Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems

Sonia Mahjoub; Fai&#231;al Mnif; Nabil Derbel

doi:10.1007/s11633-015-0880-3

April 2015

Article Contents

Sonia Mahjoub, Faiçal Mnif and Nabil Derbel. Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems. International Journal of Automation and Computing, vol. 12, no. 2, pp. 134-141, 2015. doi: 10.1007/s11633-015-0880-3

Cite as: Sonia Mahjoub, Faiçal Mnif and Nabil Derbel. Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems. International Journal of Automation and Computing, vol. 12, no. 2, pp. 134-141, 2015. doi: 10.1007/s11633-015-0880-3

Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems

Advanced Control and Energy Management Laboratory, University of Sfax, Sfax Engineering School, Sfax 3038, Tunisia;
Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat, Oman

Received: 2014-03-23

Abstract

In this paper, first-order and second-order sliding mode controllers for underactuated manipulators are proposed. Sliding mode control (SMC) is considered as an effective tool in different studies for control systems. However, the associated chattering phenomenon degrades the system performance. To overcome this phenomenon and track a desired trajectory, a twisting, a super-twisting and a modified super-twisting algorithms are presented respectively. The stability analysis is performed using a Lyapunov function for the proposed controllers. Further, the four different controllers are compared with each other. As an illustration, an example of an inverted pendulum is considered. Simulation results are given to demonstrate the effectiveness of the proposed approaches.
- Underactuated manipulator,
- sliding mode control,
- twisting algorithm,
- super-twisting algorithm,
- inverted pendulum

References

[1]	A. Isidori. Nonlinear Control Systems, Berlin, Germany: Springer Verlag, 1995.
[2]	W. Perruquetti, J. P. Barbot. Sliding Mode Control in En-gineering, FL, USA: CRC Press, 2002.
[3]	J. A. Burton, A. S. I. Zinober. Continuous approximation of variable structure control. International Journal of Systems Science, vol. 17, no. 6, pp. 875-885, 1986.
[4]	H. Lee, V. I. Utkin. Chattering suppression methods in sliding mode control systems. Annual Reviews in Control, vol. 31, no. 2, pp. 179-188, 2007.
[5]	K. C. Veluvolu, Y. C. Soh, W. Cao. Robust observer with sliding mode estimation for nonlinear uncertain systems. IET Control Theory Applications, vol. 1, no. 5, pp. 1533-1540, 2007.
[6]	A. Levant. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, vol. 76, no. 9, pp. 924-941, 2003.
[7]	A. Levant. Chattering analysis. IEEE Transactions on Au-tomatic Control, vol. 55, no. 6, pp. 1380-1389, 2010.
[8]	J. A. Moreno, M. Osorio. Strict Lyapunov functions for the super twisting algorithm. IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 1035-1040, 2012.
[9]	I. Nagesha, C. Edwardsb. A multivariable super-twisting sliding mode approach. Automatica, vol. 50, no. 3, pp. 984-988, 2014.
[10]	I. Fantoni, R. Lozano. Non-linear Control for Underactu-ated Mechanical Systems, London, UK: Springer, pp. 21-42, 2002.
[11]	B. Jakubczyk, W. Respondek. On the linearization of con-trol systems. Bult. Acad. Polon. Sei. Math., vol. 28, pp. 517-522, 1980.
[12]	J. Zhao, M. W. Spong. Hybrid control for global stabi-lization of the cart pendulum system. Automatica, vol. 37, no. 12, pp. 1941-1951, 2001.
[13]	W. D. Chang, R. Hwang, J. G. Chsieh. A self-tuning PID control for a class of nonlinear systems based on the Lya-punov approach. Journal of Process Control, vol. 12, no. 2, pp. 233-242, 2002.
[14]	D. Voytsekhovsky, R. M. Hirschorn. Stabilization of single-input nonlinear systems using higher-order term compen-sating sliding mode control. International Journal of Robust and Nonlinear Control, vol. 18, no. 4-5, pp. 468-480, 2008.
[15]	C. Aguilar, R. Hirschorn. Stabilization of an Inverted Pen-dulum, Report on Summer Natural Sciences and Engineer-ing Research Council of Canada, Research Project, Queens University at Kingston, Department of Mathematics and Statistics, 2003.
[16]	S. V. Emel'yanov, S. V. Korovin, L. V. Levantovsky. Higher-order sliding modes in control systems. Differential Equations, vol. 29, no. 11, pp. 1627-1647, 1993.

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Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems

Advanced Control and Energy Management Laboratory, University of Sfax, Sfax Engineering School, Sfax 3038, Tunisia;

Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat, Oman

Received: 2014-03-23

Key words:

Abstract: In this paper, first-order and second-order sliding mode controllers for underactuated manipulators are proposed. Sliding mode control (SMC) is considered as an effective tool in different studies for control systems. However, the associated chattering phenomenon degrades the system performance. To overcome this phenomenon and track a desired trajectory, a twisting, a super-twisting and a modified super-twisting algorithms are presented respectively. The stability analysis is performed using a Lyapunov function for the proposed controllers. Further, the four different controllers are compared with each other. As an illustration, an example of an inverted pendulum is considered. Simulation results are given to demonstrate the effectiveness of the proposed approaches.

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Reference (16)

[1]	A. Isidori. Nonlinear Control Systems, Berlin, Germany: Springer Verlag, 1995.
[2]	W. Perruquetti, J. P. Barbot. Sliding Mode Control in En-gineering, FL, USA: CRC Press, 2002.
[3]	J. A. Burton, A. S. I. Zinober. Continuous approximation of variable structure control. International Journal of Systems Science, vol. 17, no. 6, pp. 875-885, 1986.
[4]	H. Lee, V. I. Utkin. Chattering suppression methods in sliding mode control systems. Annual Reviews in Control, vol. 31, no. 2, pp. 179-188, 2007.
[5]	K. C. Veluvolu, Y. C. Soh, W. Cao. Robust observer with sliding mode estimation for nonlinear uncertain systems. IET Control Theory Applications, vol. 1, no. 5, pp. 1533-1540, 2007.
[6]	A. Levant. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, vol. 76, no. 9, pp. 924-941, 2003.
[7]	A. Levant. Chattering analysis. IEEE Transactions on Au-tomatic Control, vol. 55, no. 6, pp. 1380-1389, 2010.
[8]	J. A. Moreno, M. Osorio. Strict Lyapunov functions for the super twisting algorithm. IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 1035-1040, 2012.
[9]	I. Nagesha, C. Edwardsb. A multivariable super-twisting sliding mode approach. Automatica, vol. 50, no. 3, pp. 984-988, 2014.
[10]	I. Fantoni, R. Lozano. Non-linear Control for Underactu-ated Mechanical Systems, London, UK: Springer, pp. 21-42, 2002.
[11]	B. Jakubczyk, W. Respondek. On the linearization of con-trol systems. Bult. Acad. Polon. Sei. Math., vol. 28, pp. 517-522, 1980.
[12]	J. Zhao, M. W. Spong. Hybrid control for global stabi-lization of the cart pendulum system. Automatica, vol. 37, no. 12, pp. 1941-1951, 2001.
[13]	W. D. Chang, R. Hwang, J. G. Chsieh. A self-tuning PID control for a class of nonlinear systems based on the Lya-punov approach. Journal of Process Control, vol. 12, no. 2, pp. 233-242, 2002.
[14]	D. Voytsekhovsky, R. M. Hirschorn. Stabilization of single-input nonlinear systems using higher-order term compen-sating sliding mode control. International Journal of Robust and Nonlinear Control, vol. 18, no. 4-5, pp. 468-480, 2008.
[15]	C. Aguilar, R. Hirschorn. Stabilization of an Inverted Pen-dulum, Report on Summer Natural Sciences and Engineer-ing Research Council of Canada, Research Project, Queens University at Kingston, Department of Mathematics and Statistics, 2003.
[16]	S. V. Emel'yanov, S. V. Korovin, L. V. Levantovsky. Higher-order sliding modes in control systems. Differential Equations, vol. 29, no. 11, pp. 1627-1647, 1993.

Second-order Sliding Mode Approaches for the Control of a Class of Underactuated Systems

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