Model-Based Disturbance Rejection via Sliding-Mode Control: Continuous-Time Formulation and Proper Implicit Discretization

We propose a robust exact sliding-mode controller for reference tracking with model-based disturbance rejection, where the disturbance is generated by a linear time-invariant model driven by a bounded input and may be bidirectionally coupled with the plant. The controller embeds the internal model principle and its error dynamics mirror those of the robust exact differentiator, ensuring finite-time convergence. We develop a proper implicit discretization for sampled-data implementation that preserves stability and accuracy. In the nominal case, which contains a disturbance generated by a $p$-th order integrator chain, the method achieves a control accuracy of $T^{p+1}\bar{D}$, with discretization time $T$ and disturbance bound $\bar{D}$. This accuracy is shown to relate to the fundamental lower bound by a factor depending only on the order of the disturbance model. Simulations demonstrate the efficacy of the proposed methods.