. 60(1), 160–169 (2013)
8. 8.
Aghababa, M.P.: Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems. Int. J. Control 86(10), 1744–1756 (2013)
9. 9.
Aghababa, M.P.: A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems. Nonlinear Dyn. 78(3), 2129–2140 (2014)
10. 10.
Chen, L.P., Wu, R.C., He, Y.G., Chai, Y.: Adaptive sliding-mode control for fractional-order uncertain linear systems with nonlinear disturbances. Nonlinear Dyn. 80(1–2), 51–58 (2015)
11. 11.
Aghababa, M.P.: A switching fractional calculus-based controller for normal non-linear dynamical systems. Nonlinear Dyn. 75(3), 577–588 (2014)
12. 12.
Li, R.H., Chen, W.S.: Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems. Nonlinear Dyn. 76(1), 785–795 (2014)
13. 13.
Yang, N.N., Liu, C.X.: A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control. Nonlinear Dyn. 74(3), 721–732 (2013)
14. 14.
Zhang, L.G., Yan, Y.: Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control. Nonlinear Dyn. 76(3), 1761–1767 (2014)
15. 15.
Aghababa, M.P.: A fractional sliding mode for finite-time control scheme with application to stabilization of electrostatic and electromechanical transducers. Appl. Math. Model. 39(20), 6103–6113 (2015)
16. 16.
Yin, C., Chen, Y.Q., Zhong, S.M.: Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50(12), 3173–3181 (2014)
17. 17.
Yin, C., Stark, B., Chen, Y.Q., Zhong, S.M., Lau, E.: Fractional-order adaptive minimum energy cognitive lighting control strategy for the hybrid lighting system. Energy Build. 87, 176–184 (2015)
18. 18.
Chen, D.Y., Liu, Y.X., Ma, X.Y., Zhang, R.F.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67(1), 893–901 (2012)
19. 19.
Yin, C., Zhong, S.M., Chen, W.F.: Design of sliding mode controller for a class of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 356–366 (2012)
20. 20.
Yuan, J., Shi, B., Ji, W.Q.: Adaptive sliding mode control of a novel class of fractional chaotic systems. Adv. Math. Phys. 2013 (2013). doi:
21. 21.
Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3), 437–445 (2011)
22. 22.
Liu, L.P., Han, Z.Z., Li, W.L.: Global sliding mode control and application in chaotic systems. Nonlinear Dyn. 56(1–2), 193–198 (2009)
23. 23.
Yin, C., Cheng, Y.H., Chen, Y.Q., Stark, B., Zhong, S.M.: Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems. Nonlinear Dyn. 82(1), 1–14 (2015)
24. 24.
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.Y., Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)Google Scholar
25. 25.
Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: State variables and transients of fractional order differential systems. Comput. Math. Appl. 64(10), 3117–3140 (2012)
26. 26.
Yu, Y.G., Li, H.X., Wang, S., Yu, J.Z.: Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Solitons Fract. 42(2), 1181–1189 (2009)
27. 27.
Li, C.G., Chen, G.R.: Chaos in the fractional order Chen system and its control. Chaos Solitons Fract. 22(3), 549–554 (2004)
28. 28.
Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354(4), 305–311 (2006)
29. 29.
Wang, X.Y., Wang, M.J.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos Interdiscip. J. Nonlinear Sci. 17(3), 033,106–033,111 (2007)
30. 30.
Zhang, Y.B., Zhou, T.S.: Three schemes to synchronize chaotic fractional-order Rucklidge systems. Int. J. Mod. Phys. B 21(12), 2033–2044 (2007)
31. 31.
Feng, Y., Han, F., Yu, X.: Chattering free full-order sliding-mode control. Automatica 50(4), 1310–1314 (2014)
32. 32.
Mobayen, S.: An adaptive chattering-free PID sliding mode control based on dynamic sliding manifolds for a class of uncertain nonlinear systems. Nonlinear Dyn. 1–8 (2015)Google Scholar
33. 33.
Cong, B.L., Chen, Z., Liu, X.D.: On adaptive sliding mode control without switching gain overestimation. Int. J. Robust Nonlinear Control 24(3), 515–531 (2014)
34. 34.
Wei, Y.H., Gao, Q., Peng, C., Wang, Y.: A rational approximate method to fractional order systems. Int. J. Control Autom. Syst. 12(6), 1180–1186 (2014)