Zero-Crossing Time Lower Bounds for the Homogeneous Super-Twisting Algorithm
Abstract
High-frequency switching and chattering are well-known problems in sliding-mode control. Although the super-twisting algorithm produces a continuous control input, the sliding variable may still cross zero repeatedly near the sliding manifold. This paper studies how fast these crossings can occur and whether they can accumulate in finite time for the scalar homogeneous super-twisting algorithm. Two lower bounds on the time required to return to zero are derived, together with a bound on the time needed to complete a full excursion. These results give minimum separation times and upper bounds on the number of crossings associated with excursions above a prescribed amplitude. Although zero crossings may accumulate in finite time, the amplitudes of the complete excursions considered here must vanish and satisfy a summability condition. Thus, crossings associated with excursions above any prescribed nonzero amplitude cannot accumulate in finite time. When the perturbation-rate bound is zero, the estimates recover the scaling imposed by homogeneity. For the classical super-twisting algorithm, the square-root scaling also holds for nonzero perturbation-rate bounds. Numerical simulations illustrate the results.
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BibTeX
@misc{gutierrezoribio2026zerocrossing,
title = {{Zero-Crossing Time Lower Bounds for the Homogeneous Super-Twisting Algorithm}},
author = {Diego Gutiérrez-Oribio and Ioannis Stefanou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14285},
eprint = {2610.14285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14285},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Zero-Crossing Time Lower Bounds for the Homogeneous Super-Twisting Algorithm
AU - Diego Gutiérrez-Oribio
AU - Ioannis Stefanou
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14285
UR - https://arxiv.org/abs/2610.14285
AB - High-frequency switching and chattering are well-known problems in sliding-mode control. Although the super-twisting algorithm produces a continuous control input, the sliding variable may still cross zero repeatedly near the sliding manifold. This paper studies how fast these crossings can occur and whether they can accumulate in finite time for the scalar homogeneous super-twisting algorithm. Two lower bounds on the time required to return to zero are derived, together with a bound on the time needed to complete a full excursion. These results give minimum separation times and upper bounds on the number of crossings associated with excursions above a prescribed amplitude. Although zero crossings may accumulate in finite time, the amplitudes of the complete excursions considered here must vanish and satisfy a summability condition. Thus, crossings associated with excursions above any prescribed nonzero amplitude cannot accumulate in finite time. When the perturbation-rate bound is zero, the estimates recover the scaling imposed by homogeneity. For the classical super-twisting algorithm, the square-root scaling also holds for nonzero perturbation-rate bounds. Numerical simulations illustrate the results.
ER -