On the Discretization of Sliding-Mode Like Controllers

Stability of a discretized continuous sliding-mode based state feedback control is proved using an L 2-gain analysis result for linear continuous-time systems with sampled-data output. It has been shown before that a strictly proper linear continuous-time system with sampled-data output has finite L 2-gain. This gain converges to the L 2-gain associated with the continuous-time output when the sampling period approaches +0. This result is incorporated in the analysis of the discretized sliding-mode based control applying techniques from non-linear L 2-gain theory. The result is then compared to a Lyapunov function analysis based approach. In contrast to the Lyapunov function technique, the sampling-time constraint vanishes for a stable plant if no control is used. Numerical results are demonstrated for a particular example, the control of the non-linear inverted pendulum.