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Dynamic sliding mode control and higher order sliding mode are studied. Dynamic sliding mode control adds additional dynamics, which can be considered as compensators. The sliding system with compensators is an augmented system. These compensators (extra dynamics) are designed for achieving and/or improving the system stability, hence obtaining desired system behaviour and performance. Higher order sliding mode control and dynamic sliding mode control yield more accuracy and also reduce and/or remove the chattering resulting from the high frequency switching of the control. It is proved that certain J-trajectories reach a sliding mode in a finite time. A sliding mode differentiator is also considered.

This paper considers an adaptive backstepping algorithm for designing the control for a class of nonlinear continuous uncertain processes with disturbances that can be converted to a parametric semi-strict feedback form. Sliding mode control using a combined adaptive backstepping sliding mode control (SMC) algorithm, is also studied. The algorithm follows a systematic procedure for the design of adaptive control laws for the output tracking of nonlinear systems with matched and unmatched uncertainty.

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In discrete-time systems, instead of having a hyperplane as in the continuous case, there is a countable set of points comprising a so-called lattice; and the surface on which these sliding points lie is the latticewise hyperplane. In this paper the concept of multivariable discrete-time sliding mode is clarified and new sufficient conditions for the existence of the sliding mode are presented. A new control design using the properties of discrete sliding is proposed, and the behavior of the system in the sliding mode is studied. Furthermore, the stabilization of discrete-time systems and an optimal sliding lattice are considered. Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/25/2015 Terms of Use: http://asme.org/terms A suitable Lyapunov candidate function is V(k)ϭx(k) T Px(k) where P is the u.p.d.s. solution of Lyapunov equation ͑32͒. Then the triangle and Cauchy-Schwartz inequalities give, from ͑33͒, Journal of Dynamic Systems, Measurement, and Control DECEMBER 2000, Vol. 122 Õ 797 Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/25/2015 Terms of Use: http://asme.org/terms Journal of Dynamic Systems, Measurement, and Control DECEMBER 2000, Vol. 122 Õ 801 Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/25/2015 Terms of Use: http://asme.org/terms