A class of continuous controllers termed Robust Integral of the Signum of the Error (RISE) have been published over the last decade as a means to yield asymptotic convergence of the tracking error for classes of nonlinear systems that are subject to exogenous disturbances and/or modeling uncertainties. The development of this class of controllers relies on a property related to the integral of the signum of an error signal. A proof for this property is not available in previous literature. The stability of some RISE controllers is analyzed using differential inclusions. Such results rely on the hypothesis that a set of points is Lebesgue negligible. This paper states and proves two lemmas related to the properties.
@article{arxiv.1306.3432,
title = {Supporting Lemmas for RISE-based Control Methods},
author = {Rushikesh Kamalapurkar and Joel A. Rosenfeld and Justin Klotz and Ryan J. Downey and Warren E. Dixon},
journal= {arXiv preprint arXiv:1306.3432},
year = {2026}
}