Can predefined-time stability be guaranteed for discrete-time systems? Novel theorems and a sliding mode realization.

While the theories of predefined-time (PdT) stability have been well established for continuous-time systems, research on discrete-time systems remains an open problem. This paper investigates stability control for nonlinear discrete-time perturbed systems and proposes the discrete-time PdT stability concept. Based on this concept, this paper develops two Lyapunov theories-the Lyapunov theory for PdT stability and that for practical PdT stability. Furthermore, this paper constructs a PdT discrete-time sliding mode controller for a class of second-order perturbed systems, which guarantees PdT stability throughout the reaching and sliding phases. Additionally, this paper presents an evaluation of the proposed discrete-time sliding mode control method, centered on an analysis of its key performance indicators. The results of numerical simulations and physical scenario-based simulations verify the method's effectiveness.