Abstract
In this paper a collection of control algorithms which are able to give rise to 2-sliding modes have been presented, and for each of them the sufficient convergence conditions are given. Furthermore, the real sliding behaviour is briefly considered, and, in some cases, the upper bound of the convergence time is given.
2-sliding mode control seems to be an effective tool for the control of uncertain nonlinear systems since it overcomes the main drawbacks of the classical sliding mode control, i.e., the chattering phenomenon and the large control effort. Its real implementation implies very simple control laws and assures an improvement of the sliding accuracy with respect to real 1-sliding mode control.
The main difficulty in using 2-sliding mode controllers is the tuning of the parameters which characterize the various algorithms. Their values depend on the bounds of the uncertain dynamics and on the chosen sliding manifold, and only sufficient conditions for the convergence to the sliding behaviour are known. These conditions are very conservative, and, in practice, the parameters are heuristically tuned. It depends on the engineer’s experience to define which of the presented algorithms is more suitable for the specific control problem, even if, in the authors’ opinion, the super-twisting and the sub-optimal ones seem to be able to cover a large class of control problems with a remarkable implementation easiness.
Preview
Unable to display preview. Download preview PDF.
References
G. Bartolini, P. Pydynowski, “An improved chattering free VSC scheme for uncertain dynamical systems,” IEEE Trans. Automatic Control, vol. 41, pp. 1220–1226, 1996.
G. Bartolini, A. Ferrara, E. Usai, “Second order VSC for non linear systems subjected to a wide class of uncertainty conditions”, Proc. of the 1996 IEEE Int. Workshop on Variable Structure Systems — VSS'96, Seiken Symposium No. 19, pp. 49–54, Tokyo, Japan, December 1996.
G. Bartolini, A. Ferrara, E. Usai, “Applications of a sub-optimal discontinuous control algorithm for uncertain second order systems”, Int. J. of Robust and Nonlinear Control, vol. 7, no. 4, pp. 299–319, 1997.
G. Bartolini, A. Ferrara, E. Usai, “Output Tracking Control of Uncertain Nonlinear Second-Order Systems”, Automatica, vol. 33, no. 12, pp. 2203–2212, 1997.
G. Bartolini, A. Ferrara, A. Pisano, E. Usai, “Adaptive reduction of the control effort in chattering free sliding mode control of uncertain nonlinear systems”, Applied Mathematics and Computer Science, vol. 8, no. 1, pp. 51–71, 1998.
G. Bartolini, A. Ferrara, E. Usai, “Chattering Avoidance by Second Order Sliding Mode Control”, IEEE Trans. Automatic Control, vol. 43, no. 2, pp. 241–246, 1998.
G. Bartolini, A. Ferrara, E. Usai, “Real-time output derivatives estimation by means of higher order sliding modes”, Proc. of CESA’ 98 IMACS Multiconference, pp. 782–787, Nabeul-Hammamet, Tunisia, April 1998.
G. Bartolini, A. Ferrara, E. Usai, V. I. Utkin, “Second order chattering-free sliding mode control for some classes of multi-input uncertain nonlinear systems”, Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, Alghero, Italy, June 1998, to be published.
G. Bartolini, A. Pisano, E. Usai, “Digital Second Order Sliding Mode Control of SISO Uncertain Nonlinear Systems', Proc. of the 1998 American Control Conference ACC'98, vol. 1, pp. 119–124, Philadelfia, Pensilvania, June 1998.
ML.W. Chang, “A MIMO sliding control with second order sliding condition,” ASME W.A.M., paper no. 90-WA/DSC-5, Dallas, Texas, 1990.
M. Corless, and G. Leitmann, “Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems,’ IEEE Trans. Automatic Control, vol. 26, pp. 1139–1141, 1981.
M. Corless, G. Leitmann, “Exponential Convergence for Uncertain Systems with Component-Wise Bounded Controllers”, Robust Control via variable structure and Lyapunov techniques. F. Garofalo and L. Glielmo eds., Lecture Notes in Control and Information Science no. 217, pp. 175–196, Springer-Verlag, London, 1996.
S.V. Emel'yanov, S.K. Korovin, L.V. Levantovsky, “Higher-Order Sliding Modes in the Binary Control Systems”, Soviet Physics, Dokady, vol. 31, no. 4, pp. 291–293, 1986.
S.V. Emel'yanov, S.K. Korovin, L.V. Levantovsky, “Drift Algorithm in Control of Uncertain Processes”, Problems of Control and Information Theory, vol. 15, no. 6, pp. 425–438, 1986.
A.F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic Publishers, Dordrecht, 1988.
A. Isidori, Nonlinear control systems, Springer-Verlag, New York, 1989.
D.E. Kirk, Optimal control theory, Prentice Hall, Englewood Cliffs, NJ, 1970.
G. Leitmann, The calculus of variations and optimal control, Plenum Press, New York, 1981.
L. Fridman, A. Levant, “Higher Order Sliding Modes as a Natural Phenomenon in Control Theory”, in Robust Control via variable structure and Lyapunov techniques. F. Garofalo and L. Glielmo eds., Lecture Notes in Control and Information Science no. 217, pp. 107–133, Springer-Verlag, London, 1996.
A. Levant (Levantovsky L.V.), “Sliding order and sliding accuracy in sliding mode control”, Int. J. of Control, vol. 58, no. 6, pp. 1247–1263, 1993.
A. Levant, “Robust exact differentiation via sliding mode technique”, Automatica, vol.34, no. 3, pp. 379–384, 1998.
A. Levant, “Arbitrary-order sliding modes with finite time convergence”, Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, Alghero, Italy, June 1998, to be published.
C. Milosavljevic, “General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable systems”, Automation Remote Control, vol. 43, no. 1, pp. 307–314, 1985.
K.S. Narendra, A.M. Annaswamy, Stable adaptive systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.
H. Sira-Ramirez, “On the sliding mode control of nonlinear systems,” System and Control Letters, vol. 19, pp. 303–312, 1992.
V.I. Utkin, Sliding modes in control and optimization, Springer Verlag, Berlin, 1992.
M. Zhihong, A.P. Paplinski, H.R. Wu, “A robust MIMO terminal sliding mode control for rigid robotic manipulators”, IEEE Trans. Automatic Control, vol. 39, no. 12, pp. 2464–2468, 1994.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this chapter
Cite this chapter
Bartolini, G., Ferrara, A., Levant, A., Usai, E. (1999). On second order sliding mode controllers. In: Young, K., Ă–zgĂĽner, Ăś. (eds) Variable structure systems, sliding mode and nonlinear control. Lecture Notes in Control and Information Sciences, vol 247. Springer, London. https://doi.org/10.1007/BFb0109984
Download citation
DOI: https://doi.org/10.1007/BFb0109984
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-85233-197-9
Online ISBN: 978-1-84628-540-0
eBook Packages: Springer Book Archive