Improved sampled-data accuracy for higher-order disturbance models

Determine whether, for disturbance-model orders p>1, a discrete-time controller can achieve a control-error accuracy band smaller than T^{p+1}D in the nominal sampled-data reference-tracking problem with an integrator-chain disturbance.

Background

The paper establishes a lower bound on the best possible worst-case sampled-data control accuracy for an integrator-chain disturbance with bounded unknown input. The proposed proper implicit robust exact sliding-mode controller achieves an accuracy of order T{p+1}D. The lower bound coincides with this accuracy order for p=0 and p=1, but is strictly smaller for p>1. Consequently, the authors explicitly leave unresolved whether the dependence among consecutive discrete disturbance values can be exploited by another causal discrete-time controller to obtain a narrower accuracy band for higher-order disturbance models.

References

However, for $p>1$, according to Proposition~\ref{prop:best-possible-accuracy}, there might exist a discrete-time controller that achieves a smaller accuracy band than $T{p+1}D$.

— Model-Based Disturbance Rejection via Sliding-Mode Control: Continuous-Time Formulation and Proper Implicit Discretization  (2610.02890 - Andritsch et al., 2 Oct 2026) in Remark following Proposition 1, Section 5 (Control Accuracy in the Nominal Case)