Abstract
In this paper, we propose a new sliding mode control for discrete time nonlinear uncertain systems. The uncertainties include both parametric uncertainties in the state model and external disturbances. To recover the lost invariance and robustness properties of discrete sliding mode control, we develop a disturbance estimation scheme to compensate the system uncertainties without affecting the control law. This control approach ensures the stability of the closed loop system as well as chattering reduction. The performance of the proposed controller is applied to control the motion of a cart-inverted pendulum used as a typical benchmark of nonlinear systems. The stabilization problem of the inverted pendulum system is to design a controller to keep the pendulum in its unstable equilibrium point in the presence of disturbances and parameters variation. The simulation result shows the effectiveness of the control design.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. J. Åström and K. Furuta, “Swinging up a pendulum by energy control,” Automatica, vol. 36, no. 2, pp. 287–295, 2000. [click]
N. Muskinja and B. Tovornik, “Swinging up and stabilization of a real inverted pendulum,” IEEE Transactions on Industrial Electronics, vol. 53, no. 2, pp. 631–639, 2006. [click]
B. A. Elsayed, M. A. Hassan, and S. Mekhilef, “Fuzzy swinging-up with sliding mode control for third order cartinverted pendulum system,” International Journal of Control, Automation, and Systems, vol. 13, no. 1, pp. 1–11, 2015.
V. I. Utkin and H. C. Chang, “Sliding mode control on electro-mechanical systems,” Mathematical Problems in Engineering, vol. 8, no. 4–5, pp. 451–473, 2002. [click]
K. D. Young, V. I. Utkin, and Ö. Özgüner, “A control engineer’s guide to sliding mode control,” IEEE Transactions on Control System Technology, vol. 7, no. 3, pp. 328–342, 1999. [click]
C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor & Francis, Ltd, 1998.
M. Rubagotti, A. Estrada, F. Castanos, A. Ferrara, and L. Fridman, “Integral sliding mode control for nonlinear systems with matched and unmatched perturbations,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2699–2704, 2011.
J. Liu, S. Vazquez, L. Wu, A. Marquez, H. Gao, and L. G Franquelo, “Extended state observer-based sliding-mode control for three-phase power converters,” IEEE Transactions on Industrial Electronics, vol. 64, no. 1, pp. 22–31, 2017. [click]
J. Liu, W. Luo, X. Yang, and L. Wu, “Robust model-based fault diagnosis for PEM fuel cell air-feed system,” IEEE Transactions on Industrial Electronics, vol. 63, no. 5, pp. 3261–3270, 2016. [click]
W. Gao and J. Hung, “Variable structure control of nonlinear systems: a new approach,” IEEE Transactions on Industrial Electronics, vol. 40, no. 1, pp. 45–55, 1993. [click]
J. J. E. Slotine, “Sliding controller design for non-linear systems,” International Journal of Control, vol. 40, no. 2, pp. 421–434, 1984. [click]
G. Wheeler, C. Su, and Y. Stepanenko, “A sliding mode controller with improved adaption laws for the upper bounds on the norm of uncertainties,” Automatica, vol. 34, no. 12, pp. 1657–1661, 1998. [click]
H. Sira-Ramírez, “On the dynamical sliding mode control of nonlinear systems,” International Journal of Control, vol. 57, no. 5, pp. 1039–1061, 1993. [click]
A. G. Bondarev, S. A. Bondarev, N. Y. Kostylyeva, and V. I. Utkin, “Sliding modes in systems with asymptotic observers,” Automation and Remote Control, vol. 46, no. 5, pp. 679–684, 1985.
A. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” International Journal of Control, vol. 76, no. 9, pp. 924–941, 2003. [click]
G. Bartolini, A. Pisano, E. Punta, and E. Usai, “A survey of applications of second order sliding mode control to mechanical systems,” International Journal of Control, vol. 76, no. 9–10, pp. 875–892, 2003. [click]
S. Ding and S. Li, “Second-order sliding mode controller design subject to mismatched term,” Automatica, vol. 77, pp. 388–392, 2017.
S. P. Bhat and D. S. Bernstein, “Continuous finite-time stabilization of the translational and rotational double integrators,” IEEE Transactions on Automatic Control, vol. 43, no. 5, pp. 678–682, 1998. [click]
S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM Journal on Control and Optimization, vol. 38, no. 3, pp. 751–766, 2000. [click]
A. Polyakov and L. Fridman, “Stability notions and Lyapunov functions for sliding mode control systems,” Journal of the Franklin Institute, vol. 351, no. 4, pp. 1831–1865, 2014.
S. Huang and Z. Xiang, “Finite-time stabilization of switched stochastic nonlinear systems with mixed odd and even powers,” Automatica, vol. 73, pp. 130–137, November 2016.
S. Huang and Z. Xiang, “Finite-time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching,” International Journal of Robust and Nonlinear Control, vol. 26, no. 10, pp. 2136–2152, July 2016. [click]
J. Mao, S. Huang and Z. Xiang, “Adaptive practical finitetime stabilization for switched nonlinear systems in pure feedback form,” Journal of the Franklin Institute, vol. 354, no. 10, pp. 3971–3994, 2017.
O. Kaynak and A. Denker, “Discrete-time sliding mode control in the presence of system uncertainty,” International Journal of Control, vol. 57, no. 5, pp. 1177–1189, 1993. [click]
A. Bartoszewicz, “Discrete-Time Quasi-Sliding-Mode Control Strategies,” IEEE Transactions on Industrial Electronics, vol. 45, no. 4, pp. 633–637, 1998. [click]
Y. Lin, Y. Shi, and R. Burton, “Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system,” IEEE/ASME Transactions on Mechatronics, vol. 18, no. 1, pp. 1–10, 2013.
H. Ma, J. Wu, and Z. Xiong, “Discrete-time sliding-mode control with improved quasi-sliding-mode domain,” IEEE Transactions on Industrial Electronics, vol. 63, no. 10, pp. 6292–6304, 2016. [click]
A. Z. Sarpturk, S. Y. Istefanopulos, and O. Kaynak, “On the stability of discrete-time sliding mode control systems,” IEEE Transactions on Automatic Control, vol. 32, no. 10, pp. 930–937, 1987.
D. Milosavljevic, “General conditions for the existence of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems,” Automation and Remote Control, vol. 46, no. 3, pp. 307–314, 1985.
K. Furuta, “Sliding mode control of a discrete system,” System and Control Letters, vol. 14, no. 2, pp. 145–152, 1990.
C. C. Cheng, M. H. Lin, and J. M. Hsiao, “Sliding mode controller design for linear discrete-time systems with matching perturbations,” Automatica, vol. 36, no. 8, pp. 1205–1211, 2000.
H. Romdhane, K. Dehri, and A. S. Nouri, “Stability analysis of discrete input output second order sliding mode control,” International Journal of Modelling, Identification and control, vol. 22, no. 2, pp. 159–169, 2014.
A. Bartoszewicz and P. Le´sniewski, “Reaching law approach to the sliding mode control of periodic review inventory systems,” IEEE Transactions on Automation Science and Engineering, vol. 11, no. 3, pp. 810–817, 2014. [click]
S. Qu, X. Xia, and J. Zhang, “Dynamics of discrete-time sliding mode control under uncertain systems with a disturbance compensator,” IEEE Transactions on Industrial Electronics, vol. 61, no. 7, pp. 3502–3510, 2014. [click]
K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, New Jersey, 1996.
W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure control systems,” IEEE Transactions on Industrial Electronics, vol. 42, no. 2, pp. 117–122, 1995. [click]
J. Kim, S. Oh, D. Cho, and J. Hedrick, “Robust discretetime variable structure control methods,” Journal of Dynamic Systems, Measurement, and Control, vol. 122, no. 4, pp. 766–775, 2000. [click]
E. A. Misawa, “Discrete-time sliding mode control: the linear case,” Journal of Dynamic Systems, Measurement, and Control, vol. 119, no. 4, pp. 819–821, 1997. [click]
J. H. Williams Jr., Fundamentals of Applied Dynamics, John Wiley & Sons, New York, 1996.
J. Zhao and M. W. Spong, “Hybrid control for global stabilization of the cart pendulum system,” Automatica, vol. 37, no. 12, pp. 1941–1951, 2001. [click]
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Sung Jin Yoo under the direction of Editor PooGyeon Park. The authors appreciate the Associate Editor and anonymous reviewers for their constructive comments on the basis of which the presentation of this paper has been greatly improved.
Jalel Ghabi received the Master degree in Automatic Control and Industrial Computing from University of Sfax in 2003. In 2009, he obtained his Ph.D. degree in Automatic from University of Monstir. He is currently an associate professor at University of Kairouan, Tunisia. His research interests include Sliding mode control, Robust predictive control, Chaos control.
Hedi Dhouibi received the Master degree in Industrial Maintenance from University of Tunisa in 1999. In 2005, he obtained his Ph.D. degree in Automatic Control from the Institute of Sciences and Technologies of Lille, France. He is currently a professor at University of Kairouan. His research interests include Intelligent control, Diagnostics and fault detection.
Rights and permissions
About this article
Cite this article
Ghabi, J., Dhouibi, H. Discrete Time Sliding Mode Controller Using a Disturbance Compensator for Nonlinear Uncertain Systems. Int. J. Control Autom. Syst. 16, 1156–1164 (2018). https://doi.org/10.1007/s12555-017-0185-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-017-0185-0