Supporting Lemmas for RISE-based Control Methods
Rushikesh Kamalapurkar, Joel A. Rosenfeld, Justin Klotz, Ryan J. Downey, Warren E. Dixon
TL;DR
RISE-based control methods rely on the integral $\int_0^{x} f'(y) \mathrm{sgn}(f(y)) dy$ to establish Lyapunov-based stability, but a rigorous proof was previously unavailable. This paper provides the missing proof of the integral identity $\int_0^{x} f'(y) \mathrm{sgn}(f(y)) dy = |f(x)| - |f(0)|$, along with Lebesgue measure-zero lemmas for points where $f=0$ and $f'\neq 0$ and generalizations, and a constructive bound via the Mean Value Theorem. It further derives a vector-valued bound: there exists a continuous strictly increasing $\rho$ such that $\|f(x) - f(x_d)\| \le \rho(\|x - x_d\|) \|x - x_d\|$ for $f:\mathbb{R}^n \to \mathbb{R}^m$, with an explicit construction using $G_1$ and $G_2$. These results strengthen the theoretical underpinning of RISE controllers, enabling robust analysis under disturbances and measurement noise.
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.