Lyapunov Stability and Performance Analysis of the Implicit Discrete Sliding Mode Control

Discrete-time sliding mode controllers with an implicit discretization of the signum function are considered. With a proper choice of the equivalent part of the control, the resulting controller is shown to be Lyapunov stable with finite-time convergence of the sliding variable to 0. The convergence of the control input, as the sampling period goes to 0, to the continuous-time one is shown. The robustness with respect to matching perturbations is also investigated. The discretization performance in terms of the error order is studied for different discretizations of the equivalent part of the input. Numerical and experimental results illustrate and support the analysis.