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

TL;DR

This work tackles disturbance rejection for general nonautonomous nonlinear systems subjected to unknown-frequency trigonometric disturbances. It introduces a canonical nonlinear internal model that, when merged with an adaptive disturbance observer, turns disturbance rejection into an adaptive stabilization problem for an augmented plant, without solving regulator equations. The approach achieves global asymptotic convergence of disturbance estimates and, under persistent excitation, exponential convergence of both disturbance and parameter estimates; it also supports estimation of disturbance derivatives and online frequency identification. The method is validated on a numerical example involving a flexible-joint planar robot manipulator, illustrating effective disturbance estimation, frequency identification, and trajectory tracking with rejection.

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.

Figures (6)

• Figure 1: Estimation error of $\hat{\xi}_2(t)$.
• Figure 2: Estimation error of $\hat{d}(t)$.
• Figure 3: Estimation error of $\theta(t)$.
• Figure 4: Estimation errors of $\hat{\omega}_1(t)$ and $\hat{\omega}_2(t)$.
• Figure 5: Position tracking error.