A Canonical Internal Model for Disturbance Rejection for a Class of Nonlinear Systems Subject to Trigonometric-Polynomial Disturbances

TL;DR

This paper introduces a new canonical internal model for disturbance rejection in certain nonlinear systems affected by trigonometric-polynomial disturbances, enabling adaptive stabilization without solving regulator equations.

Contribution

It presents a direct synthesis method for internal models from the plant and exosystem, broadening applicability to nonlinear systems and improving disturbance rejection techniques.

Findings

Global asymptotic convergence of disturbance estimates

Exponential convergence under persistent excitation

Successful simulation on a robotic manipulator

Abstract

In this paper, we propose a novel framework for disturbance rejection in a class of nonautonomous nonlinear systems affected by trigonometric-polynomial disturbances. The core of our approach is the design of a canonical internal model that directly converts the disturbance rejection problem into an adaptive stabilization problem for an augmented system. Unlike conventional methods, this internal model is synthesized directly from the given nonlinear plant and the knowledge of the exosystem, without relying on the solution of the regulator equations. This makes the approach applicable to a significantly broader class of nonautonomous nonlinear systems. Furthermore, we develop an adaptive disturbance observer comprising the canonical nonlinear internal model, a Luenberger-type state observer, and a parameter adaptation law. This observer ensures global asymptotic convergence of the disturbance estimate to the true disturbance without requiring persistent excitation (PE). Under the PE condition, both the disturbance estimation error and the parameter estimation error converge exponentially. By incorporating the disturbance estimate as a feedforward compensation signal, we establish sufficient conditions for achieving global trajectory tracking and asymptotic disturbance rejection. The effectiveness of the proposed method is demonstrated through a numerical simulation of a flexible-joint robotic manipulator.