Sliding mode control in discrete-time and difference systems
Wide use of digital controllers has placed onto the research agenda the generalization of sliding mode control methodology to discrete-time control systems. In the first studies, control algorithms intended for continuous- time systems were applied to discrete-time problems; resulting in chattering since the switching frequency can not exceed that of sampling. Then methods for reducing chattering were developed in many publications.
However, the fundamental question — what is the sliding mode in discrete-time systems? — was not considered. Discontinuous control in continuous-time systems may result in sliding in some manifold, while it results in chattering in discrete-time systems. The sliding mode may be originated in discrete-time systems with continuous control after a finite time interval, while any manifold with state trajectories may be reached asympotically only in continuous-time systems with continuous control (precisely speaking for systems governed by differential equations with Lipschitzian right-hand sides).
Design methods for sliding mode control for finite and infinite dimensional discrete-time and difference systems have been developed in this chapter. They enables decoupling of the overall dynamics into independent partial motions of lower dimension, and low sensitivity to system uncertainties. For all systems the motions are free of chattering, which has been the main obstacle for certain applications of discontinuous control action in systems governed by discrete and difference equations.
KeywordsSlide Mode Control Sliding Mode State Trajectory Admissible Domain Discontinuous Control
Unable to display preview. Download preview PDF.
- Drakunov, S.V., Izosimov, D.B., Luk'yanov A.G., Utkin V.A. and Utkin V.I. 1990a, Block control principle I. Automation and Remote Control, 51, 601–609Google Scholar
- Drakunov, S.V., Izosimov, D.B., Luk'yanov A.G., Utkin V.A. and Utkin V.I. 1990b, Block control principle II. Automation and Remote Control, 51, 737–746Google Scholar
- Drakunov, S.V., Utkin, V.I. 1990, Sliding mode in dynamic systems. International Journal of Control 55, 1029–1037Google Scholar
- Furuta, K. 1990, Sliding mode control of a discrete system. Systems and Control Letters, 14, 145–152Google Scholar
- Gantmacher, F.R. 1959, The theory of matrices, Vol.1, Chelsia, New YorkGoogle Scholar
- Kokotović, P.V., O'Malley, R.B., Sannuti, P. 1976, Singular perturbations and order reduction in control theory. Automatica 12, 123–132Google Scholar
- Kotta, U. 1989, Comments on the stability of discrete-time sliding mode control systems. IEEE Transactions on Automatic Control 34, 1021–1022.Google Scholar
- Kwakernaak, H., Sivan R. 1972, Linear optimal control systems, Wiley Interscience, New YorkGoogle Scholar
- Milosavljević, C. 1985, General conditions for the existence of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems. Automation and Remote Control 46, 679–684Google Scholar
- Sarpturk, S.Z., Isteganopolis, Y., Kaynak O. 1987, On the stability of discrete-time sliding mode control systems. IEEE Transactions on Automatic Control 10 930–932Google Scholar
- Spurgeon, S.K. 1991, Sliding mode control design for uncertain discrete-time systems. Proc IEEE Conference on Decision and Control,, Bighton, England, 2136–2141Google Scholar
- Utkin, V.I. 1992, Sliding modes in control and optimization, Springer-Verlag, Berlin.Google Scholar
- Utkin, V.I., Orlov, Y.V. 1990, Theory of infinite-dimensional control systems with sliding modes, Nauka, Moscow (in Russian)Google Scholar