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IMA Journal of Mathematical Control and Information 2001 18(2):161-187; doi:10.1093/imamci/18.2.161
© 2001 by Institute of Mathematics and its Applications
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A new approach to discretization applied to a continuous, nonlinear, sliding-mode-like control using nonsmooth analysis

Guido Herrmann1, Sarah K. Spurgeon1 and Christopher Edwards1

1 Control Systems Research Group, University of Leicester, LE1 7RH, UK. Email: gh17@sun.engg.le.ac.uk, Email: eon@le.ac.uk, Email: ce@sun.engg.le.ac.uk

This paper introduces a new approach to discretization of nonlinear control laws with a Lipschitz property. The sampling time is defined as a parameter, which must be selected sufficiently small so that the closed-loop system is stable. In contrast to similar results, the stabilizing effect of the control is taken into account. This can result in less conservative constraints on the minimum sampling frequency. The discretization techniques are explained on a general nonlinear model and applied to the discretization of a novel nonlinear, robust sliding-mode-like control law. Similar robustness features as for continuous control are demonstrated. Nonsmooth Lyapunov functions are used for the discretized sliding-mode-like control introducing cone shaped regions of the state space. One of these cone shaped regions coincides with a cone shaped layer around the sliding mode defined by the continuous sliding-mode-like control. A stability theorem using nonsmooth Lyapunov functions is provided.

Keywords: discretized nonlinear; continuous-; time control; sliding-; mode control approaches; nonsmooth analysis.


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