Supporting Lemmas for RISE-based Control Methods

TL;DR

This paper proves key lemmas related to the integral property underlying RISE controllers, enhancing the theoretical foundation for their stability analysis in nonlinear systems with disturbances.

Contribution

It provides the first formal proofs of lemmas crucial for the stability analysis of RISE-based control methods, filling a gap in the existing literature.

Findings

Proved two lemmas related to the integral property of RISE controllers.

Enhanced theoretical understanding of RISE controller stability.

Supports rigorous stability analysis using differential inclusions.

Abstract

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.