Chattering suppression in hybrid non-homogeneous sliding mode differentiator via implicit-Euler discretization

The chattering phenomenon is a problem in sliding mode technique under discrete-time implementation. In this paper, a novel discrete-time algorithm for the hybrid non-homogeneous sliding mode differentiator is presented to prevent chattering in output signals. To be specific, an equivalence framework is employed linking the traditional set-valued signum function with the multilayer set-valued mapping to reduce discretization complexity. The stability of the system is analyzed using a combined Lyapunov approach and input-to-state stability theory. In addition, the discretization procedure is based on the equivalence relation between the set-valued function and the saturation function, as well as the implicit-Euler method. Numerical simulations demonstrate that the proposed algorithm completely suppresses chattering (0 amplitude), outperforming both the sigmoid function method (2.5 × 10^−7 amplitude) and the boundary layer method (2.0 × 10^−7 amplitude), while achieving improved filtering performance and enhanced disturbance rejection, thereby yielding more reliable output signals.