A New Type of Nonlinear Disturbance Rejection

TL;DR

This paper develops a quantitative framework for nonlinear disturbance rejection, extending linear high-pass concepts to nonlinear systems and providing explicit performance guarantees.

Contribution

It introduces a gain inequalities approach to nonlinear disturbance rejection, generalizing linear high-pass filters and integral feedback properties.

Findings

Quantitative disturbance rejection guarantees for nonlinear systems.

Extension of high-pass filter concept to nonlinear resistors.

Non-asymptotic output bounds for disturbance rejection.

Abstract

Asymptotic disturbance rejection (equivalently tracking) for nonlinear systems has been studied only in qualitative terms (the state is asymptotically stable under bounded disturbances). We show how to prove quantitative performance guarantees for the nonlinear servomechanism problem. Our technique originates by applying a gain inequalities point of view to an ad fontes reexamination of the linear problem: what is the nonlinear equivalent of a sensitivity transfer function with a zero at the origin? We answer: a nonlinear input-output system is high-pass if its output is stable with respect to the \emph{derivative} of the input. We first show that definition generalizes high-pass resistor-capacitor circuit analysis to accommodate nonlinear resistors. We then show that this definition generalizes the steady-state disturbance rejection property of integral feedback controllers for linear systems. The theoretical payoff is that low-frequency disturbance rejection is captured by a quantitative, non-asymptotic output cost bound. Finally, we raise theoretical questions about compositionality of nonlinear operators.