Generalized delta-operator sliding-mode control with adaptive uncertainty envelopes

This paper addresses the robust control of uncertain sampled-data systems. Conventional shift-operator models become numerically ill-conditioned at high sampling frequency. To overcome this difficulty, the Delta-operator framework is adopted. It preserves mathematical consistency between continuous-time and discrete-time representations. An adaptive sliding-mode control (SMC) strategy and a generalized discrete-time reaching law are proposed within this framework. The reaching law unifies various reaching dynamics. A dual-layer adaptive uncertainty envelope is constructed to reject complex disturbances. The first layer compensates for the slowly-varying mean of the lumped disturbance; the second layer bounds the fast residual. A Lyapunov-based analysis proves closed-loop stability. The sliding-mode variable converges to a quasi-sliding mode (QSM) band in finite steps. The framework applies to systems with matched and projected unmatched disturbances. Simulations on a switched reluctance motor (SRM) and an inverted pendulum-cart (IPC) indicate that the strategy achieves robust tracking precision, rapid convergence, and substantial chattering reduction.