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International audienceIn this paper it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an event-capturing, or time-stepping algorithm) allows for multiple switching surfaces (i.e., a sliding surface of codimension ⩾2). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied

In this paper, a novel discrete-time implementation of sliding-mode control systems is proposed, which fully exploits the multivaluedness of the dynamics on the sliding surface. It is shown to guarantee a smooth stabilization on the discrete sliding surface in the disturbance-free case, hence avoiding the chattering effects due to the time-discretization. In addition, when a disturbance acts on the system, the controller attenuates the disturbance effects on the sliding surface by a factor (where is the sampling period). Most importantly, this holds even for large. The controller is based on an implicit Euler method and is very easy to implement with projections on the interval [ 1, 1] (or as the solution of a quadratic program). The zero-order-hold (ZOH) method is also investigated. First-and second-order perturbed systems (with a disturbance satisfying the matching condition) without and with dynamical disturbance compensation are analyzed, with classical and twisting sliding-mode controllers.

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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 finitetime 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.