Zero-Crossing Time Lower Bounds for the Homogeneous Super-Twisting Algorithm
Diego Gutiérrez-Oribio, Ioannis Stefanou
eess.SY
Oct 9, 2026 · v1
math.OC
TL;DR
The algebraic theorem chain behind the excursion-time, cumulative, and accumulation bounds is formalized and machine-checked in Lean 4 with Mathlib, with a public repository.
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.
Problem
In the super-twisting sliding-mode algorithm, the sliding variable can cross zero repeatedly near the sliding manifold even though the control is continuous. The open questions are how fast these crossings can occur and whether they can accumulate in finite time for the scalar homogeneous super-twisting algorithm.
Approach
The authors derive two lower bounds on the time for the sliding variable to return to zero after an excursion of a given amplitude, plus a bound on the duration of a full excursion. The bounds depend only on the excursion amplitude, the controller gains, and the known uncertainty bounds. The algebraic theorem chain behind the main results was formalized in Lean 4 and machine-checked with Mathlib. Numerical simulations with the classical super-twisting algorithm illustrate the bounds.
Results
The bounds give minimum separation times and upper bounds on the number of crossings for excursions above a prescribed amplitude. Zero crossings may accumulate in finite time, but the excursion amplitudes must vanish and satisfy a summability condition, so crossings tied to excursions above any nonzero amplitude cannot accumulate. With a zero perturbation-rate bound the estimates recover the scaling imposed by homogeneity, and for the classical algorithm the square-root scaling also holds with nonzero perturbation-rate bounds.
